Nuclear Isomers: A Primer 3030786749, 9783030786748

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Ashok Kumar Jain Bhoomika Maheshwari Alpana Goel

Nuclear Isomers A Primer

Nuclear Isomers

Ashok Kumar Jain · Bhoomika Maheshwari · Alpana Goel

Nuclear Isomers A Primer

Ashok Kumar Jain Amity Institute of Nuclear Science and Technology Amity University, Noida, India

Bhoomika Maheshwari Amity Institute of Nuclear Science and Technology Amity University, Noida, India

Alpana Goel Amity Institute of Nuclear Science and Technology Amity University, Noida, India

ISBN 978-3-030-78674-8 ISBN 978-3-030-78675-5 (eBook) https://doi.org/10.1007/978-3-030-78675-5 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are solely and exclusively licensed 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 Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Dedicated to NAMAN

Foreword

Being in a “long-lived excited state” sounds like a good place to be, and nuclear isomers live up to expectations. Ashok Jain, Bhoomika Maheshwari and Alpana Goel have done a remarkable job in bringing together many of the strands associated with these remarkable states of matter. Isomer science is essentially an interdisciplinary endeavour, linking nuclear physics, atomic physics and astrophysics, together with applications such as medical imaging, energy storage and super-accurate time keeping. In this book, we are introduced to the key features, and taken on extended tours in selected areas. It is all based on the micro world where quantisation rules. Every excited quantum state has a lifetime associated with its existence, but typical lifetimes are a trillionth of a second or less. By contrast, isomers live for much longer, some even for years. One exceptional isomer is an excited state of the technetium-99 nucleus, with a half-life of six h. It is the most highly used isomer of all—for imaging work in hospitals—but its security of supply is not guaranteed. If you would like to dig deeper into the whys and wherefores of nuclear isomers, then keep reading. It has got a lot to do with “spin”, that wonderful property of quantum objects that we like to ignore. Spin can tell us a great deal about what is inside an object (compare spinning a raw egg with a hard-boiled egg) and it turns out that atomic nuclei can have many different spin states, not only spinning at different speeds, but also spinning in different ways. It’s a bit more complicated than a gyroscope, but that’s part of the fun. This book will indulge your curiosity. Philip Walker University of Surrey, Guildford, UK

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Preface

Nuclear isomers are the long-lived excited states of nuclei. Therefore, they constitute the meta-stable landscape of nuclei. The first isomer was observed as early as 1921 by Otto Hahn. The year 2021, therefore, marks 100 years of the discovery of isomers. We are, therefore, quite pleased to celebrate the centenary of nuclear isomers with this book. The main aim of this book is to introduce the specialized field of nuclear isomers to the beginners as well as the graduate students in the field. It also provides an overview of the subject and may be used as a reference book by active researchers. Since the first discovery, the number of isomers has been growing steadily at a rapid pace due to better experimental capabilities. This number now stands at 2581 for the isomers having 10 ns as upper limit for half-lives. This has opened up new windows to observe and decipher the underlying nuclear structure and interactions. Further, the isomers are beginning to be seen as potential energy storage devices with a host of applications in the medical field. An ultra-precise nuclear clock is almost a certainty now, which will open up a wide range of new practical applications. The dream of a gamma ray laser may also become a reality provided we hit upon the right kind of isomer. There could be many underlying reasons for the long lives of excited nuclear states, which constitute the basis of classification of isomers. Accordingly, they are broadly classified into spin isomers, shape isomers, fission isomers, and K-isomers. However, we have specially brought out the Seniority isomers as a separate class, which were so far clubbed with the spin isomers. Besides this, we also highlight the extremely low energy isomers as a separate class, which are eliciting huge interest due to the current focus on the 8 eV isomer in 229 Th. Novel angular momentum couplings, nuclear shapes, pairing, etc. conspire together to impart a huge range to isomeric half-lives, explaining which still remains an elusive goal. We also highlight several unresolved issues and open problems throughout the book and hope that it

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Preface

is useful for active researchers. We will appreciate if readers point out any errors or important omissions, which may have crept in despite our best efforts. Noida, India

Ashok Kumar Jain Bhoomika Maheshwari Alpana Goel

Acknowledgements

The authors are grateful to many of their colleagues for providing valuable inputs and literature on isomers. Prof. Philip M. Walker (Surrey) and Prof. Suresh C. Pancholi (New Delhi) went through the entire book very painstakingly, and provided very useful comments and suggestions almost continuously, which helped us in resolving many issues and improve the book. Phil also agreed to write a very succinct and illuminating Foreword to the book, and we thank him for the same. Dr. Balraj Singh (McMaster) provided many useful inputs on several topics in the book and also sent us many papers related to isomers. We would like to put on record the names of Rudrajyoti Palit (TIFR), Gopal Mukherjee (VECC), N. Madhavan (IUAC), Timo Dickel and Wolfgang Plass (GSI), and Costel Petrache (Paris), for providing the details of important works related to isomers. Swati Garg (SJTU) was quick to respond to our requests for new or updated data on isomers. Prof. B. Ananthanarayan approached one of us (AKJ) to write this Springer Monograph on nuclear isomers, and implanted the seed for the book. We thank him for initiating the project and supporting it throughout. The timeline of the book continued to extend like the famous “tail of Hanuman” in the well known Indian mega-epic of Ramayana, largely due to the Covid19 disruptions. The Springer support team, however, remained positive and fully supportive. We thank them for their patience. Finally, we are thankful to the Amity University UP and more specifically to the Founder President Dr. Ashok K. Chouhan, Chancellor Dr. Atul K. Chauhan, and Vice Chancellor Dr. (Mrs.) Balvinder Shukla for providing the facilities to complete the work. We also thank the Science and Engineering Research Board (SERB, Govt. of India) for financial support in the form of a core research grant on nuclear isomers to AKJ and AG.

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Contents

1 An Overview of Nuclear Isomers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Why a Primer on Nuclear Isomers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 What are Nuclear Isomers? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Early History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Where are the Isomers Found? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Definition and Scope of Isomers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Half-Life of Isomers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Classification of Isomers and Hindrance Mechanisms . . . . . . . . . . . . 1.8 Systematic Features of Isomers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8.1 Half-Life Systematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8.2 Spin Systematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8.3 Multipolarity Systematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8.4 Role of Pairing in Isomeric Energies . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 1 2 3 4 5 6 6 9 9 11 12 15

2 Spin Isomers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Isomeric Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Internal Conversion and Isomeric Half-Life . . . . . . . . . . . . . . . . . . . . 2.3 Islands of Spin Isomers Near Magic Numbers . . . . . . . . . . . . . . . . . . 2.3.1 g9/2 Spin Isomers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 h 11/2 Spin Isomers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 i 13/2 Spin Isomers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 High-Spin Isomers Near the Proton Drip Line . . . . . . . . . . . . . . . . . . 2.5 Spin Isomers in 180m Ta and the Only Natural Isomer . . . . . . . . . . . . . 2.6 Spin Isomers in 208 Pb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 E5 Decaying Spin Isomers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17 17 20 22 23 24 25 27 27 30 31 32

3 Seniority Isomers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Seniority and Seniority Isomers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Single-j Quasi-spin Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Decay Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Magnetic Moments and g-Factors . . . . . . . . . . . . . . . . . . . . . .

35 35 37 40 45 xiii

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Contents

3.2 Examples of Seniority Isomers and Their Moments . . . . . . . . . . . . . . 3.2.1 Seniority Isomers in N = 50 Isotones . . . . . . . . . . . . . . . . . . . 3.2.2 Seniority Mixing in 72,74 Ni Isotopes . . . . . . . . . . . . . . . . . . . . 3.2.3 Seniority Isomers in 128 Pd and 126 Pd . . . . . . . . . . . . . . . . . . . . 3.2.4 Seniority Isomers in Pb Isotopes . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46 46 47 49 50 50

4 Generalized Seniority Isomers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Multi-j Quasi Spin Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Decay Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Group Theoretical Understanding . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Excitation Energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Generalized Seniority in the Sn Isotopes . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 The 10+ , 13− , and 15− Isomers . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Comparison of Sn, Pb and N = 82 Isomers . . . . . . . . . . . . . . 4.3 First 2+ and 3− States in Sn, Cd, Te Isotopes . . . . . . . . . . . . . . . . . . . 4.3.1 Twin Asymmetric B(E2) Parabola in Sn Isotopes . . . . . . . . . 4.3.2 Twin Asymmetric B(E2) Parabola in Cd and Te Isotopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Inverted B(E3) Parabola in Sn Isotopes . . . . . . . . . . . . . . . . . 4.3.4 Inverted B(E3) Parabola in Cd and Te Isotopes . . . . . . . . . . . 4.4 Isomeric Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Quadrupole Moment of 11/2− States . . . . . . . . . . . . . . . . . . . 4.4.2 Generalized Seniority Schmidt Model . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51 51 54 56 57 57 61 63 64 64

5

66 67 68 70 70 73 77

K -Isomers in Deformed Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 The K -Quantum Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Deformed Nilsson Model and High-K States . . . . . . . . . . . . . . . . . . . 5.2.1 Quasi-particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 High-K MQP States and Isomers . . . . . . . . . . . . . . . . . . . . . . . 5.3 Calculation of MQP States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Three-Quasiparticle States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 MQP States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Some General Features of High-K States . . . . . . . . . . . . . . . . . . . . . . 5.4.1 K -Isomer in 250 No . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 K -Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.3 K -Isomeric Rotational Band . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Theoretical Treatments Used for K -Isomers . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79 79 81 83 84 85 88 89 90 91 92 94 94 98

6 Shape and Fission Isomers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Double-Hump Barrier and the Shell Corrections . . . . . . . . . . . . . . . . 6.2 Discovery of the Fission Isomers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Additional Features of Fission Isomers . . . . . . . . . . . . . . . . . . . . . . . . 6.4 The Low-Lying 0+ Shape Isomers . . . . . . . . . . . . . . . . . . . . . . . . . . . .

101 101 104 107 108

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6.5 Half-Lives of the Shape Isomers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.1 Projected Shell Model Calculations . . . . . . . . . . . . . . . . . . . . . 6.5.2 Other Microscopic Calculations . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

112 112 113 114

7 Unusual Isomers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Examples of Unusual Isomers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 High Energy Isomers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.2 Extremely Low Energy (ELE) Isomers . . . . . . . . . . . . . . . . . . 7.1.3 Very High-Spin Isomers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.4 Very Long-Lived Isomers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.5 Highest Multipolarity Isomers . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.6 100% Proton Decaying Isomers . . . . . . . . . . . . . . . . . . . . . . . . 7.1.7 Highest Quasi-particle Isomer . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 ELE Isomers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 A Specific Case of ELE Isomer in 229 Th . . . . . . . . . . . . . . . . . . . . . . . 7.4 β-Decaying Isomers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

117 117 117 118 118 118 119 119 119 119 121 125 127

8 Experimental Methods, Applications, Future Prospects . . . . . . . . . . . . 8.1 Experimental Methods and Applications . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 Gamma Ray Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.2 Recoil/Fragment Mass Analyzers . . . . . . . . . . . . . . . . . . . . . . . 8.1.3 Mass Measurement Techniques . . . . . . . . . . . . . . . . . . . . . . . . 8.1.4 Highly Charged Ion Storage Rings . . . . . . . . . . . . . . . . . . . . . . 8.1.5 Isomeric Targets and Isomeric Beams . . . . . . . . . . . . . . . . . . . 8.1.6 Medical Applications of Isomers . . . . . . . . . . . . . . . . . . . . . . . 8.1.7 Isomer Depletion by External Triggers . . . . . . . . . . . . . . . . . . 8.1.8 Moments and g-Factor Measurements . . . . . . . . . . . . . . . . . . . 8.2 Future Applications of Isomers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Isomer Battery and Gamma Ray Laser . . . . . . . . . . . . . . . . . . 8.2.2 Nuclear Clock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

131 131 132 132 133 134 134 135 136 137 139 139 139 140

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

Chapter 1

An Overview of Nuclear Isomers

Keywords Nuclear isomers · Definition · Early history · Classification of isomers · Hindrance mechanisms · Systematic properties

1.1 Why a Primer on Nuclear Isomers Nuclear isomers currently occupy the centre stage in nuclear physics research. Recent developments in nuclear detectors, rare beam facilities, reaction analyzers, and digital electronics as well as computational power, make it possible to observe new isomers, and measure their detailed properties. New techniques to investigate these metastable states continue to emerge. New measurements on isomers are being made almost every week, and better data are being published much more regularly now. It is, therefore, prudent to answer some basic questions related to isomers in simple terms. What are nuclear isomers? Where are they found? What are their different varieties? How to understand the formation of nuclear isomers? How can we possibly predict the existence of an isomer? Why is there, suddenly, so much of interest in this area? We attempt to answer many such questions in this book.

1.2 What are Nuclear Isomers? Nuclear isomers are those excited states of nuclei, which decay rather slowly, and therefore, have a relatively longer life-time on the nuclear time-scale. That is why these are also termed as meta-stable states. The life-time of these meta-stable states varies from nanoseconds to millions of year s. In fact, there are many examples, where the ground state of a nucleus may not be the longest-lived state, and an excited state lives longer than the ground state [1]. This is one of the many peculiar charac© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. K. Jain et al., Nuclear Isomers, https://doi.org/10.1007/978-3-030-78675-5_1

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1 An Overview of Nuclear Isomers

teristics of nuclear isomers which makes it very exciting to study them. But a brief history first.

1.3 Early History In 1917, Soddy [2] remarked, “We can have isotopes with the identity of atomic weight, as well as of chemical character, which are different in their stability and mode of breaking up”. This may be taken as the first vague hint to what could be a nuclear isomer. Only six years had elapsed since the discovery of the nucleus, and the picture of a nucleus itself was in its infancy in 1917. It was, therefore, treated as a mere speculation, which made sense only after the experimental observation by Otto Hahn in 1921 [3], while studying the radioactive Uranium salt displaying two halflives. This information remained unique for several years, until artificial radioactivity by using neutron-capture opened up a new way to find many more similar situations. In 1935, Szilard and Chalmers [4] found the first indication of isomerism in the Indium isotopes, while two independent groups, one of Kurchatov [5] and the other of Fermi [6], actually established the isomerism in Br isotopes in the same year. Weizsacker [7] made a breakthrough in theoretical understanding of isomers in 1936; he demonstrated a connection between the slow decay rate (and, therefore, a long half-life), and a large change in angular momentum and small decay energy during de-excitation. This is how the nuclear isomerism was understood at large, and led to the first review of isomers by Segre and Helmholtz [8] in 1949. Additional inputs to the occurrence of isomers in terms of shell structure were made by Feenberg [9], Feenberg and Hammack [10], and Nordheim [11] in 1949 itself. Axel and Dancoff [12] presented the first classification of isomers in 1949. In parallel, Goldhaber and Sunyar [13] also reported the classification of isomers which eventually resulted in a review on “Nuclear Isomers and Shell Structure” by Goldhaber and Hill [14]. As the isomer data grew in number, it gradually began to play a crucial role in the development of various nuclear models [15–19]. A well known historical example of isomers is the 8− state in 180 Hf with 5.5 h half-life. In 1953, Bohr and Mottelson [19] identified a group of levels in 180 Hf as the very first rotational band ever seen in a nucleus. This ground rotational band, populated by the long-lived 8− isomeric state of 180 Hf, opened up a new field of rotational phenomena in nuclei, which resulted in the development of the nuclear collective models [20, 21]. This also led to the discovery of a new type of isomers, the K -isomers, found in deformed nuclei [22, 23]. Isomers, therefore, began to play a crucial role in the growth of nuclear structure physics of spherical as well as deformed nuclei.

1.4 Where are the Isomers Found?

3

Fig. 1.1 The chart of nuclear isomers on the nuclear landscape. The vertical lines represent the neutron and proton magic numbers. The projection on the side panels also carry the lines for respective magic numbers of protons and neutrons

8 4

100

E (MeV)

12

n Proto

0 60

140

er

Numb

100 60

20 20

n

tro

u Ne

r

e mb Nu

1.4 Where are the Isomers Found? Feenberg summarized nearly 100 isomers in his book published in 1955 [17], mostly near the magic numbers and in spherical nuclei, highlighting a strong correlation between occurrence of isomers and the placement of intruder orbitals close to the Fermi surface in the strong coupling shell model [18]. These unique-parity intruder orbitals carry large angular momentum and opposite parity, and give rise to highspin excitation near the ground state; both the conditions supportive of a hindered transition, and formation of isomers. Although isomers are spread throughout the nuclear chart, a clustering of isomers near the magic numbers is visible in Fig. 1.1. New isomers in different regions of nuclear landscape, excitation energies and spin continue to be discovered at an increasing pace with the availability of several stateof-the-art experimental techniques, and prompted by possibilities of novel practical applications [22–29]. Our recently updated version of the “Atlas of Nuclear Isomers” lists about 2581 isomers with a lower limit of half-life of 10 ns [1]. It is very prudent to classify the isomers by the number of neutrons and protons they carry, since the odd-A, even-even and odd-odd isomers have their own typical features. This also shows up in their numbers. The number of odd-odd isomers is almost 846 which is nearly double the number of even-even isomers which is almost 444. There are 1291 isomers in odd-A nuclei (comprising both odd-Z even-N, and even-Z odd-N), almost equal to the 1290 isomers in even-A nuclei. The data presented in the Atlas constitute the backbone of most of the discussions in this book.

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1 An Overview of Nuclear Isomers

1.5 Definition and Scope of Isomers A normal excited state of a nucleus is expected to decay in approximately 10−13 seconds or less. One may argue that a time-scale larger than this could be used to define an isomer. Life-time measurements were earlier limited by the electronic techniques which could measure a life-span up to few nanoseconds. However, the limit of measurements has been continuously lowered over the years with an advancement in experimental capabilities. This also has had an impact on the half-life definition of isomers. The very first version of Nubase in 1997 [30] listed an isomer as an excited state having a half-life greater than 1 ms. The ENSDF (Evaluated Nuclear Structure Data File) database [31] also adopted the same value. Later on, Nubase 2003 [32] as well as Nubase 2012 [33] adopted a lower limit of 100 ns. However, these limits were adopted purely as a matter of convenience and not for some fundamental reasons. Some authors in recent works have started to label states with a half-life of several picoseconds also as isomers. We have based our discussions in this book on the “Atlas of Nuclear Isomers” [1], which has kept the lower limit of isomers as 10 ns. It is, therefore, important to take care of this limitation when discussing the systematic features of isomers as it may become necessary sometimes to include lower half-lives to complete the systematic. Isomer research has seen a rapid growth in the last couple of decades due to a qualitative leap in the experimental facilities. However, most of the theoretical efforts in understanding isomers have continued to use the standard nuclear models like the shell model, which are often quite sufficient. Dracoulis [22] highlighted the unlimited scope of isomers as a probe of studying nuclear reactions and nuclear structure, particularly in populating states that are normally inaccessible. In the past two decades, extensive isomeric studies have been carried out to investigate the unexplored regions of nuclear chart, exotic nuclei, nuclei of astrophysical interest etc. Nuclear isomers have also enabled highly sensitive spectroscopy, which opens up the decay paths preceding and following their formation. Further, their multidisciplinary and promising applications in medical imaging and therapy, nuclear batteries, gamma-ray lasers etc., have prompted a huge interest in this field [24–27]. The recent atlas of nuclear isomers [1] presents the most comprehensive catalogue of isomers along with their various spectroscopic properties like energies, spin and parity, decays, multipolarities, gamma-energies etc. with a lower limit on half-life at 10 ns. These data reveal many novel systematic features across the whole nuclear landscape, which also throw light on the underlying physics. The habitat of nuclear isomers now covers the whole nuclear landscape with changing characteristics from region to region as shown in Fig. 1.1. Most of the isomers decay via isomeric transitions (IT) i.e. gamma decay and /or, internal conversion. However, there are many isomers known to have other decay modes like beta decay (β) [34–39], alpha decay (α) [40, 41] and, spontaneous fission. Rare modes of decay like proton decay ( p) [42], double proton decay (2 p) [43], electronic capture delayed proton decay ( p) [44], beta delayed neutron decay (β − n) [45], and beta

1.5

Definition and Scope of Isomers

5

delayed proton decay (β + p) [37–39] etc., have also been observed depending on their location in the nuclear landscape. These peculiar decays eventually lead to a better insight into the structure of the complex nuclei, and offer a testing ground for the nuclear wave-functions, location of single-particle orbitals, and also nature of the nuclear force. Nature of hindrance to the various decay modes results in their classification into the spin isomers in spherical nuclei, the seniority isomers in semi-magic nuclei, the K -isomers in axially-deformed nuclei, the fission isomers in heavy-mass nuclei, and the shape isomers in nuclei with shape co-existence. While we discuss all kinds of isomers, our focus will be on two broad categories: isomers in spherical or nearly spherical nuclei and isomers in deformed nuclei. We will focus on the observations as well as theoretical understanding of these isomers. We propose to highlight the open questions at each stage as we proceed.

1.6 Half-Life of Isomers Knowledge and understanding of the half-life of an isomer is very crucial, and probably the most difficult, both in theory and experiment. Half-lives of isomers vary from ns to year s, and is often very difficult to measure. Theoretically, the gamma-ray partial half-life of an isomer broadly depends on three quantities, all related to the isomeric state and the state to which it decays. These three quantities are, the angular momentum and parity of the isomeric state and the final state into which it decays, the decay energy between the two states, and the matrix element of the operator responsible for the transition between the two states. Smaller is the matrix element, narrower becomes the decay path and smaller is the probability of transition and, therefore, larger is the half-life. Similarly, smaller is the decaying energy, smaller is the decay probability and, therefore, larger is the half-life. Multipolarity (λ) of the transition is controlled by the difference in the angular momenta of the two states. Larger is the multipolarity, weaker is the decay. This may be represented by a simple relation given by, T1/2 ∝ 1/| f |Tλ |i|2 (E γ )(2λ+1)

(1.1)

The matrix element carries the nuclear structure information implicit in the wavefunctions of the initial and the final states involved in the decay, and the interaction operator causing the transition. Calculating the matrix element is often the most difficult part as it involves a knowledge of actual wave-functions.

6

1 An Overview of Nuclear Isomers

1.7 Classification of Isomers and Hindrance Mechanisms Theoretical interpretations as spin isomers, K -isomers, seniority isomers, shape and fission isomers have been made on the basis of the physical mechanisms of hindrance in their respective decays. We have shown a typical example of each type of isomer in Fig. 1.2. It may be noted that a combination of two or more such factors may sometimes come into play and it may not be easy to delineate the effects very clearly in such cases. The most dramatic example of spin isomers is the longest-lived 9− , 180m Ta isomer having a half-life greater than 1015 years, and may be treated as almost stable (see Fig. 1.2). Its gamma decay to the lower-lying states is highly hindered due to the large angular momentum difference, and never observed. It is purely a spin isomer, or a “spin trap”. There is another 9− isomer known in 210 Bi, which decays in about 3 × 106 years, since it is able to undergo alpha decay. In an example of K -isomer, the 8− isomer in the deformed 180 Hf can decay to the + + 8 /6 states but is hindered in its decay due to another physics reason. In this case the K -quantum number, representing the projection of the angular momentum on the symmetry axis, changes drastically from K = 8 to K = 0, implying a change in the inclination of the angular momentum. Angular momentum is a vector quantity and its conservation, therefore, involves both magnitude and direction. This is the reason for the occurrence of a K -isomer. The other two isomers in Fig. 1.2 arise due to sudden changes in the shape of the nucleus during the decay. The self-conjugate nucleus 72 Kr has a 0+ isomeric state, which is considered to be due to oblate-prolate shape co-existence. The oblate-prolate shape mixing gives rise to the observed transition. It is considered to be an example of a shape isomer. A different kind of shape isomer is observed in 242 Am, which undergoes spontaneous fission. The 14 ms isomer is trapped in the highly deformed second minimum on the way to fission, and is known as a fission isomer. Existence of an isomeric state itself is indicative of a very unusual situation, which may allow us to manipulate it for different applications by exploiting the basics of its existence and decay properties. Those isomers, which live longer than their respective ground states, are particularly interesting. This feature provides significant experimental advantages to study those nuclei where spectroscopy is hard to perform like in the super-heavy region or, the neutron-rich region.

1.8 Systematic Features of Isomers It is a great learning experience to begin by looking at the 3D-plot of isomers in Fig. 1.1 (an updated version from the atlas [1]), where isomeric excitation energies have been plotted on the vertical scale and the proton and neutron numbers in the horizontal plane. The solid black lines represent the magic numbers for protons and neutrons. The 2D-projections at the back (blue) and the side (yellow) show the

E (MeV)

2.0

2.2

2.4

0.0

0.4

0.8

242Am

72Kr

1-

0-

3-

2-

0+

γ−decay

(2+,3-)

Shape isomer

shape transition

Fission isomer

0+

2+

SF~100%

14 ms

26.3 ns

0.00

0.04

0.08

0.0

0.4

0.8

1.2

1+

2+

2+ 0+

4+

6+

Spin isomer λ=7

180Ta

K=0

180Hf

8+

K=8

λ=8

5.47 h

unobserved gammas

16 9- >4.5x10 y

K- isomer

ΔK > λ

8-

Systematic Features of Isomers

Fig. 1.2 Some typical examples of various types of isomers

0.00

0.04

0.08

E (MeV)

E (MeV)

E (MeV)

1.8 7

8

1 An Overview of Nuclear Isomers

No. of isomers

Fig. 1.3 The bar chart of various isomeric half-lives. Note that the lower limit on the half-life of isomers is 10 ns

Fig. 1.4 The long-lived isomers in nuclear chart. The zig-zag line represents the path of beta-stable nuclei. The dashed lines represent the magic numbers Proton Number

80

h d y

40

0 0

40

80

120

160

Neutron Number

shadow-plots of excitation energies as a function of the neutron number (N ) and the proton number (Z ), respectively. The plot looks like a copy of the Segre chart with multiple data points at a given isotope/isotone. Two salient features emerge from the Atlas plotted in Fig. 1.1: (i) the excitation energies of the isomers rise significantly near the magic numbers; (ii) the number of isomers also rises near the magic numbers. There are also a few faint peaks appearing in the middle of the magic numbers. The rise in the number of isomers around the magic numbers is linked to the shell structure, where the spin-orbit splitting is known

1.8

Systematic Features of Isomers

9

to push the high-j unique-parity orbitals just below the magic numbers among the orbitals of opposite parity, giving rise to high angular momentum excitation closer to the respective ground state having a parity opposite to the isomeric state. The rise in excitation energy is linked to the large energy gaps encountered at the magic numbers. Besides the well known spherical magic numbers at Z = N = 20, 28, 50, 82, and N = 126, we do observe faint peaks at other nucleon numbers also. We thus notice a distinct rise in the excitation energies near Z = 66 − 68, N = 106, and N = 142 − 146. These nucleon numbers seem to coincide with the shell structure in deformed nuclei, and reflect the gaps in the Nilsson scheme near these particle numbers [46, 47]. The faint peaks observed in the middle of the closed shells in Fig. 1.1 thus correspond to the deformed nuclei, where only K (the projection of angular momentum on the symmetry axis) remains a good quantum number for axially-symmetric shapes. When the high K -value orbitals come close to the Fermi energy at specific nucleon numbers, these may give rise to the K -isomers.

1.8.1 Half-Life Systematic Nuclear isomers span a huge range of half-lives from ns to year s. We divide the halflives into nanoseconds (ns), micr oseconds (µ s), milliseconds (ms), seconds (sec), minutes (min), hour s (h), days (d) and year s (y) ranges, which contain isomers with half-life range of 10–999 ns, 1–999 (µ s), 1–999 ms, 1–59 sec, 1– 59 min, 1–23 h, 1–364 d and >1 y, respectively. Figure 1.3 depicts a bar chart of the isomeric half-lives (data used from atlas [1]), where one can see continuous and almost exponential decline in the number of isomers with increasing half-lives; a slight increment in the numbers is, however, noticed for the range of seconds. Most of the isomers, thus, lie in the ns and µs range. We have plotted the longest-lived isomers in Fig. 1.4, having half-lives from hours to years on the N − Z plot. It is highly interesting that all of them lie on and around the line of beta-stability. Further, most of these isomers belong to the category of odd-odd nuclei. For example, 11 out of 12 isomers having half-lives in y , are doublyodd systems. Specific isomers which live longer than respective ground state may be very helpful in searches of super-heavy nuclei or, neutron rich nuclei, and may also play an important role in the astrophysical nucleosynthesis processes.

1.8.2 Spin Systematic Isomers carry a spin-parity depending upon their structural surroundings in terms of the occupied orbitals and configurations, and other orbitals in their vicinity. It is very useful at this stage to recollect the single-particle shell model picture of orbitals as shown in Fig. 1.5. We remind ourselves of the shell structure and the magic gaps that show up in the orbital scheme, and the effect of strong spin-orbit coupling. Also,

10

1 An Overview of Nuclear Isomers

Fig. 1.5 Schematic single-particle levels from the shell model

we must take note of the presence of unique-parity intruder orbitals having high-j values, which fall at the top of each group of levels forming the shell and just below the magic gaps. While all the orbitals in a shell belong to a given principal quantum number N , the high-j orbital lying at the top of each shell has N value higher by one unit. It, therefore, has a different parity than the rest of the orbitals in the shell. This unique-parity nature of the high-j orbital accords it a special status, enabling it to play a controlling role in many nuclear phenomena including the formation of isomers near the magic numbers. This orbital picture is also a very successful guide to understand the configuration of a large number of isomers. We may also remind ourselves that the spherical shell model scheme shown here should be replaced by the deformed shell model scheme (or, the Nilsson scheme), when discussing the isomers in deformed mass regions. Further, the actual ordering of the levels may sometimes differ slightly, and depends on the mass region and various other factors. When we look at their spin systematic simply in terms of number of isomers for a given spin, we find some interesting features, as shown in Fig. 1.6 (using the data from atlas [1] and its recent update). One would expect the largest number of isomers at high-spin values, and indeed, we do find maxima in the number of

1.8

Systematic Features of Isomers

11

Negative Parity 120

Integer Half-integer

No. of isomers

70

20

Positive Parity

70

20

0

4

8

12

16

20

24

28

32

36

Spin Fig. 1.6 Number of isomers corresponding to various spins. The peaks are generally associated with the unique-parity intruder orbitals

isomers at spins 9/2+ , 11/2− and 13/2+ for positive, negative, and positive parities, respectively. These correspond to the 1-quasiparticle (qp) states for the g9/2 , h 11/2 and i 13/2 orbitals, respectively. However, it is highly interesting that a large number of isomers occur at spin 1/2 for both the parities. This may be understood as 1/2 is the lowest possible spin, and decay from 1/2 is more likely to occur to a higher spin state, causing a hindered decay. Spin values higher than 11/2 and 13/2 most likely arise from a combination of two, or more number of quasi-particles. For example, the peaks in the Fig. 1.6 at 8+ or, 8− may be due to the (1g9/2 )2 , or (1h 9/2 )2 , or, 1g9/2 ⊗ 1 f 7/2 , or h 9/2 ⊗ f 7/2 2−qp combinations. The spin-parity values and excitation energies of isomers may also provide a clue to the placement of the high-j single-particle orbitals in the shell model scheme.

1.8.3 Multipolarity Systematic Most of the nuclear isomers are known to decay via isomeric transitions (IT), a term commonly used to denote gamma decay as well as internal conversion decay. However, other decay modes like β-decay and α-decay are also observed, particularly at the limits of nuclear binding. Isomers are observed to decay by IT having multipolarities from E1, M1 to M4, E5. Large half-life generally signals a large

12

1 An Overview of Nuclear Isomers

difference in the angular momenta of the initial and final states, identifying them as spin isomers [7, 24]. In an apparent contradiction, there exist a large number of isomers which decay by low multipolarity transitions such as E1, M1 or E2, M2 transitions (multipolarity as low as 1 or 2) [1]. Such cases highlight the subtle role of structural peculiarities in their origin, which may eventually be related to the additional quantum numbers like seniority in spherical nuclei and K in axially-deformed nuclei, or any other physical phenomenon like magnetic rotation [48]. We also note that certain multipolarities are parity changing and others parity preserving during the decay. Thus, M2, M4, E1, E3, E5 transitions connect states differing in parity. The high-j unique-parity orbitals just below the magic numbers often connect with lower-lying or ground states by such transitions, and are more likely to lead to isomers. It is indeed observed that large number of spin isomers have a clustering of isomers near magic numbers which decay by parity changing transitions like M2, E3, M4, and E5 as compared to M3 and E4 decays. It is observed that the number of isomers decaying by parity changing transitions is much more than those isomers which maintain the same parity in decay. Also, we note that the number of M4 decaying isomers is quite large compared to the E5 decaying isomers, although both M4 and E5 may compete with each other often.

1.8.4 Role of Pairing in Isomeric Energies Pairing of identical nucleons plays a very important role in the nuclear binding, and in a number of nuclear phenomena like moment of inertia and its variation, backbending and band-crossing, rotational alignment, super-deformed bands etc. As we shall discuss in this book, the seniority isomerism is also a subtle manifestation of symmetry of the pairing Hamiltonian. We present a very simple empirical manifestation of pairing gap in nuclear isomers. We have plotted in Fig. 1.7, the variation of isomeric excitation energies with respect to proton/neutron number for even-mass nuclear isomers. The upper two panels show the plots for the odd-odd isomers as a function of proton number, and neutron number, respectively. The lower two panels plot the isomer excitation energies for the even-even isomers versus proton number, and neutron number, respectively. The vertical dashed lines represent the respective proton/neutron magic numbers. We have also plotted an average pairing energy in MeV, approximated by the for√ mula  = 12/ A MeV by a solid line. It is known that the average pairing energy decreases with increasing mass number. It is indeed amazing to see that nearly all the odd-odd isomers lie below the pairing energy line, as shown in the upper two panels in Fig. 1.7. On the other hand, all the isomers in the even-even nuclei lie above the pairing energy line as shown in the lower two panels of Fig. 1.7. Thus, nearly all the isomers in even-even nuclei are formed by breaking a pair, while isomers in odd-odd nuclei already have a 2-qp configuration available near the ground state, even without breaking a pair. The odd-odd isomers,

0

80

80

A

0

0

40

80

80

120

120

Neutron Number (N)

even-even

40

odd-odd

160

160

Systematic Features of Isomers

Fig. 1.7 The variation of isomeric excitation energies with proton and neutron number for even- A nuclei. The solid line represents the average pairing energy variation calculated by √12 MeV. The vertical dashed lines mark the magic numbers

Proton Number (Z)

0

40

0

0

4

4

12

0

8

even-even

40

8

12

0

4

4

12

8

odd-odd

8

12

E(MeV)

1.8 13

14

1 An Overview of Nuclear Isomers

12

even-odd

8

E (MeV)

4

0 0

40

80

120

160

Neutron Number (N) 12

odd-even 8

4

0 0

40

80

Proton Number (Z) Fig. 1.8 The variation of isomeric excitation energies with proton and neutron number for odd- A nuclei. The solid line represents an average pairing energy variation calculated by √12 MeV. The A vertical dashed lines mark the magic numbers

which lie above the pairing energy are few in number, and most likely are formed by breaking an additional pair and, therefore, are 4-qp isomers. This also suggests that there is a great scope of finding new isomers in odd-odd nuclei. We also notice a group of isomers in the even-even nuclei, which are lying below the pairing energy line. There are seven E0 isomers in the even-even nuclei between Z = 30 and 45, which lie much below the pairing gap line in Fig. 1.7. All the seven cases correspond to 0+ → 0+ transition and represent shape isomers arising due to shape coexistence. This empirical observation is probably the strongest evidence which supports the shape coexisting nature of these 0+ isomers. Besides these, there are three more cases of 0+ isomers, viz. 12 Be, 32 Mg, and 44 S, which also appear to be the cases of shape isomerism at very low energies; these cases are discussed in Sect. 6.4. Two more cases are seen to lie below the pairing energy line. These are 138 I and 210 Hg, one odd-odd and one even-even, having a 3− isomer. 210 Hg is lying almost on the pairing energy line. But 138 I lies very low at 67.9 keV. Such a low lying 3− excitation in 138 I lacks a full explanation.

1.8 Systematic Features of Isomers

15

Similarly, Fig. 1.8 exhibits the variation of the isomeric excitation energies for odd-A nuclei versus the proton, neutron and mass numbers, respectively. In the upper panel, we plot the data for even Z -odd N nuclei as a function of odd neutron number. In the lower panel, we plot the data for odd Z -even N isomers as a function of odd proton number. The most impressive features of the isomers is the passage of the pairing energy line through the clear openings in the data of both the graphs. These gaps arise in odd-A nuclei in going from 1-qp to 3-qp isomeric states. It may be stressed that the pairing energy exhibits many more features than the average variation plotted here, and is critically influenced by the shell structure due to its intimate relationship with the level density. Therefore, we do expect similar effects on isomer excitation energies also, which should be examined. It may also be pointed out here that the pairing Hamiltonian plays a very crucial role in the formation of seniority isomers. These isomers will be discussed in detail later on.

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34. 35. 36. 37. 38. 39.

40. 41.

42.

43. 44.

45.

46. 47. 48.

Chapter 2

Spin Isomers

Keywords Spin isomers · Isomeric transitions · Internal conversion · Weisskopf estimates · Island of isomers · Role of high-j intruder orbitals · Examples Spin isomers or spin traps represent the most dominant class of isomers observed across the whole chart of nuclides. When the decay process is primarily suppressed by a large change in angular momentum and/or small decay energy, it leads to a longer half-life for the excited state, and hence, the spin isomers. Their occurrence and decay may be understood in terms of the nuclear shell structure, and the selection rules arising from the theory of electromagnetic decay. However, decay by internal conversion becomes very important for smaller decay energies and may modify the transition rates and half-lives significantly.

2.1 Isomeric Transitions The term, isomeric transition (IT), in a nucleus is used to refer to the decay of an excited state by emission of a photon as well as by internal conversion [1]. Weizsacker [2] was the first to provide a theoretical explanation of the long-life of spin isomers, in terms of higher order of multipolarity required for the isomeric decay. Higher the multipolarity of a particular transition, lower is the transition probability, and longer is the half-life. If the decay energy is also small, it is an added advantage. The theory of gamma decay leads to the well known selection rules, which originate from the conservation of angular momentum and parity during the decay. The transition from an initial state i, having angular momentum Ji , to the final state f , having angular momentum Jf , can take place only if the emitted gamma pho which translates ton carries away an angular momentum L such that Jf = Ji + L, into the condition, |Ji − J f | ≤ L ≤ |Ji + J f |, where L, Ji and J f are the angular momentum quantum numbers of the photon, the initial and the final states, respec© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. K. Jain et al., Nuclear Isomers, https://doi.org/10.1007/978-3-030-78675-5_2

17

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2 Spin Isomers

tively. The photon has an intrinsic spin of 1, so the transition Ji = 0 → J f = 0, which requires a L = 0 photon emission, is completely forbidden. The value of L determines the multipolarity of the emitted gamma radiation. Besides, the nuclear states are also labelled by an additional good quantum number, parity. Since parity must be conserved in nuclei, it helps in deciding the electric or, the magnetic nature of a transition. An L-multipole transition is electric in nature if πi π f = (−1) L , and, is magnetic in nature if πi π f = (−1) L+1 , where πi , π f are the parities of the initial and the final states, respectively [11]. Let the transition probability per unit time of the emitted gamma decay from an initial nuclear state i to the final nuclear state f , be denoted by T f i . The mean life. The time t of the transition is given by 1/T f i and its half-life is given by t1/2 = ln2 Tfi transition probability can be shown to be,

M T fαL i

L +1 2 = 0  L[(2L + 1)!!]2



Eγ c

2L+1

| < J f m f | Oˆ αL M |Ji m i > |2

(2.1)

where α denotes the type of the field, either electric (E) or magnetic (M). The Oˆ αL M is the nuclear operator related to the multipole radiation field αL M. Since the magnetic substates are all degenerate, it is useful to redefine the transition probability by averaging over the initial substates (m i ) and summing over all the final substates (m f ) and M, T fαL i =

 1 T αL M 2Ji + 1 m Mm f i i

=

f

L +1 2 0  L[(2L + 1)!!]2



Eγ c

2L+1 B(αL; Ji → J f )

(2.2)

where the reduced transition probability is defined as, B(αL; Ji → J f ) =

1 | < J f || Oˆ αL ||Ji > |2 2Ji + 1

By using the numerical values of the constants like MeV-fm, one can get

e2 4π0 c

=

1 137.04

(2.3) and c = 197.33

2L+1 Eγ B(E L)[e2 f m 2L ]sec−1 (2.4) 197.33  2L+1 Eγ L +1 T fMi L = 6.080 × 1020 B(M L)[(μ N /c)2 f m 2L−2 ]sec−1 L[(2L + 1)!!]2 197.33 T fEi L = 5.498 × 1022

L +1 L[(2L + 1)!!]2



(2.5)

2.1 Isomeric Transitions

19

Table 2.1 Reduced and total transition probabilities in Weisskopf units for electric and magnetic transitions up to L = 5. The gamma energies E γ to be used in MeV TW (EL) (sec−1 )

BW (EL) (e2 f m 2L )

EL

A2/3

E1 E2 E3 E4 E5 ML M1 M2 M3 M4 M5

0.06446 0.05940 A4/3 0.05940 A2 0.0628 A8/3 0.0693 A10/3 BW (ML) ((μ N /c)2 f m 2L−2 ) 1.790 1.650 A2/3 1.650 A4/3 1.746 A2 1.926 A8/3

1.023 ×1014 E γ 3 A2/3 7.265 ×107 E γ 5 A4/3 3.385 ×101 E γ 7 A2 1.0649 ×10−5 E γ9 A8/3 2.395 ×10−12 E γ11 A10/3 TW (ML) (sec−1 ) 3.184 ×1013 E γ 3 2.262 ×107 E γ 5 A2/3 1.054 ×101 E γ 7 A4/3 3.276 ×10−6 E γ9 A2 7.367 ×10−13 E γ 11 A8/3

In case of decay to many possible final states, the transition probabilities are additive. One may, therefore,  sum over the final states f , to obtain the total half-life 1 = f t f 1 . Here t f 1/2 is the partial half-life for transition to a of the initial state t1/2 1/2 particular final state f . For simplicity, one can estimate the reduced transition probabilities by introducing some approximations in the wave functions. To start with, the radial wave function is assumed to be constant inside the nucleus and zero outside. The radial integral then acquires a simple value in terms of R = R0 A1/3 = 1.2 A1/3 fm. One, therefore, obtains the Weisskopf single-particle estimate for the reduced transition probability as, BW (E L) =

1.22L 4π



3 L +3

2 2L

A 3 e2 f m 2L

(2.6)

for electric transitions, and 10 × 1.22L−2 BW (M L) = π



3 L +3

2 A

2L−2 3

(μ N /c)2 f m 2L−2

(2.7)

for magnetic transitions. By substituting these expressions, one obtains the transition probabilities per unit time in Weisskopf units (W.U.) (TW ). For completeness, the Table 2.1 provides a few numeric values of the reduced and total transition probabilities in Weisskopf units for both electric and magnetic transitions up to L = 5. These are also known as the single-particle estimates as it is assumed in the derivation that only one nucleon changes its single-particle state in the nucleus during the decay. If the observed transition is actually single-particle in nature, then the

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2 Spin Isomers

respective total transition probability per unit time is expected to be around 1 W.U . It is, therefore, a normal practice to quote the decay probabilities in terms of the Weisskopf estimates, which allows one to make useful physics inferences. It becomes clear from the Eqs. (2.4) and (2.5) that the gamma decay probability diminishes for small decay energy and/or, large multipolarity L of the transition. For a given gamma decay energy and mass number, the gamma decay transition probability goes down by six orders of magnitude for the next higher multipole order. Obviously, it is more likely to have an isomer if E3, or M3 decay is involved rather than E2, or M2 decay. Therefore, occurrence of the spin isomers may be considered a ‘normal’ feature in the excitation spectrum of nuclei, when high-j orbitals lie in the proximity of the Fermi energy. This is the situation near the magic numbers. Therefore, a clear link can be established between the shell effects associated with the spherical magic numbers and the island of isomers.

2.2 Internal Conversion and Isomeric Half-Life When an excited nucleus interacts electromagnetically with one of the electrons in an atomic orbital and ejects it from the atom, it is termed as the process of internal conversion (IC). This process usually competes with gamma decay except in the case where the atom is fully ionised, and there are no electrons to be ejected. This situation becomes very important in stellar atmosphere where fully ionised atoms are present, and the half-life may change drastically [11]. IC becomes more and more favourable for smaller decay energies and plays a critical role in low energy isomers that we discuss in later chapters. IC also becomes critical in the 0+ → 0+ (E0) transitions, where both the initial and final states have zero spin and positive parity, so that a gamma decay is not possible, and IC is the only possible mode of decay. It may be pointed out that IC may also have a contribution from nuclear structure part besides the well known electromagnetic part. This is important for the E0 transitions and also hindered E1 and M1 transitions. The review by Church and Weneser [3] discusses these aspects in detail. This is particularly relevant for the E0 isomers, where the gamma decay is totally absent. The competition between internal conversion and gamma decay is quantified in the form of the total internal conversion coefficient (α) which is defined as the ratio α = Ne /Nγ where Ne is the number of conversion electrons emitted and Nγ is the number of gamma-rays emitted in the same time from a decaying nucleus. The total IC coefficient is a sum of the partial conversion coefficients for K-shell, L1 sub-shell, L2 sub-shell ionization etc. and may be written as α = α K + α L1 + α L2 + ...etc. The partial conversion coefficients are needed in case of partial ionization of atoms. A comparison of the theoretical values with the experimental values of the internal conversion coefficients (ICC), is commonly used to fix the multipolarities and mixing ratios of gamma transitions. This is very helpful in constructing the correct level/decay schemes, and in the decay heat calculations of spent nuclear fuel [4].

2.2 Internal Conversion and Isomeric Half-Life

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IC being an electromagnetic process, it is possible to calculate the IC coefficients (α) rather precisely for a given Z and associated gammas by using the BrIcc calculator [5]. The BrIcc values are based on the relativistic Dirac-Fock model, whose results are summarized in [6]. On taking the ICCs into account, the total transition probabilities become 2L+1  Eγ L +1 B(E L)(1 + α)sec−1 L[(2L + 1)!!2 ] 197.33 2L+1  L +1 Eγ = 6.080 × 1020 B(M L)(1 + α)sec−1 L[(2L + 1)!!2 ] 197.33

T fEi L = 5.498 × 1022

(2.8)

T fMi L

(2.9)

where B(E L) and B(M L) are in units of [e2 f m 2L ] and [(μ N /c)2 f m 2L−2 ], respectively, and L is the multi-polarity of transition. For the particular cases of E1 and E2 transitions, the total transition probabilities can be calculated by using the formulas: T (E1) = 1.587 × 1015 E γ 3 × B(E1) × (1 + α)sec−1 T (E2) = 1.223 × 10 E γ × B(E2) × (1 + α)sec 9

5

−1

(2.10) (2.11)

Here B(E1) and B(E2) are the reduced transition probabilities for E1 and E2 transitions, respectively. The values of E γ are in the units of MeV. Therefore, the T (E L) is obtained in the units of second−1 . The corresponding half-lives (in the units of seconds) will be given by γ

T 1 = 0.693/T (E L)

(2.12)

2

In case of multiple gammas decaying from a nuclear state, the total half-life can be written in terms of partial half-lives. Therefore, the total half-life in seconds (which can be directly compared to the measured half-life) may be obtained as γ

T 21 = T 1 i × B.R.i

(2.13)

2

γ

where T 1 i is the partial half-life for any given γi transition and B.R.i is the branching 2 ratio for the respective γi , and may be calculated by Iγi (1 + αi ) B.R.i =  i Iγi (1 + αi )

(2.14)

where αi and Iγ i are the IC coefficients and intensities for the respective γi . The Relativistic self-consistent-field Dirac-Fock model has been used in the Raine code [6] to calculate the ICC, which have been tested against the experimental data [7, 8], and the results have been found to be accurate within 1% for transition energies above 1 keV of the atomic shell binding energies. For gamma transition energies close to the shell binding energies, the kinetic energy of outgoing electrons is very

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2 Spin Isomers

Half-life (seconds)

1E16

E1 (No ICC) E2 (No ICC) M1 (No ICC) E1 (ICC) E2 (ICC) M1 (ICC)

1000000

1E-4

1E-14 0.0

5.0x10 3

1.0x10 4

1.5x10 4

2.0x10 4

Eγ (eV) Fig. 2.1 Half-life variation for E1, E2 and M1 transitions versus very low arbitrary values of gamma transition energies (E γ in eV) for a situation typical of 230 Th, with and without the correction due to the internal conversion coefficient (ICC). The values with ICC correction are plotted up to 1 keV above the atomic shell binding energy only, the limit of the BrIcc code

low and electron correlation effects may become significant. This is why BrIcc can only give ICC’s for transition energies 1 keV above the binding energy. Figure 2.1 presents the variation of calculated half-lives corresponding to the E1, E2 and M1 transitions for very low arbitrary values of gamma transition energies (E γ ) in a situation typical of 230 Th. The half-lives are obtained with and without ICC, and plotted in the figure. A significant decrease in the half-lives, when ICC are taken into account, is observed. Therefore, the correction due to internal conversion is very crucial in calculating the isomeric half-lives.

2.3 Islands of Spin Isomers Near Magic Numbers A knowledge of the single-particle shell model level scheme and the associated magic numbers (see Fig. 1.5), along with the gamma decay selection rules is sufficient to figure out the occurrence of high-spin isomers near magic nuclei. If we are dealing with the even-even or, odd-odd nuclei, then the rules to figure out the allowed and favored spins, like the Nordheim rules [9] and Brennan-Bernstien rules [10] may help [11], because we may have 2-quasiparticle (qp) or 4-qp excitations as isomers. In odd-A nuclei, we may come across 3-qp or 5-qp excitations as isomers. We now present several pictorial representations which demonstrate the correlation of islands of high-spin isomers and the magic numbers.

2.3 Islands of Spin Isomers Near Magic Numbers

23

2.3.1 g9/2 Spin Isomers We depict in Fig. 2.2, the occurrence of isomers having spins 9/2+ , 8+ , and 21/2+ . The zig-zag line has been plotted by joining the most stable isotopes for the known elements, and represents the corridor of stability. It serves the purpose of identifying the location of isomers with respect to the line of beta-stability. We try to understand the occurrence of these isomers in terms of the filling of the g9/2 orbital. Single unpaired particle or, hole in g9/2 orbital can give rise to a spin 9/2 and positiveparity isomer. We have shown in the three panels, the area of influence of the g9/2 orbital occupancy. The horizontal blue lines enclose the odd-proton area of influence, and the vertical blue lines, the odd-neutron area of influence of the g9/2 orbital. It is rather obvious that most of the 9/2+ isomers are indeed clustered in the influencearea of g9/2 orbital. Ideally, g9/2 orbital begins to fill at nucleon number 40 and should get filled up by nucleon number 50. However, the 9/2+ isomers begin to be observed quite early from N = 35 (59 Cr) [12]. The heaviest 9/2+ isomers observed in this cluster have Z = 55 (121 Cs and 123 Cs) beyond the magic number 50. However, a non-existence of 9/2+ isomers is noted between Z = 41 (89 Nb) and Z = 53 (119 I), where no isomers have been observed except in two Rh isotopes (99 Rh and 101 Rh) having Z = 45. This is mainly because the 9/2+ single-particle level becomes the ground state in nearly all the nuclides in this region. Another cluster of 9/2+ isomers is observed in the rare-earths, and lies in the deformed region where the K -isomers are expected (which will be discussed in the next chapter). This is the area of influence of the i 13/2 orbital and the deformed single-particle orbital having K = 9/2 appears to be playing an important role in these isomers. A lone 9/2+ isomer observed in 219 Rn lies at an unusually low energy of 4.47 keV and has a highly converted decay to the 5/2+ ground state. The Z = 86 and N = 133 numbers overlap with the octupole collectivity region starting at Z = 88, N = 134. The closely lying g9/2 , j15/2 neutron orbitals and f 7/2 , i 13/2 proton orbitals near the Fermi energy generate the octupole correlations in nuclei of this mass region [13]. However, the spin assignment 9/2 is tentative in nature, as is the 7/2 assignment to the level next to it [14]. It would be very useful if these spin assignments are ascertained experimentally. The other two panels in Fig. 2.2 depict the 8+ and the 21/2+ isomers, which most likely arise from the 2-qp and 3-qp g9/2 configurations, respectively. It is interesting that most of these isomers are clustered around N = 50, N = 82, Z = 82 and N = 126. While the N = 50 2-qp and 3-qp isomers have some overlap with the occurrence of 9/2+ isomers, the 2-qp and 3-qp isomers near N , Z = 82 and N = 126 have very little overlap with the 9/2+ isomers. These clusters need detailed investigation. There are individual cases which defy a general explanation. For example, there is a lone 8+ and a lone 21/2+ isomer lying in the actinide region. Similarly, there is a lone 9/2+ isomer lying beyond N = 126. The lone 8+ isomer in 256 Es is a betadecaying isomer having 7.6 h half-life; its spin assignment is tentative and excitation

24

2 Spin Isomers

9/2+ isomers 80

g9/2

i13/2

40

Proton Number (Z)

0 80

8+ isomers g9/2

i13/2

21/2+isomers g9/2

i 13/2

40

0 80

40

0 0

40

80

120

160

Neutron Number (N) Fig. 2.2 Occurrence of the 9/2+ , 8+ , 21/2+ isomers, mostly originating from the g9/2 orbital and also the i 13/2 orbital

energy is not known. The 21/2+ isomer in 257 Rf (Z = 104, N = 153) at 1083 keV is most likely a 3-qp isomer. However, its spin-parity both are tentative in nature and need to be confirmed.

2.3.2 h11/2 Spin Isomers Next, we look into the isomers originating from the h 11/2 orbital. The area of influence of h 11/2 orbital is also shown by the blue solid lines and all the 11/2− isomers fall in this region. We display the isomers having spins 11/2− , 10+ , and 27/2− , arising from 1-qp, 2-qp, and 3-qp states respectively, in the three panels of Fig. 2.3. As we can see from Fig. 2.3, there are plenty of 11/2− isomers from Z = 40 to 80 and N = 60 to 126. These are spread out much more evenly except for a gap around N = 100 to

2.3 Islands of Spin Isomers Near Magic Numbers

25

j15/2

-

80

11/2 isomers

h11/2

40

Proton Number (Z)

0 80

+

10 isomers

h11/2

40

0

80

-

27/2 isomers

h11/2

40

0 0

40

80

120

160

Neutron Number (N) Fig. 2.3 Occurrence of the 11/2− , 10+ , 27/2− isomers, mostly originating from the h 11/2 orbital

110, where there are no 11/2− isomers. This is the region where 11/2[505] neutron orbital becomes a hole state lying much below the Fermi energy [13]. Lower two panels of Fig. 2.3 plot the 10+ and 27/2− isomers arising from the 2-qp and 3-qp configurations in h 11/2 orbital, respectively. These are rather easily interpreted once the 1-qp isomers are understood.

2.3.3 i 13/2 Spin Isomers We show in Fig. 2.4, the 13/2+ isomers on the N − Z plot, which are all clustered around the area of influence of i 13/2 neutron orbital. Most of these isomers are found near Z = 82 and interestingly, many of them lie above Z = 82, indicating that the influence of i 13/2 orbital extends beyond Pb isotopes. It is rather puzzling to see a lone 13/2+ isomer in 69 Cu, having Z = 29. It is most likely a 3-qp structure lying at 2.74 MeV.

26

2 Spin Isomers

+ 13/2 isomers

80

i13/2 40

Proton Number (Z)

0

+ 12 isomers

80

i13/2

40

0

+ 33/2 isomers

80

i13/2

40

0 0

40

80

120

160

Neutron Number (N) Fig. 2.4 Occurrence of the 13/2+ , 12+ , 33/2+ isomers, mostly originating from the i 13/2 orbital

We plot in the lower two panels, the 2-qp and 3-qp isomers having spin 12+ and 33/2+ respectively. Most of these are easily understood as 2-qp or, 3-qp counterparts of the 1-qp 13/2+ isomers in neighbouring nuclei. However, again we have some 12+ and 33/2+ isomers lying below N = 82 and some lying near N = 30 to 60. The 12+ isomers having masses lower than the rare-earths have high excitation energies and may have a multi-quasiparticle (MQP) structure. For example, 52 Fe, 84 Kr, 98 Cd, 106 Cd, and 134 Te, all have excitation energies in the range of 4.6 to 6.9 MeV. Higher lying 33/2+ isomers may be easily interpreted as 3-qp isomers due to the i 13/2 orbital. We pay special attention to the 12+ isomers in some of the nuclides listed here. Incidentally, all these nuclides are in close proximity of magic numbers and have either two-particle or, two-hole neutrons/protons or, both. The 12+ isomer in 52 Fe (Z = N = 26) undergoes beta-decay as well as E4 transition to the lower 8+ state. As noted by A. Gadea et al. [15], there is an inversion of states so that 12+ comes below 10+ and a spin trap is created which decays by E4 transitions to the two lower-lying 8+ states. The high multipolarity of gamma decay is the main cause of the large half-life of 45.9 sec of the 12+ isomer. It is clearly a

2.3 Islands of Spin Isomers Near Magic Numbers

27

spin isomer. The shell model calculations reported by Gadea et al. [15] indicate that the two E4 transitions in 52 Fe are highly hindered compared to the E4 transitions in other nuclei in this region. A similar 12+ isomer lying at 6635 keV in 98 Cd has a half-life of 224 ns and decays by an E4 transition to the 8+ seniority isomer. It is a two-proton hole nucleus (Z = 48, N = 50). Incidentally, the E4 decay to 8+ is the strongest although a decay to 10+ also exists. Due to a large E4 gamma energy, the 12+ isomer has been interpreted as a core excitation state by Park et al. [16] and others.

2.4 High-Spin Isomers Near the Proton Drip Line Role of spin and seniority isomers in the drip line neutron-deficient nuclei near the doubly magic 100 Sn was discussed quite early by Grawe et al. [17]. These isomers have proved to be a sensitive probe of single-particle energies, neutron-proton residual interaction, and shell structure. In an interesting prediction based on large scale shell model calculations, a high lying 16+ isomer in the N = Z nucleus, 96 Cd was predicted and termed as “spin gap” isomer. This isomer was observed for the first time in an experiment conducted at GSI. Fragmentation of 124 Xe high energy beam on 9 Be target was analysed by the FRS, stopped in the Rising setup, and gamma rays detected by Euroball array [18]. As envisaged in the calculations [17], the isoscalar part of the neutron-proton interaction was found to play an important role in the origin of the isomer.

2.5 Spin Isomers in 180m Ta and the Only Natural Isomer A well known example of spin isomers is found in the doubly-odd nucleus 180 Ta, which has the only naturally occurring isomer having a spin 9− and a half-life of >1015 y , see Fig. 1.2. There exist only two lower-lying states, the 1+ ground state, and the 2+ first-excited state. The lowest possible multipolarity of the gamma transition is λ = 7. The decay is so slow that it remains unobserved making 180m Ta the only isomer found in nature, as its half-life exceeds the age of the universe. Because of its unusual characteristics, the odd-odd 180 Ta has attracted the attention of many researchers in the fields of nuclear astrophysics and nuclear structure [19–25]. Actually, 180m Ta exists in nature, only in its J π = 9− isomeric form with an isotopic abundance of 0.012% at an excitation energy E x = 77.2 keV [36]. The ground state of 180 Ta is unstable and decays with half-life of 8.154 h and has zero natural abundance [36]. Rest of the Ta found in nature consists of 181 Ta. Even after continuous efforts made during the past two decades, the unusually small natural abundance of 180m Ta in isomeric form is still not fully understood. Figure 2.5 shows a partial level scheme of 180 Ta and its decay branches. The unstable ground state undergoes β − decay to 180 W and electron capture to 180 Hf.

28

2 Spin Isomers

Table 2.2 Various measurements on the half-life of 9− isomer in 180 Ta Technique [Ref.] Lower half-life limit in y EC β− Mass Spec. [26] Mass Spec. [27] γ-spec. NaI [28] γ-spec. NaI [29] γ-spec. Ge(Li) [30] γ-spec. Ge(Li) enriched in Ta [31] γ-spec. HPGe enriched in Ta [32] γ-spec. HPGe [33] γ-spec. Sandwich HPGe [34] γ-spec. Sandwich HPGe [35]

Total

– 4.6 ×109 2.3 ×1013 1.5 ×1013 2.1 ×1013 5.6 ×1013

×1011

9.9 – 1.7 ×1013 – – 5.6 ×1013

– 4.5 ×109 9.7 ×1012 – – 2.8 ×1013

3.0 ×1015

1.9 ×1015

1.2 ×1015

1.7 ×1016 4.45 ×1016

1.2 ×1016 3.65 ×1016

7.2 ×1015 2.0 ×1016

2.0 ×1017

5.8 ×1016

4.5 ×1016

Fig. 2.5 The decay level scheme of 180 Ta

The isomeric state with J π = 9− is due to the spin-aligned Nilsson configuration π9/2[514] ⊗ ν9/2[624] while the ground state spin of 1+ is anti-aligned configuration π7/2[404] ⊗ ν9/2[604]. Due to large spin difference, a depopulation of 180m Ta can only occur by photo-excitation into excited states which have a decay branch into the ground state. Several measurements have been performed in the past to search for the decay of 180m Ta [26–35], as listed in the Table 2.2, which lead to only a lower limit on the

2.5 Spin Isomers in 180m Ta and the Only Natural Isomer

29

Table 2.3 Various isomers in 180 Ta with a lower limit on half-life as 10 ns Energy (keV) Jπ T1/2 Multipolarity 77.2 (12) 107.85 (4) 177.65 (3) 356.68 (6) 463.24 (6) 520.14 (8) 594.43 (16) 1452.40 (18) 2588.37 (22) 3679.0 (11) 4171.0+X

9− 0− 8+ 7+ 7− 4+ (5) 15− 18(+) (22− ) (23, 24, 25)

> 4.5E+16 y 19.2(7) ns 70.0(14) ns 42(3) ns 31.2(19) ns 37.4(20) ns 16.1(19) ns 31.2(1) μs 22(2) ns 2.0(5) μs 17(5) μs

E1 E1 + M2 M1 E1 + M2 M1 M1, E2 E1 (E2)

Decay %β= ?, % = ? %I T = 100 %I T = 100 %I T = 100 %I T = 100 %I T = 100 %I T = 100 %I T = 100 %I T = 100 %I T = 100

half-life. In a recent measurement, a limit of T12 > 4.5 × 1016 y has been established for its half-life [35]. We may also point out that 180 Ta has at least 10 other isomers having half-life greater than 10 ns, and these are listed in the Table 2.3. There are many isomers having half-life less than 10 ns. This is a rather unusual collection of isomeric states in a single nucleus. Isomeric states having a large spin difference from the lower-lying states can not be depopulated or populated directly in photo-induced reactions due to the low transfer of angular momentum by photons. Therefore, photo-activation process has to proceed indirectly by the resonant excitation of a higher lying excited state, i.e. an intermediate state. Its decay can feed the isomeric or ground state, by electromagnetic transition. There is a lack of monochromatic and tunable photon sources of high spectral intensity at MeV energies. However, a Bremsstrahlung irradiation facility having high-current and low-energy electron accelerators can be used to improve photo-activation experiments [37]. Alternatively, 180 Ta has been studied by γ-spectroscopy in several experiments [38–43]. Walker et al. [43] investigated the decay of isomer indirectly through a K = 5 band, and presented a concrete suggestion of a connection between levels and transitions and the intermediate states observed in photo-activation experiments. Its induced decay by x-rays was first attempted by Collins et al. [44]. On the other hand, Loewe et al. [45] and Schlegel et al. [46] performed in-beam Coulomb excitation experiments. Schlegel et al. [46] studied the depopulation of the J π = 9− isomer in 180 Ta to the J π = 1+ ground state by Coulomb excitation. However, they could not identify the intermediate state even after populating ground state following Coulomb excitation. All scattering experiments have been limited by the extremely low natural abundance of 180 Ta. The astrophysical consequences for a possible sprocess production was discussed by Belic et al. [47]. Later on, Belic et al. [19, 47] reported the release of its energy by resonant photo-excitation to higher lying

30

2 Spin Isomers

states. They reported a photo-induced depopulation experiment using the enriched Ta material with the high-current Dynamitron accelerator at Stuttgart with a set up optimized for the off-line activation [37]. This set up resulted in extracting the intermediate states down to 1 MeV. They also performed additional nuclear resonance fluorescence measurements with the enriched target to search for the decay branches of intermediate states back to the 9− isomer. Hult et al. [33] studied the decay of isomer in 180 Ta in an underground experiment. Mohr et al. [23] reported various activities of 180 Ta in stellar environments. Dracoulis et al. [48, 49] tried to explore the connections between the high- and low-K bands of 180 Ta in comparison to the s-process nucleus 176 Lu.

2.6 Spin Isomers in 208 Pb The isomerism in and around doubly magic nuclei has been the subject of many researches as it helps in evaluating the shell-model description of nuclear systems. A large number of long-lived excited states have been identified in 208 Pb by the recent work of Broda et al. [50]. Three new isomers were established along with the previously known 535(35) ns high-spin and high-excitation isomer of 10+ spin at 4.8 MeV energy [50, 51]. The new isomers (with a half-life ≥ 10 ns) were found for the first time in [50] at the energies of 10.3, 11.3 and 13.7 MeV, as the 20− isomer with half-life of 22 (3) ns, 23+ isomer with half-life of 12.7 (15) ns and 28− isomer with half-life of 60 (6) ns, respectively. The 13.7 MeV energy of the 28− isomer is the highest measured energy of an isomer (with half-life ≥ 10 ns). It is to be noted that there exist a few more long-lived excited states in this nucleus which have a half-life < 10 ns, such as a 4.6 MeV, 8+ isomer with half-life of 3.2 ns [51]. Two cases where the half-life was not measured [50], but calculated assuming the enhancement expected for the E3 transitions being considered, are the 6.4 MeV, 13− isomer with half-life of 0.188 ns and the 6.7 MeV, 14− isomer with half-life of 0.42 ns. These high-spin one particle-one hole excitations in the 13− and 14− states 1 ) configuration and exhibit interesting are associated with a pure neutron ( j15/2 i 13/2 deexcitation scheme. Most of the states in 208 Pb involve one particle-one hole excitations and are well understood by using the shell-model, which provides a detailed structural composition [50, 52–54]. The most complex isomeric decay can be seen for the 3.2 ns, 4611 keV 8+ isomer with seven γ-decay branches out of which five were established by Broda et al. [50] for the first time. This 8+ isomeric decay also involves the fast j15/2 → g9/2 E3 transition to lower-lying 5− state (See Fig. 2.6). The 10+ isomer has two E2 and one E3 decay branches, where extra hindrance to E2 transitions 1 ) wavefunction in the structure strongly supports the mixing of neutron ( j15/2 f 5/2 of this isomer responsible for E3 transition to the lower-lying 7− state. The largest enhancement (56 W.U.) was observed for the 28− isomeric transition, which was tentatively proposed to have the involvement of two octupole phonons. Such studies were extensively compared with the shell model calculations which were found to

2.6 Spin Isomers in 208 Pb

31

Fig. 2.6 The partial decay level scheme of 208 Pb. The gammas (in keV) corresponding to 10+ and 8+ isomers are shown explicitly, among which the thicker arrows refer to the strongest intensity gammas. The color coding is just used to view clearly, both the isomers and the corresponding gammas

be in line with each other. Some discrepancies were reported in the higher part of the level scheme. These high-spin isomeric investigations, overall, provide a guidance to improve/extend the shell model calculations for higher-spin states. Besides this, the rotational interpretation of yrast and near-yrast states in 208 Pb was also considered by [50], based on the approximate linear dependence of level energies on J versus the conventional J (J + 1) (where J is spin) dependence. The associated moment of inertia was found to be a factor of 2 smaller than the expected value for a rigid 208 Pb sphere.

2.7 E5 Decaying Spin Isomers The highest multipolarity decay leading to isomers is the E5 decay. As expected, all of the known cases have a long half-life due to the hindrance caused by large multipolarity. We present here a brief description of those cases which decay by purely E5 gamma photon.

32

2 Spin Isomers 186

Re: Four long-lived excited states are known in the odd-odd nucleus 186 Re within the half-life limit of 10 ns. These are all low lying states as expected in an odd-odd nucleus (Z = 75, N = 111). They are the 99 keV, (3)− isomer with half-life of 25.5 ns, 148 keV, (8+ ) isomer with half-life of 2 × 105 y, 314 keV, (3)+ isomer with half-life of 24 ns, and approximately 330 keV, (5)+ isomer with half-life of 17 ns. The (8+ ) isomer with 2 × 105 y half-life decays with an E5 transition. 192 Ir: The 11− isomer in 192 Ir (Z = 77, N = 115) with 241 y half-life at 168.14 keV energy also decays with an E5 transition. 178 Hf: The 16+ isomer in 178 Hf (Z = 72, N = 106) with 31 y half-life at 2.44 MeV energy also decays with E5 transition. 113 Cd: The 11/2− isomer in 113 Cd (Z = 48, N = 65) with 13.89 y half-life at 263.54 keV energy, also decays by E5 transition. We note that at least one of them, the 16+ isomer in 178 Hf, has a high excitation energy of 2.44 MeV. Few additional cases are also known where the decay is by M4 + E5: these are 89 Y, 90 Y, 133 Xe and 137 Ba.

References 1. 2. 3. 4. 5. 6. 7. 8.

9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.

20.

E. Segre, A.C. Helmholtz, Rev. Mod. Phys. 21, 271 (1949) C.F. von Weizsacker, Naturewissenschaften 24, 813 (1936) E.L. Church, J. Weneser, Ann. Rev. Nucl. Sc. 10, 193 (1960) A.L. Nichols, Beta Decay and Decay Heat, I.A.E.A. Report No. INDC (NDS) 499 (2006) T. Kibedi, T.W. Burrows, M.B. Trzhaskovskaya, P.M. Davidson, C.W. Nestor, Jr., Nucl. Instr. Meth. A 589, 202–229 (2008). http://bricc.anu.edu.au/ I.M. Band, M.B. Trzhaskovskaya, C.W. Nestor Jr., P.O. Tikkanen, S. Raman, At. Data. Nucl. Data Tables 81, 1 (2002) S. Raman, C.W. Nestor Jr., A. Ichihara, M.B. Trzhaskovskaya, Phys. Rev. C 66, 044312 (2002) T. Kibedi, T.W. Burrows, M.B. Trzhaskovskaya, C.W. Nestor, Jr., P.M. Davidson, Proc. Intern. Conf. Nuclear Data for Science and Technology, Nice, France, April 22–27, 2007. O. Bersillon, F. Gunsing, E. Bauge, R. Jacqmin, S. Leray (Eds.), p. 57 (2008) EDP Sciences, 2008 L.W. Nordheim, Phys. Rev. 78, 294 (1950) M.H. Brennan, A.M. Bernstein, Phys. Rev. 120, 927 (1960) R.R. Roy, B.P. Nigam, Nuclear Physics (New Age International Ltd., New Delhi, 1967) A.K. Jain, B. Maheshwari, S. Garg, M. Patial, B. Singh, Nucl. Data Sheets 128, 1 (2015); S. Garg, B. Maheshwari, A. Goel, A.K. Jain, B. Singh, update to be published A.K. Jain , R.K. Sheline, P.C. Sood, K. Jain, Rev. Mod. Phy. 62, 393 (1990) Evaluated Nuclear Structure Data File http://www.nndc.bnl.gov/ensdf/ A. Gadea et al., Phys. Lett. B 619, 88 (2005) J. Park et al., Phys. Rev. C 96, 044311 (2017). Pub. Note Phys. Rev. C 96, 049901 (2017). Erratum Phys. Rev. C 97, 019901 (2018) H. Grawe, A. Blazhev, M. Gorska, R. Grzywacz, H. Mach, I. Mukha, Eur. Phys. J. A 27, 257 (2006) B.S. Narasingh et al., Phys. Rev. Lett. 107, 172502 (2011) D. Belic, C. Arlandini, J. Besserer, J. de Boer, J.J. Carroll, J. Enders, T. Hartmann, F. Kappler, H. Kaiser, U. Kneissl, E. Kolbe, K. Langanke, M. Loewe, H.J. Maier, H. Maser, P. Mohr, P. von Neumann-Cosel, A. Nord, H.H. Pitz, A. Richter, M. Schumann, F.-K. Thielemann, S. Volz, A. Zilges, Phys. Rev. C 65, 035801 (2002) A.P. Tonchev, J.F. Harmon, Appl. Radiat. Isot. 52, 873 (2000)

References 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46.

47. 48. 49. 50. 51. 52. 53. 54.

33

H. Utsunomiya et al., Phys. Rev. C 67, 015807 (2003) S. Goko et al., Phys. Rev. Lett. 96, 192501 (2006) P. Mohr, F. Käppeler, R. Gallino, Phys. Rev. C 75, 012802(R) (2007) T. Hayakawa, P. Mohr, T. Kajino, S. Chiba, G.J. Mathews, Phys. Rev. C 82, 058801 (2010) K.L. Malatji et al., Phys. Lett. B 791, 403 (2019) P. Eberhardt, J. Geiss, C. Lang, W. Herr, E. Merz, Z. Naturforschg. 10a, 796 (1955) P. Eberrhardt, P. Signer, Z. Naturforschg A 13, 1004 (1958) E.R. Bauminger, S.G. Cohen, Phys. Rev. 110, 953 (1958) K. Sakamoto, Nucl. Phys. A 103, 134 (1967) G. Ardisson, Radiochem. Radioanal. Lett. 29, 7 (1977) E.B. Norman, Phys. Rev. C 24, 2334 (1981) J.B. Cumming, D.E. Alburger, Phys. Rev. C 31, 1494 (1985) M. Hult, J. Gasparro, G. Marissens, P. Lindahl, U. Watjen, P.N. Johnston, C. Wagemans, M. Kohler, Phys. Rev. C 74, 054311 (2006) M. Hult et al., Appl. Rad. Isot. 67, 918 (2009) B. Lehnert, M. Hult, G. Lutter, K. Zuber, Phys. Rev. C 95, 044306 (2017) E.A. Mccutchan, Nuclear Data Sheets 126, 151 (2015) D. Belic et al., Nucl. Instrum. Methods Phys. Res. A 463, 26 (2001) G.D. Dracoulis et al., Phys. Rev. C 53, 1205 (1996) G.D. Dracoulis et al., Phys. Rev. C 58, 1444 (1998) T.R. Saitoh et al., Nucl. Phys. A 660, 121 (1999) G.D. Dracoulis et al., Phys. Rev. C 62, 037301 (2000) C. Wheldon et al., Phys. Rev. C 62, 057301 (2000) P.M. Walker, G.D. Dracoulis, J.J. Carroll, Phys. Rev. C 64, 061302(R) (2001) C.B. Collins et al., Phys. Rev. C 37, 2267 (1988) M. Loewe et al., Z. Physik A 356, 9 (1996) Ch. Schlegel, P. von Neumann-Cosel, J. de Boer, J. Gerl, M. Kaspar, I. Kozhoukharov, M. Loewe, H.J. Maier, P.J. Napiorkowski, I. Peter, M. Rejmund, A. Richter, H. Schaffner, J. Srebrny, M. Wurkner, H.J. Wollersheim, The EBGSI96-Collaboration Eur. Phys. J. A 10, 135 (2001) D. Belic et al., Phys. Rev. Lett. 83, 5242 (1999) G.D. Dracoulis, A.I.P. Conf, Proc. 1269, 295 (2010) G.D. Dracoulis, F.G. Kondev, G.J. Lane, A.P. Byrne, M.P. Carpenter, R.V.F. Janssens, T. Lauritsen, C.J. Lister, D. Seweryniak, P. Chowdhury, Phys. Rev. C 81, 011301(R) (2010) R. Broda et al., Phys. Rev. C 95, 064308 (2017) N. Roy et al., Phys. Lett. B 221, 6 (1989) B.A. Brown, Phys. Rev. Lett. 85, 5300 (2000) A. Heusler, R.V. Jolos, T. Faestermann, R. Hertenberger, H.-F. Wirth, P. von Brentano, Phys. Rev. C 93, 054321 (2016) M. Rejmund, M. Schramm, K.H. Maier, Phys. Rev. C 59, 2520 (1999)

Chapter 3

Seniority Isomers

Keywords Seniority · Quasi-spin scheme · Seniority isomers · Isomeric decay properties · Isomeric moments · Examples · Seniority mixing This chapter is devoted to the seniority isomers, which are generally found in spherical or nearly spherical semi-magic nuclei. It is rather unusual that a quantity like seniority introduced purely on theoretical grounds, merely to distinguish states having same quantum numbers, should become so important to understand a class of nuclear excitation.

3.1 Seniority and Seniority Isomers The concept of seniority was first introduced by Racah in 1943 [1], in the third of the celebrated series of four papers on complex atomic spectra, as a purely mathematical tool to distinguish those many-electron states, which have the same values of L, S, and J , representing the orbital, spin, and total angular momentum quantum numbers, respectively. In most simple terms, seniority v may be defined as the number of unpaired particles required to generate a given state. The symbol v was used by Racah to denote seniority and has its origin in phonetic beginning with v for seniority in Hebrew. Later on, Seniority was introduced in nuclear physics by Racah [2], Flowers [3] and Talmi [4] almost simultaneously. A detailed description of the concept of seniority and related symmetry schemes may be found in the books of Talmi [4], Rowe and Wood [5], and Kota and Devi [6]. A simple physical exposition of seniority has also been presented by Casten in his book [7], which underlines the powerful nature of this concept in exploring a wide range of observations in semi-magic nuclei. The talk by Racah in 1958 [8], however, presents a very simple yet profound physics understanding of the purely mathematical concept of seniority. Racah empha© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. K. Jain et al., Nuclear Isomers, https://doi.org/10.1007/978-3-030-78675-5_3

35

36

3 Seniority Isomers

sized the key role of short-range attractive pairing interaction which diagonalizes an operator Q, giving the number of paired identical nucleons. In his own words, “these ‘saturated’ pairs, being coupled to zero angular momentum, possess particular symmetry properties and either do not contribute at all or contribute in a very simple way to many characteristic properties of a state, like angular momentum and multipole moments. A number of saturated pairs can, therefore, be added to any given system of particles without changing these properties very much. Thus we can consider a state as consisting of a number, v, of non-saturated particles which determines most of the properties of the state, plus a number of saturated pairs. We can construct a series of states for systems containing v, v + 2, v + 4, etc. particles, each differing from the others only by the number of additional saturated pairs and all having very similar properties. The number v of non-saturated particles is called the seniority number because it gives the smallest number of particles needed for building a state with a given set of properties and therefore specifies the simplest configuration which contains such a state”. The concept of good seniority states leads us to a special class of wave functions which are the eigen states of the pairing Hamiltonian. Seniority quantum number is thus intimately linked to the symmetry of the pairing Hamiltonian. Seniority, which corresponds to the number of unpaired nucleons, has been instrumental in explaining several complex phenomena in an elegant manner. Several structural and spectroscopic properties in nuclei can be described in terms of seniority selection rules. This symmetry, therefore, brings forth a simple physical understanding of several interesting features in the semi-magic nuclei, like particle number independence of energy and the parabolic variation of B(E2) values. The seniority quantum number for identical nucleons in single-j shell, may be related to dynamical symmetry of the Symplectic group Sp(2 j + 1), according to the classification U (2 j + 1) ⊃ Sp(2 j + 1) ⊃ SU (2). Accordingly, all the states of the j n configuration constitute an irreducible representation [1n ] of U (2 j + 1), where n is the number of particles in the j-shell. This reduces to the irreducible representation [1v ] of symplectic group Sp(2 j + 1) with allowed values of seniorities v = n, n − 2, ..., 1, or, 0. This implies that the state with an odd-particle in the valence orbital having pure-j, always has the lowest seniority as v = 1. On the other hand, the ground states of even-even nuclei always have the lowest seniority v = 0, i.e. all the particles are paired to J = 0, where J is the total angular momentum quantum number of the nucleus [4–11]. As an example, for n = 2 particles in j = 9/2 orbit, the allowed levels are J = 0, 2, 4, 6 and 8. While J = 0 state has a seniority v = 0, the other states having J = 2 to 8 have the seniority v = 2, because at least one pair must be broken to generate the first excited state having non-zero angular momentum, as shown in Fig. 3.1. In general, the lower seniority states lie lower in energy. It is easy to show that seniority v is exactly conserved for two particles in a pure- j configuration having j ≤ 7/2. This description becomes complicated for higher- j values. However, as we shall see, seniority appears to remain a reasonably useful quantum number for orbitals having j as high as j = 11/2, 13/2 etc.

3.1 Seniority and Seniority Isomers

37

Fig. 3.1 Schematic level scheme for (9/2)2 configuration in single-j shell. The levels are identified by seniority quantum number on extreme right. The transitions are either seniority changing (v = 2) or seniority conserving (v = 0)

Van Isacker [12] has discussed the validity of seniority for j > 7/2 and more specifically for j = 9/2 orbital. He goes on to show that for two states with different seniority having same J , and any reasonable nuclear interaction, the off-diagonal matrix element is small compared to the energy difference between the two states. For example, the two J = 2+ states having a seniority of v = 2 and v = 4 have a splitting of about 1 MeV, while the off-diagonal matrix element is only of the order of a few tens of keV. However, the proof that seniority mixing is negligible for the J = 4+ and 6+ states of a (9/2)4 system is more subtle and crucial for the existence of seniority isomers. There are three states for each of these angular momenta, two of which, with v = 2 and v = 4, are close in energy and could possibly mix easily (Fig. 3.2). The v = 4 members of these closely-spaced doublets are the so-called ‘solvable’ J = 4+ and 6+ states discussed in refs. [11, 13, 14] which have exact seniority v = 4 for any interaction. As a consequence, seniority mixing only arises through the mixing between the v = 2 and the higher-lying v = 4 members of the triplet, and this mixing was found to be small. Thus, seniority mixing begins to occur for higher-j values, yet, one or more of the multiple states still remain rather pure in seniority. This result is very useful when trying to understand the isomers arising from orbitals having j values higher than 7/2. Such higher-j orbitals are generally the intruder orbitals near the shell closures, and carry an added advantage because of their unique-parity nature.

3.1.1 Single-j Quasi-spin Scheme Kerman [15, 16] and Helmers [17] proposed the quasi-spin algebra for single-j orbital, simplifying the description of seniority. The quasi-spin scheme introduces a

38

3 Seniority Isomers

Fig. 3.2 Schematic representation of seniority mixing in J = 4+ and 6+ states for (9/2)4 system

pair creation operator S +j for a single-j shell, which may be defined as  S +j

=

2j + 1 + A ( j j; J = 0, M = 0) 2

More explicitly, 1 (−1) j−m a +jm a +j,−m 2 m  = (−1) j−m a +jm a +j,−m

S +j =

(3.1)

m>0

This operator creates a paired-state with J = 0 while acting on the vacuum state. Similarly, one may define the pair annihilation operator, which annihilates a pair and is a Hermitian conjugate of the pair creation operator. Therefore, the pair annihilation operator for a single-j shell may be defined as 

S −j

2j + 1 A( j j; J = 0, M = 0) 2 1 = (−1) j−m a j,−m a jm 2 m  = (−1) j−m a j,−m a jm =

m>0

(3.2)

3.1 Seniority and Seniority Isomers

39

A state with maximum seniority vmax in the single-j configuration j v may, therefore, be defined by (3.3) S −j | j v , vmax , J, M = 0 as there are no more pairs to annihilate. If we add pair of particles coupled to J = 0 to this state, n−v (3.4) (S +j ) 2 | j v , v, J, M are we get a state with the same seniority v in the j n configuration, where n−v 2 the number of pairs. This means that the states of j n configuration with n even, or odd have the seniorities v as even, or odd, respectively. The commutation relation between the pair creation operator and its Hermitian conjugate results in, [S +j , S −j ] =

 m

1 (a +jm a jm ) − (2 j + 1) = nˆ j −  = 2S 0j 2

(3.5)

where S 0j =

1 + 2j + 1 a a jm − 2 m jm 4

(3.6)

The operator S 0j is, thus, defined in terms of a number operator n j and a constant term. This further satisfies the relations, [S 0j , S +j ] = S +j , [S 0j , S −j ] = −S −j

(3.7)

Therefore, the operators S +j , S −j and S 0j follow the SU (2) Lie algebra, similar to the angular momentum (spin) algebra due to which this scheme was given the name as quasi-spin scheme. More algebraic details may be found in [4, 5, 7, 9, 10]. In order to relate the scheme with pairing interaction, Racah [1] introduced a special Hamiltonian, 2S +j S −j , which represents a constant pairing. Good seniority states may be obtained as the eigen states of this pairing Hamiltonian, H pair = −2G S +j S −j = −(2 j + 1)G A+ ( j j; J = 0, M = 0)A( j j; J = 0, M = 0)

(3.8)

where G is pairing strength. The eigen values of this pairing Hamiltonian may simply be obtained as: H pair = −2G S +j S −j = −2G(S 2j − S 0j (S 0j − 1)) Therefore,

(3.9)

40

3 Seniority Isomers





1 H pair (n, v) = −G 2s(s + 1) − ( − n)( + 2 − n) 2 n−v = −G (2 + 2 − n − v) 2

(3.10)

where quasi-spin s = 21 ( − v) and S 0j = 21 (n − ), the number operator n = n j , and pair degeneracy (total number of possible pairs)  = 21 (2 j + 1). The limiting value of  corresponds to n2 for a filled-up j n configuration. The eigen values in Eq. (3.10) directly correspond to the eigen states having seniority v in j n configuranumber of pairs. The eigen value will simply become equal to −nG, tion with n−v 2 i.e. −(2 j + 1)G for a v = 0 state from Eq. (3.10), which represents the fully paired J = 0 state. In other words, v can not exceed the value , and v =  represents the middle of the j-shell. The semi-magic nuclei are long known to have states with maximum pairing, and thus, have the lowest seniority. For many particles in the single-j seniority scheme, the states with different seniorities in the j n configuration become orthogonal. If we operate S +j on a state with v and J in j v configuration, we will obtain a state with same seniority v as well as J in the j v+2 configuration. Similarly, if we annihilate a pair in the j v+2 configuration having same value of J by using S −j and it has zero eigenvalue of pairing Hamiltonian, then the seniority of that state will be v + 2. There can be two or more independent states with same seniority v and same value of J in a given j v configuration. An orthogonal basis for such states must be chosen along with some additional quantum number α to characterize the basis states. Interestingly, any two such orthogonal states of times. j v configuration remain orthogonal even after S +j operation applied for n−v 2 Thus, the understanding obtained in j v configuration may serve well for the states in any j n configuration which is the beauty of the seniority scheme. We now proceed to discuss the decay properties such as reduced transition probabilities, g-factors, Q-moment etc. for isomers and other excited states by using the seniority scheme.

3.1.2 Decay Properties On the basis of the commutation relations of quasi-spin operators with creation operator a +jm and annihilation operator (−1) j−m a j,−m = a˜ jm , it follows that these operators become the κ = 21 and κ = −1 components, respectively, of an irreducible 2 quasi-spin tensor of rank s = 21 . The commutation relations of quasi-spin operators with a quasi-spin tensor operator Tκ(s) tell that the higher rank of these tensor operators may be obtained by linear combinations of products of creation and annihilation operators. In order to obtain a κ component operator, the products may contain x annihilation and x + 2κ creation operators resulting in the fact that only κ = 0 component will survive between the states having same number of particles. The operators for electromagnetic moments and transition probabilities are composed of single-particle operators, which are Hermitian in nature. According to

3.1 Seniority and Seniority Isomers

41

Wigner-Eckart theorem, the matrix elements of odd-tensor single-particle operators vanish in j n configuration. Since odd-tensor operators can not have s = 1 rank with κ = 0 component, they can only behave as quasi-spin scalars with rank s = 0. As a consequence, the matrix elements of odd-rank tensor operator vanish between states with different values of s, i.e. different seniority (v) values. This may be expressed as follows:  j n vl J ||T (k=odd) || j n v l  J  

s=0  s, S 0j |Tκ=0 |s , S 0j    s 0 s s−S 0j s=0 = (−1) s||T ||s −S 0j 0 S 0j

=

(3.11)

where s = s  due to the quasi-spin scalar nature. This eventually requires v = v  , since s = 21 ( − v), s  = 21 ( − v  ) and S 0j = 21 (n − ). The involved 3j-symbol would be equal to one. So, a similar relation remains valid in j v configuration by simply substituting n = v, i.e. S 0j = −s. This further results in the well known particle number (n) independent variation of the non-vanishing matrix elements for single-particle odd tensor operators. This result may simply be written as:  j n vl J M|Tκ(k=odd) | j n v l  J  M   =  j v vl J M|Tκ(k=odd) | j v v l  J  M   δv,v

(3.12)

It means that such operators do not allow a change in the value of v and the seniority quantum number v remains conserved. This may be related to the behavior of magnetic transitions and magnetic dipole moments. As a consequence, the magnetic transition probabilities and the magnetic moment (or the g-factors) should exhibit a particle number independent behavior for single-j shell. On the other hand, the even-tensor operators behave as the κ = 0 component of the quasi-spin vectors having rank (s = 1). According to the Wigner-Eckart theorem, one may now have non-vanishing matrix elements between states with s and s  differing at most by 1. This further implies that the seniorities of these states, v and v  , may either be equal or differ at most by 2, otherwise the matrix elements will vanish. Moreover, the Wigner-Eckart theorem in quasi-spin space for j n configuration dictates:  j n vl J ||T (k=even) || j n v l  J  

s=1  s, S 0j |Tκ=0 |s , S 0j    s 1 s s−S 0j s=1  = (−1) (s||T ||s ) −S 0j 0 S 0j

=

(3.13)

The dependence of such matrix elements on n is due to the 3j-symbol in the quasispin scheme and is different for s  = s and s  = s ± 1 as shown below: (i) When s  = s, i.e. v  = v s=1 |s, S 0j  = s, S 0j |Tκ=0

=

2S 0j √ (s||T s=1 ||s) 2s(2s+1)(2s+2) (n−) √ (s||T s=1 ||s) 2s(2s+1)(2s+2)

(3.14)

42

3 Seniority Isomers

Similar relation may be written for j v configuration, simply by replacing n with v in Eq. (3.14). Therefore, the matrix elements of involved spherical harmonics in electric transitions having j n configuration may be related to j v configuration as follows: j

n

vl J M|YκL | j n vl  J  M  

 −n  j v vl J M|YκL | j v vl  J  M   = −v 

(3.15)

where l and l  represent the initial and final states for a given electromagnetic transition. L represents the allowed value of multi-polarity, that is the angular-momentum transfer. The above result is valid for even values of L > 0. It also implies that the matrix elements of even tensor operators having L > 0, between same seniority states are equal in magnitude but have opposite signs for n particle configurations = , ( j n ) and n hole configurations ( j 2 j+1−n ). At the middle of the shell, n = 2 j+1 2 the matrix element becomes zero and changes its sign subsequently. For the specific case of L = 0, the tensor Y0(0) will have only diagonal elements in any scheme, which are all equal and proportional to n. The simplest  way to calculate it, is to recall that Y0(0) = m,m   jm jm  | j j00a +jm a˜ jm  = √21j+1 m a +jm a jm , which is simply proportional to the number operator. (ii) When s  = s + 1, i.e. v  = v − 2, 

s,

s=1 |s S 0j |Tκ=0

+ 1,

S 0j 

s−S 0j

= (−1) 

(s||T

s=1

s 1 s+1 ||s + 1) −S 0j 0 S 0j



4(s+S 0 +1)(s−S 0 +1)

j j = − 2(2s+1)(2s+2)(2s+3) (s||T s=1 ||s + 1) = − (n−v+2)(2+2−n−v) (s||T s=1 ||s + 1) 2(2s+1)(2s+2)(2s+3)

(3.16)

Therefore, j

n

vl J M|YκL | j n v









− 2, l J M  =

(n − v + 2)(2 + 2 − n − v) 2(2 + 2 − 2v)

 (3.17)

 j v vl J M|YκL | j v v − 2, l  J  M   (L > 0, even) (iii) Similarly, when s  = s − 1, i.e. v  = v + 2, j

n

vl J M|YκL | j n v







+ 2, l J M  =



(n − v + 2)(2 + 2 − n − v) 2(2 + 2 − 2v)



 j v vl J M|YκL | j v v + 2, l  J  M   (L > 0, even)

(3.18)

3.1 Seniority and Seniority Isomers

43

The matrix elements in Eqs. (3.18) and (3.19) are off-diagonal but symmetric between n and 2 − n, resulting in equal values for j n (n particles) and j 2 j+1−n (n holes) configurations. The even rank tensors, therefore, have v = 0, or 2 in associated electric transitions for single-j shell. The reduced electric transition probabilities B(E L) between Ji and J f states in a single-j configuration along with the corresponding total pair degeneracy  = 21 (2 j + 1) may be written as, B(E L) =

 1 riL Y L (θi , φi )|| j n v l  Ji |2 | j n vl J f || 2Ji + 1 i

(3.19)

where l, l  and L can only take even values. The seniority reduction formulas for the seniority conserving v = 0 and seniority changing v = 2 transitions may be obtained as:    −n  j n vl J f || riL Y L (θi , φi )|| j n vl  Ji  = −v i  riL Y L (θi , φi )|| j v vl  Ji  × j v vl J f || i

(L > 0, even)

 j n vl J f ||

 i

riL Y L (θi , φi )|| j n v ± 2, l  Ji  =

(3.20)



 (n − v + 2)(2 + 2 − n − v) 4( + 1 − v)  v × j vl J f || riL Y L (θi , φi )|| j v v ± 2, l  Ji  i

(L > 0, even)

(3.21)

These relations imply a parabolic behavior for B(E L). To sum up, the odd-tensor operators preserve the seniority for the magnetic transitions, and lead to the particle number independent behavior for the energy difference between same seniority states. This may further be related to the constant value of magnetic moments i.e. g-factors for a given J, v state in j n configuration. On the other hand, the even-tensor operators may preserve the seniority, or may change the seniority by two for the electric transitions, v = 0, ±2. As a result, the even tensor transitions exhibit a parabolic B(E L) behavior for both seniority preserving v = 0 and seniority changing v = 2 transitions. The B(E L) parabola has a minimum at the middle of the shell for v = 0 transitions while has a maximum at the middle of the shell for v = 2 transitions. The dip at the middle of the shell for B(E L) (L > 0, even) values leads to a half-life larger than that in its neighborhood, and gives rise to the seniority isomerism. It is, therefore, expected and observed that the seniority isomers occur for electric quadrupole transitions, particularly in semi-magic nuclei, where seniority remains a good quantum number. These results

44

3 Seniority Isomers

Fig. 3.3 Schematic variation of one-body matrix elements for electric and magnetic transitions in both single-j shell and multi-j shell, respectively, by using the electromagnetic and seniority selection rules [18]

show the elegance of the seniority scheme, where the states with varying number of particles n in a shell reduce to single v seniority state. For example, Eqs. (3.20) and (3.21) relate the reduced matrix elements of electric multipole operators for the states of j n configuration to the states of j v configuration. If one obtains the solution of matrix elements in j v configuration, the full trend with changing nucleon number n can be obtained. The schematic plots for the one-body matrix elements in electric and magnetic transitions for the case of the single-j shell are shown in the top two panels of Fig. 3.3 [18]. Note that the electric tensors may have v = 0, or v = 2 non-zero matrix elements, while the magnetic tensors may have only v = 0 non-zero matrix elements. As the parity change is not possible in the single-j situation, the evenmagnetic and odd-electric transitions are forbidden in single-j shell. We also present the schematic plots for the results of the reduced transition probabilities for single-j shell, in Fig. 3.4. The odd-tensor operators conserve the seniority for the magnetic transitions, and lead to the particle number independent behavior as shown in top right panel of Fig. 3.4. On the other hand, the even-tensor operators preserve or, change the seniority by 2 for the electric transitions. As pointed out, the even-tensor transitions, therefore, show a parabolic behavior for both seniority preserving v = 0 and seniority changing v = 2 transitions. The parabola has a minimum at the middle in the case of seniority preserving v = 0 transitions, while

3.1 Seniority and Seniority Isomers

45

Fig. 3.4 Schematic variation of reduced transition probabilities for electric and magnetic transitions in both single-j shell and multi-j shell respectively, by using the electromagnetic and seniority selection rules[18]

it has a maximum at the middle in the case of seniority changing v = 2 transitions, as shown in top left panel of Fig. 3.4.

3.1.3 Magnetic Moments and g-Factors On one hand, the reduced electric even tensor transition probabilities show a parabolic trend with a dip or peak in the middle of the shell. On the other hand, the magnetic L ˆ multipole transition operator O(Mag.) M behaves as odd tensor in the single-j scheme for the electromagnetic transitions, thus conserving parity. Therefore, the reduced matrix elements for such transitions can be written as: L n v L v ˆ ˆ  j n v J f || O(Mag.) M || j v Ji  =  j v J f || O(Mag.) M || j v Ji 

(3.22)

This can further result in the particle number independent behavior of magnetic moments for identical nucleons [4], since the matrix elements of magnetic dipole moments in j n configuration can be reduced to the matrix elements of magnetic dipole moments in j v configuration without ‘n’ dependence as follows: ˆ j n  =  j v |μ| ˆ jv  j n |μ|

(3.23)

46

3 Seniority Isomers

Therefore, the magnetic moments for a given seniority v state is sufficient to know the magnetic moments of the other isotopes/isotones having the same seniority state in j n configuration. In this way, the magnetic moment (μ) of n identical nucleons in a single-j orbital giving rise to total angular momentum J , can be written as μ =

n  i

g ji = g

n 

ji = g J

(3.24)

i

The g-factor is simply the ratio of the magnetic moment μ and J . It is nearly same for all the seniority states arising from j n configuration. Also, the variation of g-factor is going to be particle number independent as per the seniority scheme. Hence, g-factor of any seniority state such as v = 2 in even-A or v = 3 in odd-A, will be approximately equal to the g-factor of a single seniority (v = 1) state for any given pure j n configuration.

3.2 Examples of Seniority Isomers and Their Moments The seniority isomers arise due to the vanishing transition probability in seniority conserving transitions at the middle of a single-j shell. These isomers have been observed in and around semi-magic nuclei, where seniority is a reasonably pure quantum number. We present some detailed experimental evidences and features, which strongly support the validity of the single-j seniority scheme. We also show the validity of the seniority scheme for higher-j values and how does it merge into multi-j seniority, which is more commonly termed as the generalized seniority. Nearly all the examples come from the semi-magic nuclei, which are easy to deal with due to only one kind of particles in the valence space.

3.2.1 Seniority Isomers in N = 50 Isotones The 8+ isomers in N = 50 isotones 92 Mo, 94 Ru, 96 Pd and 98 Cd are well known to be seniority v = 2 isomers arising from g9/2 orbital. Figure 3.5 shows the comparison of experimental and seniority calculated reduced electric transition probabilities B(E2) for these 8+ isomers in N = 50 isotones. Experimental data have been taken from ENSDF data base [19]. The calculations were done assuming a seniority v = 2 for the g9/2 orbital with a pair degeneracy  = 5. The seniority results are able to reproduce the measured B(E2) trend quite well. These 8+ isomers are hence single-j seniority v = 2 isomers. Figure 3.6 shows the g-factor i.e. magnetic moment variation of these 8+ isomers with particle number (valence protons above 90 Zr core). The constant value of experimental g-factor is in line with the seniority expectations and also lie very close to the Schmidt line of proton g9/2 orbital. Experimental data have been

3.2 Examples of Seniority Isomers and Their Moments 92

1.6

Mo

94

Ru

96

Pd

98

Cd

8+ isomers N=50 isotones

1.2

B(E2) W.U.

Fig. 3.5 A comparison of the experimental [19] and seniority trends of the reduced electric transition probabilities B(E2) with respect to particle number (n) for the seniority v = 2, 8+ isomers in N = 50 isotones arising from g9/2 orbital with a pair degeneracy  = 5

47

0.8

0.4 Exp. Seniority

0.0 2

4

6

8

Particle number (n)

taken from Stone’s table [20], and a weighted averaged value has been adopted in case of multiple measured values. Figure 3.7 presents the experimental and seniority calculated B(E2) trends for the 21/2+ isomers in odd-A N = 50 isotones. The calculated results have used seniority v = 3 configuration in proton g9/2 orbital. The calculations reproduce the observed trend very well. However, the measured value deviates from the pure seniority result for 97 Ag, which is highly neutron-deficient. No g-factor measurements are available for these v = 3, 21/2+ isomers in odd-A N = 50 isotones. These simple calculations can make dependable predictions for future measurements. Thus, we can predict that the g-factors will be of the same order as for the 8+ isomers shown in Fig. 3.6.

3.2.2 Seniority Mixing in 72,74 Ni Isotopes One would expect, similarly, that 8+ isomers should exist in the neutron-rich Ni (Z = 28) isotopes, where neutrons occupy the g9/2 shell. However, a reverse situation seems to prevail in the case of 72 Ni and 74 Ni [21]. A simple explanation of this observation was given by Isacker using seniority arguments [12]. The spectra of 70 Ni display a regular seniority like band for two neutron-particles g29/2 having excited states J = 2+ , 4+ , 6+ and 8+ . 76 Ni also has two-neutron-holes in g−2 9/2 based yrast sequence of excited states with J = 2+ , 4+ , 6+ and 8+ . However, there are no 8+ isomers in 72,74 Ni due to the occurrence of two levels for J = 4+ and J = 6+ lying quite close in energy. The 8+ states in both these isotopes are able to decay to the

48

3 Seniority Isomers

Fig. 3.6 A comparison of the experimental g-factor trend [20] and Schmidt values of proton g9/2 orbital versus particle number (n) for the seniority v = 2, 8+ isomers in N = 50 isotones

92

2.0

94

Mo

96

Ru

98

Pd

Cd

g-factor (n.m.)

1.6

1.2

8+ isomers N=50 isotones 0.8

Exp. Schmidt line

0.4

0.0 2

4

6

8

Particle number (n) 93

6

5

95

Tc

97

Rh

Ag

21/2+ isomers N=50 isotones

4

B(E2) W.U.

Fig. 3.7 A comparison of the experimental [19] and seniority trends of the reduced electric transition probabilities B(E2) with respect to particle number (n) for the seniority v = 3, 21/2+ isomers in odd-A N = 50 isotones arising from g9/2 orbital with a pair degeneracy  = 5

3

2

Exp. Seniority

1

0 2

4

6

Particle number (n)

8

3.2 Examples of Seniority Isomers and Their Moments Fig. 3.8 A comparison of the experimental [19] and seniority calculated B(E2) trends from neutron g9/2 orbital for the seniority v = 2, 8+ isomers in Pb isotopes beyond doubly-magic 208 Pb core

210

1.0

Pb

49 212

Pb

214

Pb

216

Pb

8+ isomers Z=82 isotopes core: 208Pb

0.8

B(E2) W.U.

0.6

0.4

0.2

Exp. Seniority

0.0 2

4

6

8

Particle number (n)

second 6+ state which has v = 4. This enhances the transition probability between the 8+ and 6+ states leading to a shorter half-life of the 8+ state, i.e. no isomer.

3.2.3 Seniority Isomers in 128 Pd and 126 Pd The level structures of the very neutron-rich nuclei 128 Pd and 126 Pd have been investigated for the first time by Watanabe et al. [22]. In the r-process waiting-point nucleus 128 Pd, a new isomer with a half-life of 5.8(8) µs is proposed to have a spin and parity of 8+ and is associated with a maximally aligned configuration arising from the g9/2 proton subshell with seniority v = 2. For 126 Pd, two new isomers have been identified with half-lives of 0.33(4) and 0.44(3) µs. The yrast 2+ energy is much higher in 128 Pd than in 126 Pd, while the level sequence below the 8+ isomer in 128 Pd is similar to that in the N = 82 isotone 130 Cd. The electric quadrupole transition that depopulates the 8+ isomer in 128 Pd is more hindered than the corresponding transition in 130 Cd, as expected in the seniority scheme for a semi-magic, spherical nucleus. These experimental findings indicate that the shell closure at the neutron number N = 82 is fairly robust in the neutron-rich Pd isotopes.

50

3 Seniority Isomers

3.2.4 Seniority Isomers in Pb Isotopes Figure 3.8 presents the experimental [19] and seniority calculated B(E2) trends for the 8+ isomers in even-A 210−216 Pb isotopes. These isomers can be understood in terms of v = 2 configuration in neutron g9/2 orbital beyond 208 Pb core. Seniority results very well explain the experimental B(E2) trend. The inclusion of three-body forces has been used by Gottardo et al. [23] to explain the deviation between the seniority calculated and measured trends. Only one measured g-factor value for 210 Pb is available, which lies quite far from the Schmidt value of neutron g9/2 orbital highlighting the role of extra mechanism needed to explain these 8+ seniority isomers in Pb isotopes.

References 1. G. Racah, Phys. Rev. 63, 367 (1943) 2. G. Racah, Nuclear levels and Casimir operators, Research Council of Israel, Jerusalem. L. Farkas Memorial Volume, 294 (1952) 3. B.H. Flowers, Proc. Roy. Soc. (London) A 212, 248 (1952) 4. I. Talmi, Simple Models of Complex Nuclei, Harwood Academic, 1993 and original references therein 5. D.J. Rowe, J.L. Wood, Fundamentals of Nuclear Models- Foundational Models (World Scientific Publishing, Singapore, 2010) 6. V.K.B. Kota, Y.D. Devi, Nuclear Shell Model and The Interacting Boson Model: Lecture Notes for Practitioners (Inter University Consortium for DAE Facilities, Calcutta Centre, 1996) 7. R.F. Casten, Nuclear Structure from a Simple Perspective (Oxford University Press, Oxford, 1990) 8. G. Racah, Proceedings of the Rehovoth Conference on Nuclear Structure, p. 155 (1958) 9. R.D. Lawson, Theory of the Nuclear Shell Model (Oxford University Press, New York, 1980) 10. K.L.G. Heyde, Basic Ideas and Concepts in Nuclear Physics (CRC Press, 2004); From Nucleons to the Atomic Nucleus, Create Space, 1998 11. P. Van Isacker, S. Heinze, Phys. Rev. Lett. 100, 052501 (2008) 12. P. Van Isacker, J. Phys. Conf. Ser. 322, 012003 (2011) 13. A. Esuderos, L. Zamick, Phys. Rev. C 73, 044302 (2006) 14. C. Qi, Phys. Rev. C 83, 014307 (2011) 15. A.K. Kerman, Ann. Phys. (NY) 12, 300 (1961) 16. A.K. Kerman, R.D. Lawson, M.H. Macfarlane, Phys. Rev. 124, 162 (1961) 17. K. Helmers, Nucl. Phys. 23, 594 (1961) 18. B. Maheshwari, Ph.D. Thesis, IIT Roorkee (2016) 19. Evaluated Nuclear Structure Data File http://www.nndc.bnl.gov/ensdf/ 20. N.J. Stone, www-nds.iaea.org/publications, INDC(NDS)-0658, Feb. (2014) 21. M. Sawicka et al., Phys. Rev. C 68, 044304 (2003) 22. H. Watanbe et al., Phys. Rev. Lett. 111, 152501 (2013) 23. A. Gottardo et al., Phys. Rev. Lett. 109, 162502 (2012)

Chapter 4

Generalized Seniority Isomers

Keywords Generalized seniority · Multi-j quasi-spin scheme · B(E2) trends · Q-moments · Generalized seniority Schmidt model · g-factors · Examples The real life situation in nuclei may involve many active orbitals in their valence space. These orbitals may have many j-values. It is, therefore, imperative that the seniority concept as introduced in the last chapter should be generalized to multi-j environment. We, therefore, present in this chapter a generalization of the seniority scheme, and its consequences on the seniority selection rules and occurrence of seniority isomers. While most of our focus will be on the semi-magic nuclei, we will also touch upon examples which are not semi-magic, and test the validity of generalized seniority in such cases. It is rather remarkable that imprints of the seniority scheme leaves may still be seen in situations which are not semi-magic.

4.1 Multi-j Quasi Spin Scheme An extension of the single-j seniority to the multi-j seniority has been known for a long time and may be credited to Kerman [1, 2] and Helmers [3]. However, Arima and Ichimura [4] were the first to introduce the concept of generalized seniority for multi-j shell, mainly for degenerate orbitals. Talmi further extended it to the case of several non-degenerate orbitals [5, 6]. Here, we present an extension of the single-j seniority scheme to multi-j generalized seniority scheme by simply extending the quasi-spin algebra of single-j orbital to multi-j degenerate orbitals [7]. This may be achieved by defining the multi-j pair creation operator in analogy to the single-j pair creation operator as [8]:  S +j , (4.1) S+ = j

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. K. Jain et al., Nuclear Isomers, https://doi.org/10.1007/978-3-030-78675-5_4

51

52

4 Generalized Seniority Isomers

where the summation takes care of all the orbitals in the valence space. In a way this operator sums up the effect of all the active valence orbitals. The commutation relation between the multi-j pair creation operator and its Hermitian conjugate may be written as [S + , S − ] =



(a +jm a jm ) −

jm

 1 (2 j + 1) = nˆ j −  = 2S 0 2 j j

(4.2)

The operator S 0 further satisfies the commutators, [S 0 , S + ] = S + , [S 0 , S − ] = −S −

(4.3)

As is evident from these relations, these operators, S + , S − and S 0 , are also the generators of SU (2) Lie group. Total quasi-spin of the system is now given by the sum of the partial quasi-spins corresponding to each j-orbit, s = s j1 + s j2 + s j3 + s j4 + ....

(4.4)

It, therefore, follows that the system of eigen states of S 2 , or those of 2S + S − , defines a seniority scheme for identical nucleons in multi-j shell of several degenerate orbitals. This is called the generalized seniority scheme, and embodies all the features of the single-j seniority scheme, which follow from the SU (2) algebra. Energy eigen-values of the new pairing interaction −2G S + S − = −2G(S 2 − S 0 (S 0 − 1)), which looks similar to the single-j relation, are given by   n−v 1 (2 + 2 − n − v) − G(2s(s + 1) − ( − n)( + 2 − n)) = −G 2 2 (4.5) since total quasi-spin as given in Eq. (4.4) can be obtained as s = 21 ( − v), and S 0 = 21 (n − ). The seniority in single-j shell is now modified to  the generalized seniority v in multi-j shell. Also, the total number of particles n = j n j , and pair  degeneracy,  = 21 j (2 j + 1) for the multi-j shell. It may be noted that the good generalized seniority states are now akin to the quasi-particle states, supporting a shared occupancy and corresponding mixed configuration. For example, the generalized seniority v = 0 state can be obtained by (S + )n |0 >, where n = j n j and  n S + = j S +j . Hence, each state with (S +j ) j |0 > is now mixed with various single-j n configurations to generate a generalized seniority v = 0, (S + ) |0 > state. We now recollect the properties of spherical tensor operators, which are needed for transition probabilities and moments. A single-nucleon tensor operator in multi-j space with rank k and component κ is defined as, Tκ(k) = √

 1 (k) ( j||T (k) || j  )(a +j × a˜ j  )κ 2k + 1) j, j 

(4.6)

4.1 Multi-j Quasi Spin Scheme

53

The commutation relation between multi-j pair creation operator S + and tensor operator Tκ(k) may be obtained as: [S + , Tκ(k) ] =

 [S +j  , Tκ(k) ] j

=

 1 ( j||T (k) || j  )( jm j  m  | j j  kκ) (2k + 1) j j  ,mm  



×a +jm (−1) j +m [S +j  , a j  ,−m  ]  1 (k) ( j||T (k) || j  )(a +j , ×a + = j )κ (2k + 1) j j 

(4.7)

The tensor Tκ(k) behaves as a quasi-spin scalar, if [S + , Tκ(k) ] = 0

(4.8)

which is true for j = j  . For j = j  , the right hand side of Eq. (4.7) may be expressed as = = =

1   (k) [( j||T (k) || j  )(a +j × a +j  )(k) || j)(a +j  × a +j )(k) κ + ( j ||T κ ] 2k + 1 j< j 

1   [( j||T (k) || j  ) + (−1)( j+ j −k+1) ( j  ||T (k) || j)] × (a +j × a +j  )(k) κ 2k + 1 j< j  1  [1 + (−1)k ]( j||T (k) || j  )(a +j × a +j  )(k) κ 2k + 1 j< j 

(4.9)

where the last equality may be obtained by using the property (J  ||T (k) ||J )∗ =  (−1) J −J (J ||T (k) ||J  ) for Hermitian operators. This means that the commutation + (k) [S , Tκ ] = 0 for odd k values and the odd Hermitian tensors behave as quasi-spin scalars. The result is actually a simple generalization of the result of the seniority scheme in single-j shell. To sum up, the Hermitian tensor T (k) behaves as a quasi-spin scalar for odd-k values while it behaves as the κ = 0 component of the quasi-spin vector for evenk values. These results are useful for the operators of electromagnetic transitions, usually expressed in terms of spherical harmonics Y L , which are Hermitian tensors. Furthermore, the multi-j shell may, or may not have orbitals of different parities. So, the multi-j pair creation operator may be redefined as [9] S1+ =

 (−1)l j S +j j

(4.10)

54

4 Generalized Seniority Isomers

where the orbital angular momentum l j takes care of the parity of each orbital. The  phase factor (−1)k in Eq. (4.9), now changes to (−1)l+l +k .

4.1.1 Decay Properties 

For the electromagnetic transitions, the phase factor can be rewritten as (−1)l+l +L due to the involvement of spherical harmonics Y L . L is the multipolarity and l, l  define the parities of the initial and final states involved in the electromagnetic transition. So, • Y L tensor behaves as a quasi-spin scalar, if the sum l + l  + L is odd. • Y L behaves as the κ = 0 component of the quasi-spin vector, if the sum l + l  + L is even. Interestingly, the sum l + l  + L always remains even for electric transitions irrespective of the nature of L value due to the selection rules for electromagnetic transitions. Similarly, the sum l + l  + L always remains odd for the magnetic transitions irrespective of the nature of L value. Hence, we conclude that the magnetic transitions, for both even and odd L, become quasi-spin scalar [10]. This implies that the matrix elements of magnetic transitions are particle number independent as we fill up the shell and conserve the seniority. If we define a multi-j configuration as j˜ = j ⊗ j  ⊗ ... then the matrix elements for magnetic transitions can be expressed as:  j˜n vl J M|O(Mag.) LM | j˜n v l  J  M   =  j˜v vl J M|O(Mag.) LM | j˜v v l  J  M  δv,v (4.11)  2 j+1 ˜ where the pair degeneracy  = 2 j+1 = . The result looks similar to the j 2 2 single-j shell for magnetic dipole moments. The only difference arises in terms of an additional possibility, which says that the magnetic quadrupole (like magnetic dipole) transitions/moments may also become particle number independent. Similarly, we find that the electric transitions, for both even and odd L, behave as κ = 0 component of a quasi-spin vector [10]. Therefore, the matrix elements of electric transitions between same seniority states (seniority preserving v = 0 transitions) in multi-j j˜n configuration may be written as,    − n ˜v n L ˜n    ˜  j vl J M|Yκ | j vl J M  =  j vl J M|YκL | j˜v vl  J  M   −v

(4.12)

On the other hand, the matrix elements of electric transitions between states differing in seniority by 2, (seniority changing v = 2 transitions) in multi-j j˜n configuration may be written as

4.1 Multi-j Quasi Spin Scheme

˜n

j

vl J M|YκL | j˜n v

55 







± 2, l J M  =

(n − v + 2)(2 + 2 − n − v) 2(2 + 2 − 2v)

 (4.13)

× j˜v vl J M|YκL | j˜v v ± 2, l  J  M   The reduced transition probabilities B(E L) between Ji and J f states, for a multi-j configuration j˜ = j ⊗ j  .... defined through the total pair degeneracy  = 21 (2 j˜ +  1) = 21 (2 j + 1) is given by, j

B(E L) =

 1 riL Y L (θi , φi )|| j˜n v l  Ji |2 | j˜n vl J f || 2Ji + 1 i

(4.14)

This implies that the B(E L) values exhibit a parabolic behavior in the multi-j case, irrespective of the L values (even or odd). A similar behavior was shown to be valid in single-j case, but only for even L values. Figures 3.3 and 3.4 compare the schematic variation of electromagnetic one-body matrix elements and the reduced electric/magnetic transition probabilities in both the single-j and multi-j shells [10]. The schematic results are shown for both the seniority preserving v = 0 and seniority changing v = 2 transitions. The generalized seniority reduction formulas for the reduced matrix elements with v = 0 and v = 2 transitions may also be written as follows  j˜n vl J f ||



riL Y L (θi , φi )|| j˜n vl  Ji  =

i



−n −v



 j˜v vl J f ||



riL Y L (θi , φi )|| j˜v vl  Ji 

(4.15)

i

 j˜n vl J f ||



riL Y L (θi , φi )|| j˜n v ± 2, l  Ji  =

i



 (n − v + 2)(2 + 2 − n − v) 4( + 1 − v)   j˜v vl J f || riL Y L (θi , φi )|| j˜v v ± 2, l  Ji  i

(4.16) For completeness, the involved reduced matrix elements of j˜v configuration in Eq. (4.15) can further be written in terms of fractional parentage coefficients, 3 j−, 6 j-coefficients and radial matrix elements as  j˜v vl J ||

 i

ri2 Y 2 (θi , φi )|| j˜v vl J  = v



˜ [ j˜v−1 (v1 J1 ) j˜ J |} j˜v v J ]2 (−1) J1 + j+J +2

v1 ,J1

(2J + 1)

j˜ J J1 ˜ 2 2 ˜ ( j||r Y || j) J j˜ 2

(4.17)

56

4 Generalized Seniority Isomers

where the [ j˜v−1 (v1 J1 ) j˜ J |} j˜v v J ] denotes the fractional parentage coefficients from j˜v−1 to j˜v configuration. The second last, and the last terms represent the 6 j− coefficient and the single-particle value of the given electric quadrupole operator for ˜ The latter can be related to the involved radial matrix elements arising in a given j. j˜ configuration as:  ˜ = (−1) ˜ Y || j) ( j||r 2

2

˜ 1 j− 2

(2 j˜ + 1)

  5 j˜ 2 j˜ r 2  4π − 21 0 21

(4.18)

where the second last term denotes the 3 j-symbol corresponding to spherical harmonics Y 2 in j˜ configuration and the last term r 2  represents the radial integral. With these simple yet profound changes, the picture of seniority isomerism becomes quite different as we move to the multi-j shell. We establish that the magnetic transitions now behave as quasi-spin scalars for both even and odd multipole transitions and lead to particle number independent behavior and preserve the seniority, as shown in Fig. 3.4. On the other hand, the electric transitions behave as κ = 0 component of the quasi-spin vector irrespective of even or odd L values. Therefore, the reduced electric transition probabilities between the same seniority states show a parabolic minimum at the middle of the active-shell for both even and odd multipole tensors, while a maximum at the middle is obtained for seniority changing transitions, see Fig. 3.4. This implies that one should get longer half-lives corresponding to both the even and odd tensor seniority preserving electric transitions at the middle of the multi-j shell. This simple observation leads to the discovery of a new kind of seniority isomers in Sn isotopes [11].

4.1.2 Group Theoretical Understanding We have already indicated that the quasi-spin scheme for identical nucleons in a single-j shell satisfies the SU q (2) group algebra defined in terms of the pair creation and annihilation operators, S + and S − . This interpretation has recently been expanded by Kota [12] for identical nucleons occupying r number of j-orbitals i.e. for generalized seniority. In such a situation, Kota has demonstrated the existence of 2r −1 number of pairing SU q (2) algebras. He also showed that for each set of values symplectic algebra of of the phase factor α j = (−1)l j , there exists a corresponding  Sp(2) arising from U (2) ⊃ Sp(2) with  = j (2 j + 1)/2. This one-to-one correspondence between Sp(N ) and SUq (2) leads to the same selection rules for the gamma decay between n-nucleon states having good generalized seniority, which we have summarized in the previous subsection. The choice of the phase factor α j = (−1)l j was made by Arvieu and Moszokowski [7] rather arbitrarily and out of convenience. However, this choice turns out to be most appropriate for the goodness of generalized seniority of low lying states and some specific high-j states. It also

4.1 Multi-j Quasi Spin Scheme

57

indicates that the simple pairing interaction H p = 2S + S − is an important and crucial part of the realistic effective interaction.

4.1.3 Excitation Energies The excitation energies of good generalized seniority states are expected to have a valence particle number independent behavior, similar to the good seniority states arising in single-j shell. It is rather easy to show this by extending the proof for the single-j seniority scheme by defining an effective j˜ value for a multi-j configuration. For a two-body odd-tensor interaction Vik , we can define the two-body matrix elements for 0+ (fully-coupled) state in multi-j situation as V0 =  j˜2 J = 0|Vik | j˜2 J = 0. The matrix element for a j˜n configuration may be written in terms of j˜v configuration as:  j˜n vαJ |

n 

Vik | j˜n vα J  =  j˜v vαJ |

i 64 odd-A Sn isotopes [13]. Figure 4.2 (upper panel) shows the excitation energy variation of 10+ , 27/2− and 11/2− states of Sn isotopes throughout the full isotopic chain. The 10+ and 27/2− states follow each other closely for N > 68, Sn isotopes, when the 11/2−

Fig. 4.2 Energy variation of yrast 10+ and 11/2− , 27/2− states in even-even and odd-A Sn isotopes, respectively

60

4 Generalized Seniority Isomers

Fig. 4.3 Half-life variation of 10+ and 11/2− , 27/2− states in even-even and odd-A Sn isotopes, respectively

states begin to approach the ground state, or become the ground state. On the other hand, the lower panel of Fig. 4.2 plots the energy difference of 27/2− states with the 11/2− states in odd-A Sn isotopes, and compares them with the 10+ state energies with respect to the 0+ ground states in even-A Sn isotopes. We note that the 10+ and 27/2− states now follow each other even more closely for N > 64, Sn isotopes as shown in the lower panel of Fig. 4.2. It suggests that the structural change in going from 11/2− to 27/2− states in odd-A nuclei is identical to that in going from 0+ to 10+ states in even-even nuclei. This identical nature is entirely due to the goodness of the generalized seniority and the underlying symmetry. We further present in Fig. 4.3, the half-life variation of the 10+ states in eveneven Sn isotopes and the 11/2− and 27/2− states in odd-A Sn isotopes. All the three support a peak around N = 71. This may be easily understood in terms of the generalized seniority interpretation. The g7/2 and d5/2 orbitals were considered to be frozen, as these are filled up, bringing the neutron core from N = 50 to N = 64. The remaining valence space consists of h 11/2 , d3/2 , and s1/2 , which can accommodate 18 neutrons. This valence space will be half-filled, if it has 9 more neutrons. Therefore, the transition probability parabola must touch the bottom at N = 71. The half-lives should, therefore, display a maximum near N = 71 as seen in the Fig. 4.3.

4.2 Generalized Seniority in the Sn Isotopes

61

4.2.1 The 10+ , 13− , and 15− Isomers The 119−130 Sn isotopes have been studied by using a wide variety of reactions like light ions reactions, deep inelastic reactions, or fission fragment studies by a number of authors, for example, we cite [15–24]. Many systematic features have been pointed out in the N > 64 Sn isotopes, and the 10+ and 27/2− isomeric states arising from h 11/2 neutrons, have been proposed as seniority v = 2 and v = 3 states. Pietri et al. [25] have recently identified seniority v = 4, 15− isomeric state in 128 Sn. Astier et al. [26, 27] have reported detailed level schemes in the 119−126 Sn isotopes by using the binary fission fragmentation induced by heavy ions. Iskra et al. [28] have also focused on high seniority states in neutron-rich, even-even Sn isotopes. In discussing these states, we must be careful to avoid the deformed collective states which are part of rotational bands in the even-even light mass Sn isotopes with A = 110 − 118, and are assigned a two particle - two hole proton configuration [29– 35]. We, therefore, discuss the 10+ yrast isomeric states, which are not part of any rotational structure [36]. More recently, the studies on Sn isotopes have been pushed beyond the N = 82 magic number and isomers in the N = 86 − 88 Sn isotopes have been observed by Simpson et al. [37] which throw new light on the effective interaction in neutron-rich nuclei [38]. We have also considered the application of the generalized seniority scheme to explain the recently measured trends of the B(E1) values in the 13− isomers, and the B(E2) values in the 10+ and 15− isomers in the Sn isotopes [27, 28] as shown in Figs. 4.4 and 4.5, respectively. While E1 is an odd-multipole parity changing transition, E2 is an even-multipole parity conserving transition. Although E2 decaying isomers are possible in single-j seniority scheme also, a full explanation of the whole trend could not be reached there, and E1 is simply not allowed in pure-j seniority scheme. The calculations were carried out by using the mixed configuration consisting of h 11/2 ⊗ d3/2 ⊗ s1/2 orbitals, having a pair degeneracy of  = 9. 114 Sn was assumed to be the core by freezing the lower-lying g7/2 and d5/2 orbitals. The calculated results, also shown in the Figs. 4.4 and 4.5, explain the experimental data for the E1 decaying 13− isomers and for the E2 decaying 10+ and 15− isomers in Sn isotopes. The inverted parabolic behavior leads to the occurrence of longer-lived isomers near the middle of the chosen valence space. This marks, for the first time, the discovery of a new kind of seniority isomers decaying via E1 transitions. The observed behaviour of E1 isomers in Fig. 4.4, exhibits a smoothly varying parabolic trend of the B(E1) values. This results in half-life of 13− states as: 120 Sn (4 ns), 122 Sn (40 ns), 124 Sn (6.7 ns), and 126 Sn (≤ 3 ns). Interestingly, the well known 10+ isomers were believed to be arising from the h 11/2 orbital [8]. For the first time, we have found that they also arise from the h 11/2 ⊗ d3/2 ⊗ s1/2 configuration. The 15− isomers were interpreted as v = 4 isomers arising from the same h 11/2 ⊗ d3/2 ⊗ s1/2 configuration. Iskra et al. [28] proposed the configurations of 13− and 15− isomers as v = 4 isomers arising from h 311/2 ⊗ s1/2 and h 311/2 ⊗ d3/2 configurations, respectively. Our generalized seniority results strongly

62

4 Generalized Seniority Isomers 70 XE-7

B(E1; 13 - → 12 + )W .U.

Sn isotopes

60

Exp. Ω=9 Ω=8 Ω=7

20

10

0 66

68

70

72

74

76

78

80

Neutron Number Fig. 4.4 Variation of v = 4, 13− isomers in Sn isotopes. The uncertainties in the experimental data are not shown for the clear picture, as most of them lie within the size of symbol

support the mixing of all the three orbitals i.e., h 11/2 ⊗ d3/2 ⊗ s1/2 for all three highspin isomers. The results shown in Fig. 4.5, confirm our interpretation. As  = 9 alone has been able to explain the measured systematics for all the three v = 2, 10+ and v = 4, 13− , and 15− isomers, we have calculated the halflives of these isomers by using the calculated reduced transition probabilities for  = 9, ICC [39], the experimental gamma ray energies, intensities and branching ratios [14]. We have used Eqs. (2.10) and (2.11) for calculating the total transition probabilities corresponding to the E1 and E2 transitions. The related isomeric halflives are then obtained by using Eq. (2.13), where branching ratios can be obtained using the experimental intensities and ICC as given in Eq. (2.14). Figure 4.6 presents a comparison of the experimental and the calculated half-lives in Weisskopf units (W.U.) for all the three sets of isomers in the three separate panels. The calculated trends reproduce the experimental half-lives quite well. The peaks at the middle are very obvious due to the fact that the transition probabilities show minima at the same mass numbers, irrespective of the nature of the involved electric tensors, which may be even or odd. Therefore, the significant rise in the half-lives, particularly at the middle, is due to the role of seniority, where transitions get nearly full-hindered.

4.2 Generalized Seniority in the Sn Isotopes

63

B(E2;10 + → 8 + )W .U .

2

Sn isotopes

1

0

B(E2;15 - → 13 - )W .U .

5

Exp.

4

Ω=9; h11/2 ⊗ d3/2 ⊗ s1/2

3

Ω=8; h11/2 ⊗ d3/2

2

Ω=7; h11/2 ⊗ s1/2 Ω=6; h11/2

1 0 66

68

70

72

74

76

78

80

Neutron Number (N) Fig. 4.5 B(E2) variation of v = 2, 10+ and v = 4, 15− isomers in Sn isotopes. The uncertainties in the experimental data are not shown, as most of them lie within the size of symbol

4.2.2 Comparison of Sn, Pb and N = 82 Isomers In parallel, a comparative study of seniority isomers in the semi-magic chains of Sn (Z = 50) isotopes, N = 82 isotones, and Pb (Z = 82) isotopes has also been carried out. The 10+ and 27/2− isomers have been observed in the N = 82 isotones from Z = 66 to Z = 72, and were interpreted as seniority v = 2 and v = 3 isomers arising from the h 11/2 protons [41]. Recently, the high-spin structure of N = 82 isotones having Z = 54 − 58 has also been investigated by Astier et al. [42], where the 10+ isomers were proposed to arise from the broken pairs of g7/2 and d5/2 protons. These isomers were interpreted as v = 2 and v = 3 isomers arising from h 11/2 ⊗ d3/2 ⊗ s1/2 configuration. Similarly, the 12+ and 33/2+ isomers in Pb isotopes were also proposed as v = 2 and v = 3 isomers arising from i 13/2 ⊗ f 7/2 ⊗ p3/2 configuration. The calculations have assumed the transitions to be seniority preserving v = 0 transitions. In spite of different set of orbitals involved, the features of generalized seniority persist in Pb isomers in a way very similar to the isomers in both Sn isotopes and N = 82 isotones [13].

64

4 Generalized Seniority Isomers 10

10+, v=2, E2 isomer

1 0.1

T 1/2 (W.U.)

10

(a) 15-, v=4, E2 isomer Exp. Cal.

1

1E8

(b) 13-, v=4, E1 isomer

1E7

(c) 1000000 66

68

70

72

74

76

78

80

Neutron Number (N) Fig. 4.6 Half-life variation of v = 2, 10+ and v = 4, 13− , and 15− isomers in Sn isotopes. The uncertainties in the experimental data are not shown, as most of them lie within the size of symbol

4.3 First 2+ and 3− States in Sn, Cd, Te Isotopes 4.3.1 Twin Asymmetric B(E2) Parabola in Sn Isotopes Although the discussion in this section does not directly involve any isomeric states, yet the generalized seniority exhibits its imprint in many ways, as we show. We have applied these ideas to the first excited 2+ and 3− states in the even-even Sn isotopes, and consider their B(E2) and B(E3) trends, respectively. We have first considered the long-standing puzzle of the two inverted parabolas seen in the B(E2) variation of the 2+ states in the Sn isotopes. This feature remained unexplained for long until we applied the generalized seniority scheme to this data set and were able to provide a physically consistent explanation in terms of generalized seniority, for the first time [43]. Later, state of art Monte Carlo Shell Model (MCSM) calculations [44] also claimed to explain this feature, but the physics explanation provided by us stands on its own. We expect the B(E2) transition probabilities to show a parabolic behavior with a maximum at the mid-shell for v = 2, seniority changing transitions as is the

4.3 First 2+ and 3− States in Sn, Cd, Te Isotopes

65

80

40

B(E2;0

+

+

→ 2 ) (W.U.)

60

Sn isotopes 20

Exp. Ω=10 Ω=12

0 56

64

72

80

Neutron Number (N) Fig. 4.7 Experimental and calculated B(E2) trends for the first excited 2+ states in Sn isotopes. Experimental data have been reevaluated in 2016 [43]

situation in the (0+ → 2+ ) transitions. However, the evaluated experimental data, when plotted, clearly exhibit two inverted parabola as shown in Fig. 4.7. We have also plotted the calculated results in Fig. 4.7, shown by dashed and dotted lines for generalized seniority calculations. The complete valence space consists of g7/2 , d5/2 , h 11/2 , d3/2 and s1/2 orbitals. For the generalized seniority calculations, we have divided the available valence space into two parts: j˜ = g7/2 ⊗ d5/2 ⊗ d3/2 ⊗ s1/2 ,  = 10, and j˜ = d5/2 ⊗ d3/2 ⊗ s1/2 ⊗ h 11/2 ,  = 12. For dividing the valence space, we have used the fact that the h 11/2 orbital mainly dominates after the midshell (116 Sn), while the g7/2 orbital dominates the lighter isotopes and completely freezes on crossing 108 Sn. We have taken 100 Sn as the natural choice of the core for  = 10, and 108 Sn as the core for  = 12, since the g7/2 orbital becomes filled up at that point. We have then calculated the B(E2) values by using the generalized seniority scheme, and obtained the two asymmetric parabolas as shown in Fig. 4.7. It may be noted that the two parabolas would cross each other at 116 Sn (if extrapolated), and a natural minimum would arise at the middle. Therefore, the minimum at 116 Sn represents a change in the filling of the orbitals before and after the middle, and also corresponds to the location, where g7/2 freezes out, and h 11/2 becomes active. These calculations could explain the overall trend quite well, and also provided a direct clue of the configurations and their influence on the B(E2) values. This approach helps in deciding the most active valence space for a given set of isotopes.

66

4 Generalized Seniority Isomers

The LSSM calculations so guided by generalized seniority are also able to reproduce the experimental data on B(E2) values quite well [43]. A similar conclusion was also reached by Morales et al. [44].

4.3.2 Twin Asymmetric B(E2) Parabola in Cd and Te Isotopes We have also noticed a twin asymmetric parabolic behavior of measured B(E2) values in the Cd and Te isotopes [46]. We could understand these trends, for the first time, in a manner similar to that of the Sn isotopes by using the generalized seniority scheme [47]. It may be noted that the Cd and Te isotopes are not semi-magic and this allows us to test the validity of the generalized seniority scheme as we deviate from semi-magic configuration. The Cd isotopes represent two proton hole systems and the Te isotopes are two proton particle systems in the Z = 50 closed shell. Therefore, both types of particles, neutrons and protons, play an active role in these isotopes, unlike the Sn, or the Pb isotopes where only neutrons would be active. It is, therefore, interesting to see how far the generalized seniority approach for identical particles is applicable. We have applied the same generalized seniority approach to calculate the B(E2) values for the ground 0+ states to the first 2+ transitions, in Cd and Te isotopes. The experimental data are mostly taken from the B(E2) evaluation of [46]. We have also noticed the existence of two asymmetric B(E2) parabolas for the Cd and Te isotopes as shown in Figs. 4.8 and 4.9, respectively. A dip near the middle of the plots is visible for both the Cd and Te isotopes, similar to the Sn isotopes. The generalized seniority calculations for the Cd and Te isotopes use the same multi-j configurations corresponding to  = 10 (before the middle) and  = 12 (after the middle), respectively, similar to the Sn isotopes [43]. The valence space is mainly dominated by g7/2 and d5/2 orbitals before the middle, and h 11/2 orbital after the middle. The first excited 2+ states have been taken as the generalized seniority v = 2 states for the calculations. The calculated trends depend on the square of the coefficients in Eq. (4.16), since the 0+ to 2+ transitions are generalized seniority changing v = 2 transitions. To take care of other structural effects, we have fitted one of the experimental data and restricted the values of radial integrals and involved 3 j− and 6 j−coefficients as constant, which should be the case for an interaction conserving the generalized seniority. The results are plotted in Figs. 4.8 and 4.9. We note that the generalized seniority calculated values are able to explain the overall trend of the experimental data in the Cd and Te isotopes. Asymmetry in the inverted parabola before and after the middle is again due to the difference in filling the two sets of orbitals. The dominance of g7/2 orbital shifts to the h 11/2 orbital near the middle of the shell resulting in a dip. However, the generalized seniority remains constant at v = 2 leading to the particle number independent energy variation for the 2+ states throughout the full chain.

4.3 First 2+ and 3− States in Sn, Cd, Te Isotopes

67

200

120

B(E2; 0

+

+

→ 2 ) (W.U.)

160

Cd isotopes Exp. Ω = 10 Ω = 12

80

40 56

64

72

80

Neutron Number (N) Fig. 4.8 Comparison of the experimental [46] and generalized seniority calculated B(E2) trends for the first excited 2+ states in Cd isotopes, which are not semi-magic

The generalized seniority, hence, governs the electromagnetic properties not only in Sn isotopes but also to some extent in the Cd and Te isotopes, which are not semimagic nuclei. One may note that the calculations only consider the active orbitals of N = 50 − 82 valence space. No signs of shell quenching have, therefore, been witnessed for these first excited 2+ states in all the three sets of Cd, Sn and Te isotopes. The influence of two proton holes/ particles cannot be ignored and appears to play a role in the deviations from the overall trend.

4.3.3 Inverted B(E3) Parabola in Sn Isotopes The E3 transitions are odd-tensor transitions connecting different seniority (v = 2) and different parity states. We are able to explain the general trend of the B(E3) values for the (0+ → 3− ) transitions in Sn isotopes by invoking the generalized seniority for the first time. We have obtained the B(E3) values for the (0+ → 3− ) transitions by using  = 9 and 11, corresponding to the d5/2 ⊗ h 11/2 and d5/2 ⊗ d3/2 ⊗ h 11/2 valence spaces, respectively [49]. The experimental [48] and calculated B(E3) trends are plotted in Fig. 4.10. It is quite obvious that the mixing of negative-parity h 11/2 orbital is

68

4 Generalized Seniority Isomers 200

Te isotopes Ω = 10 Ω = 12

+

→ 2 ) (W.U.)

Exp. 160

B(E2; 0

+

120

80

40 56

64

72

80

Neutron Number (N) Fig. 4.9 Same as Fig. 4.8, but for Te isotopes.

necessary to generate the 3− states, which decay via E3 transitions to the ground state. We have, therefore, calculated the complete B(E3) trend by fitting the theoretical expressions to the experimental data for 116 Sn and 118 Sn, the respective mid-points, for  = 9 and 11. We have taken 108 Sn as the core, where the core represents the n = 0 situation and the situation corresponding to the complete filling of the g7/2 orbital. The involvement of d and h orbitals also suggests the presence of octupole correlations in the 3− states, since the d and h orbitals may be connected by l = 3 interaction.

4.3.4 Inverted B(E3) Parabola in Cd and Te Isotopes Figures 4.11, and 4.12 exhibit the experimental and generalized seniority calculated B(E3) trends for Cd and Te isotopic chains. The experimental data have been taken from the compilation of Kibedi and Spear [48]. The calculations have been done by using the pair degeneracy of  = 9 and 11 corresponding to the d5/2 ⊗ h 11/2 and d5/2 ⊗ d3/2 ⊗ h 11/2 configurations, respectively. An inverted parabolic B(E3) trend is visible in both the Cd and Te isotopic chains. The calculated results could explain the gross experimental trend, wherever available. However, one can notice that the Fermi surface for the given configurations seems to be different for each chain. Therefore,

4.3 First 2+ and 3− States in Sn, Cd, Te Isotopes

Sn isotopes

25

B (E3;0 + → 3 - ) (W.U.)

69

Exp. Ω= 9 Ω = 11

20

15

10

5 56

60

64

68

72

76

Neutron Number (N) Fig. 4.10 Experimental [48] and GS calculated B(E3) trends for the first 3− states in the Sn isotopes

40

Cd isotopes Exp. Ω= 9 Ω = 11

B (E3;0 + → 3 - ) (W.U.)

35

30

25

20

15

10 52

56

60

64

Neutron Number (N) Fig. 4.11 Same as Fig. 4.10, but for Cd isotopes

68

72

70

4 Generalized Seniority Isomers

B (E3;0 + → 3 - ) (W.U.)

25

Te isotopes Exp. Ω= 9 Ω = 11

20

15

10

5 60

64

68

72

76

80

Neutron Number (N) Fig. 4.12 Same as Fig. 4.10, but for Te isotopes

the peaks belong to different neutron number in each chain, such as N = 58 − 60 for Cd isotopic chain, N = 66 for Sn isotopic chain and N = 70 for Te isotopic chain. The deviation of many points from the generalized seniority are more pronounced in Cd isotopes. More precise measurements are needed to further test the theoretical arguments.

4.4 Isomeric Moments 4.4.1 Quadrupole Moment of 11/2− States The Q-moment is usually taken as a measure of deviation from the spherical shape. We have applied the generalized seniority scheme to study the Q-moments of 11/2− states, many of them isomers, in the Cd, Sn and Te isotopes. All the experimental data, plotted in Fig. 4.13 have been adopted from [50]. The units are written as barns (b), since the electric charge e is subsumed in the definition of Q by Stone [50]. Such 11/2− states, many of them isomers and many ground states, are dominated by the unique-parity h 11/2 orbital in the neutron 50 − 82 valence space. A linear increasing trend is visible in the experimental data for nearly all the cases, particularly after

4.4 Isomeric Moments

1

Exp. Ω=9 Ω=12

0

(a) Cd isotopes

-1 1

-

Q(11/2 ) (barns)

71

0 (b) Sn isotopes

-1 1 0

(c) Te isotopes

-1 59

63

67

71

75

79

Neutron Number (N) Fig. 4.13 Quadrupole moment variation for the 11/2− states in odd-A Cd, Sn and Te isotopes. All the experimental data have been taken from [50]

N = 64 (a sub-shell closure after filling g7/2 and d5/2 orbitals). The Q-moment changes from negative to positive with a nearly zero value at 123 Sn (N = 73), 121 Cd (N = 73) and 125 Te (N = 73) in the three isotopic chains. Yordanov et al. [51] pointed out the linear behavior in the Cd isotopes beyond h 11/2 orbital. We have found a similar behavior for the Sn and Te isotopes also, as shown in Fig. 4.13 [47]. It is interesting to note that many of these states are isomeric states and many are ground states (mostly unstable). Yet, they all satisfy the same linear trend. We have carried out the calculations by assuming the 11/2− states as the generalized seniority v = 1 states arising from the multi-j configuration corresponding to the pair degeneracy  = 9, as also used by us to calculate the g-factors [52]. Equation (4.15) has been used to calculate the Q-moments by fitting one of the experimental data. The g7/2 and d5/2 orbitals are assumed to be fully occupied till N = 64, since these two orbitals lie lower in energy and get active as soon as neutrons begin to fill the N = 50 − 82 valence space. The remaining three orbitals lie higher in energy, and dominate the wave functions of the neutron-rich isotopes. The results are shown in the Fig. 4.13a–c, which exhibit the experimental and theoretical variation of the Q-moments. We obtain a linear trend as per the −n −v coefficient for all the isotopes after N = 64. The calculated trends are in line with the experimental data for all the three Cd, Sn and Te isotopic chains. The calculated

72

4 Generalized Seniority Isomers

trends also support the zero Q-value at N = 73 in all the three isotopic chains. Also, a similar range of Q-values is observed for all the three isotopic chains (from N = 65 to 81), supporting similar structure of these states on going from Cd (2 proton-holes) to Sn (the proton closed-shell) to Te (2 proton-particles). The multi-j description in terms of generalized seniority scheme appears to be essential to explain the data; occupancy of h 11/2 alone cannot explain the complete set of experimental data. For completeness, we describe the 11/2− , v = 1 states in terms of the singleparticle Q-value as expressed in Eq. (4.18), as the other factors will simply be equal to 1 in Eq. (4.17). In this way, we can get the information of involved radial integrals for the 11/2− state. For this, we have used the Eq. (4.15) and fitted the experimental Q-moment value of 117 Sn (N = 67 and n = 3 after freezing g7/2 and d5/2 ) to extract the radial integral of the 11/2− states in the Sn isotopes. We obtained a value of 42.04 f m 2 , which may be treated as a constant for the 11/2− states corresponding to  = 9. Similarly, one can find the radial integrals for 11/2− states in Cd and Te isotopes, which come out to be 45.78 and 60.05 f m 2 , by fitting the experimental values at 113 Cd and 129 Te, respectively. By assuming the charge of the effective j˜ = h 11/2 ⊗ d3/2 ⊗ s1/2 neutron to be equal to the charge of a free neutron, we may also 1/2 deduce, < r 2 > = 6.77, 6.48 and 7.75 f m for Cd, Sn and Te isotopes, respectively. On the other hand, the radial integrals for pure-j h 11/2 orbital in all the three isotopic chains change significantly resulting in a positive Q-value, and most importantly, are unable to explain the full experimental trend. We may point out that the data for 129,131 Sn (N = 79, 81) isotopes, and for 131,133 Te (N = 79, 81) isotopes deviate from the expected trend. Recent measurements reported by Yordanov et al. [53], use collienar laser spectroscopy to obtain more precise values of the moments; these measurements bring the two deviating points much closer to the trend obtained from the GS scheme. This implies that similar deviations pointed out earlier in explaining the B(E2) values of 10+ and 27/2− Sn-isomers after N = 77 may also come closer to the predicted trend [11, 13]. We stress the necessity of using a more realistic picture of non-degenerate orbitals in the generalized seniority scheme to better explain the observed values. We have also examined the Q-values in lighter Cd isotopes (before N = 64) with the generalized seniority v = 1 and  = 12 configuration as shown in Fig. 4.13a. Interestingly, the calculated values explain the data in 109,111 Cd but deviate in 113 Cd on approaching N = 65, and finally go off the observed trend for heavier Cd isotopes. This comparison further validates the N = 64 sub-shell closure where g7/2 and d5/2 orbitals freeze out; however, the possibility of mixing d5/2 orbital in the resulting wave function cannot be ruled out for the lighter (N < 64) Cd isotopes [47]. We emphasize that the same multi-j configuration has been used to explain the trend of g-factors for these 11/2− states in Sn isotopes as discussed in the following section [52], and also to describe the origin of high-spin isomers like 10+ and 27/2− isomers [11, 13]. The generalized seniority scheme appears to, therefore, work consistently in explaining all the spectroscopic properties.

4.4 Isomeric Moments

73

4.4.2 Generalized Seniority Schmidt Model We have already discussed the nature of the magnetic transition probabilities, which may have non-zero contributions from both even and odd tensor operators in a multi-j environment. The magnetic dipole moment for a given j˜ may simply be written as: ˆ j˜n  =  j˜v |μ| ˆ j˜v   j˜n |μ|

(4.21)

where j˜ = j ⊗ j  .... represents a multi-j valence shell, whose sum of shared occupancy is n, the total particle number in the valence shell. We could, therefore, write the magnetic moment of identical nucleons in the mixed-j configuration j˜n as μ = g

n 

j˜ = g J i

(4.22)

i

Therefore, the g-factors of a multi-j configuration also exhibit a particle number independent behavior, similar to the single-j case. As a result, the g-factors of all the states arising from a given multi-j configuration having identical nucleons must be equal to the g-factor of a single nucleon (the seniority v = 1 state) arising from the same multi-j configuration. If the states are of good generalized seniority then they must have a constant and particle number independent behavior of g-factor throughout any given multi-j configuration. It is useful to merge the idea of generalized seniority with the well-known Schmidt model and define a phenomenological Generalized Seniority Schmidt Model (GSSM). Schmidt model is the well known model to calculate the trend of g-factors in odd-A nuclei, particularly near the magic numbers. The g-factor expressions for single-j are well known and given by:  g=

1 j

=

1 j+1

1 g 2 s



+ (j −

1 )g 2 l

 ;j =l+

1 2

 − 21 gs + ( j + 23 )gl ; j = l −

1 2

(4.23)

where gs and gl are taken to be 5.59 n.m. and 1 n.m. for protons, while -3.83 n.m. and 0 n.m. for neutrons, respectively. We have obtained similar expressions in GSSM, ˜ by using the definition j˜ = j ⊗ j .... for multi-j situations, and replacing  1 j by j, ˜ (2 j + 1) so that the pair degeneracy of mixed configuration is given by  = 2 corresponding to the generalized seniority v, as follows:  g=

1 j˜

=

1 ˜ j+1

1 g 2 s

 + ( j˜ − 21 )gl ; j˜ = l˜ +

 −

1 g 2 s

1 2

 3 ˜ + ( j + 2 )gl ; j˜ = l˜ −

1 2

(4.24)

74

4 Generalized Seniority Isomers -0.1

Sn isotopes -0.2

g-factor (n.m.)

-0.3 -0.4

(a) 11/2

-

-0.1

Experimental GSSM Schmidt model

-0.2

-0.3

-0.4

(b) 10 60

64

68

72

76

+

80

Neutron Number (N) Fig. 4.14 g-factor trends for the 11/2− states and 10+ isomers in Sn isotopes. All the experimental data have been taken from Stone’s compilation [50]

These expressions define the GSSM. In short, the g-factor (and the magnetic dipole moment) of all the states having the same seniority arising from identical nucleons in a multi-j valence space, must be equal to the g-factor of a single nucleon (seniority v = 1) in the same valence space. If this proposition holds for various states in semi-magic nuclei having identical nucleons, then the effective interaction should be nearly diagonal for these generalized seniority states. GSSM Results in Sn Isotopes We show in Fig. 4.14, the measured g-factor trends of the 11/2− and 10+ isomers in Sn isotopes. The 11/2− isomers are observed in the Sn isotopes from A = 111 to 131. This covers a range of neutron numbers from N = 61 to 81. It may be pointed out that 11/2− becomes the ground state in three odd-A Sn isotopes from A = 123 to 127. Even then, the g-factors display a smoothly varying and nearly constant trend, suggesting a similar structure of the 11/2− isomeric states and the 11/2− ground states. The 11/2− states maybe easily interpreted as the v = 1 pure-j seniority states with h11/2 unique-parity orbital configuration. But the Schmidt model g-factors for h11/2 neutrons come out to be −0.348 n.m., quite far from the experimental values as shown in Fig. 4.14. We have calculated the g-factor trend for the 11/2− states by using the configuration j˜ = d3/2 ⊗ s1/2 ⊗ h 11/2 as used for the 10+ isomers also in the following. Interestingly, the calculated results come very close to the experimental data, as shown in Fig. 4.14. This strongly supports a mixed configuration rather than a

4.4 Isomeric Moments

75

pure h11/2 configuration, having shared occupancy of v = 1 from j˜ = d3/2 ⊗ s1/2 ⊗ h 11/2 configuration for the 11/2− states in the Sn isotopes. We present in Fig. 4.14, the observed g-factor trend of the high-spin 10+ isomers in Sn isotopes. Although only 3 values are known, a constant empirical trend is apparent. We have compared these values with the GSSM results by using the same multi-j configuration, and generalized seniority v = 2, as used for the B(E2) calculations of these isomers in the earlier sections. We have obtained a particle number independent trend, which explains the known experimental data quite well. Note that the Schmidt g-factor of pure h 11/2 configuration lies quite far from the experimental data. It may also be noted that the g-factor values of the 10+ states are nearly the same as the g-factor values of the 11/2− states, which confirms the role of configuration mixing in the generation of 11/2− states. This is also experimentally evident, as the 10+ and 27/2− isomers closely follow each other in the measured excitation energies, if one puts the 0+ and 11/2− states on equal footing. In conclusion, the 11/2− and the 10+ isomers have been understood as generalized seniority v = 1 and v = 2 states, respectively, arising from the same multi-j configuration j˜ = d3/2 ⊗ s1/2 ⊗ h 11/2 . We have noted that the existence of 10+ isomers is presently known from 116 Sn to 130 Sn involving a variation of 14 particles (with few experimental gaps in the g-factor measurements), while pure h11/2 can accommodate only 12 particles, confirming the need for a mixed configuration. These results further suggest that any realistic effective interaction for describing these states should be nearly diagonal in the generalized seniority scheme. This also leads us to some predictions. Only three measured values are known for the 10+ isomers; for the rest of the cases, we expect the g-factor values to be of the same order due to the particle number independent behavior. It would be highly interesting to measure these values. The g-factor values for the 27/2− isomers have not been measured at all. We have predicted the g-factor values of the 27/2− isomers to be of the same order as those of the 10+ isomers, due to similar configuration mixing in the wave-functions. A particle number independent g-factor behavior is also expected for the 27/2− isomers, which originate from the generalized seniority v = 3, j˜ = d3/2 ⊗ s1/2 ⊗ h 11/2 configuration. New experimental measurements are needed to confirm these predictions. GSSM Results in Pb Isotopes We have further examined the measured g-factor trends for the 13/2+ , 12+ , and 33/2+ isomers of the Z = 82 Pb-isotopes as shown in Fig. 4.15. The known measurements display a particle number independent behavior in all the three sets of isomeric states, as expected from GSSM. We have already discussed the 12+ and 33/2+ isomers in Pb isotopes as the generalized seniority v = 2 and v = 3 states arising from j˜ = p3/2 ⊗ f 7/2 ⊗ i 13/2 configuration, where the similar B(E2) parabolic trends were attributed to the goodness of generalized seniority. Since the order of the g-factor values for the three 13/2+ , 12+ , and 33/2+ isomers are nearly the same, we expect that the 13/2+ isomers also have a similar configuration mixing and a generalized seniority v = 1. Similar to the 11/2− states in the Sn isotopes discussed

76

4 Generalized Seniority Isomers

Pb isotopes

0.0 -0.2

g-factor (n.m.)

-0.4 0.0

(a) 13/2

+

Experimental GSSM Schmidt model

-0.2 -0.4

(b) 12

+

0.0 -0.2 -0.4 100

(c) 33/2 104

108

112

116

+

120

Neutron Number (N) Fig. 4.15 g-factor trends for the 13/2+ , 12+ , and 33/2+ states in Pb isotopes. All the experimental data have been taken from Stone’s compilation [50]

earlier, the 13/2+ states in Pb isotopes are expected to originate from the uniqueparity i 13/2 orbital of N = 82 − 126 valence space. Yet, it continues to follow the GSSM trend for the g-factor values. This extends the scope of applicability of Generalized seniority approach to j = 13/2, See Fig. 4.15. Note that the Schmidt values for single-j i 13/2 orbital are quite far from the experimental trend and it is necessary to invoke the generalized seniority configuration j˜ = p3/2 ⊗f7/2 ⊗ i13/2 . The results for the 12+ and 33/2+ isomers in even-even and even-odd Pb isotopes, respectively, also support the GSSM interpretation. We may note that the multij configurations used for the g-factors are the same as those used to describe the B(E2) properties. Thus, the configuration mixing suggested by generalized seniority is consistent in explaining all the electromagnetic properties for both the Sn isotopes and the Pb isotopes. Generalized seniority has emerged as a unique tool to make predictions for the gaps in the measurements as shown in the Fig. 4.15. We note that the neutrons in Z = 50 isotopes and the protons in N = 82 isotones occupy g7/2 , d5/2 , h 11/2 , d3/2 , and s1/2 orbitals in the 50 − 82 nucleon space. This allows an interesting comparison between the two. The 10+ and 27/2− isomers have already been identified as generalized seniority v = 2 and v = 3 states arising from j˜ =d3/2 ⊗ s1/2 ⊗ h11/2 configuration in the N = 82 isotones, analogous to the situation in the Sn isotopes [13]. However, there are no g-factor measurements available to compare the both. We have predicted these g-factor values to be of the order of 1.27 n.m. Measurements should be carried out to confirm these predictions.

References

77

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48.

A.K. Kerman, Ann. Phys. (NY) 12, 300 (1961) A.K. Kerman, R.D. Lawson, M.H. Macfarlane, Phys. Rev. 124, 162 (1961) K. Helmers, Nucl. Phys. 23, 594 (1961) A. Arima, M. Ichimura, Prog. Theo. Phys. 36, 296 (1966) I. Talmi, Nucl. Phys. A 172, 1 (1971) S. Shlomo, I. Talmi, Nucl. Phys. A 198, 82 (1972) R. Arvieu, S.A. Moszokowski, Phys. Rev. 145, 830 (1966) I. Talmi, Simple Models of Complex Nuclei (Harwood Academic, 1993) and original references therein I.M. Green, S.A. Moszokowski, Phys. Rev. 139, B790 (1965) B. Maheshwari, Ph.D. Thesis, IIT Roorkee (2016) B. Maheshwari, A.K. Jain, Phys. Lett. B 753, 122 (2016) V.K.B. Kota, Bulg. J. Phys. 44, 454 (2017) A.K. Jain, B. Maheshwari, Nucl. Phys. Rev. 34, 73 (2017) Evaluated Nuclear Structure Data File http://www.nndc.bnl.gov/ensdf/ P.J. Daly et al., Z. Phys. A 298, 173 (1980) B. Fogelberg, K. Heyde, J. Sau, Nucl. Phys. A 352, 157 (1981) P.J. Daly et al., Z. Phys. A 323, 245 (1986) S. Lunardi et al., Z. Phys. A 328, 487 (1987) R. Broda et al., Phys. Rev. Lett. 68, 1671 (1992) R. Mayer et al., Phys. Lett. B 336, 308 (1994) P.J. Daly et al., Phys. Scr. T56, 94 (1995) J.A. Pinston et al., Phys. Rev. C 61, 024312 (2000) C.T. Zhang et al., Phys. Rev. C 62, 057305 (2000) R.L. Lozeva et al., Phys. Rev. C 77, 064313 (2008) S. Pietri et al., Phys. Rev. C 83, 044328 (2011) A. Astier, Journal of Physics: Conf. Series 420, 012055 (2013) A. Astier et al., Phys. Rev. C 85, 054316 (2012) L.W. Iskra et al., Phys. Rev. C 89, 044324 (2014) J. Bron et al., Nucl. Phys. A 318, 335 (1979) A.V. Poelqeest et al., Nucl. Phys. A 346, 70 (1980) H. Harda et al., Phys. Lett. B 207, 17 (1988) A. Savelius et al., Nucl. Phys. A 637, 491 (1998) J. Gableske et al., Nucl. Phys. A 691, 551 (2001) M. Wolinska-Cichocka et al., Eur. Phys. J. A 24, 259 (2005) S.Y. Wang et al., Phys. Rev. C 81, 017301 (2010) N. Fotiades et al., Phys. Rev. C 84, 054310 (2011) G.S. Simpson et al., Phys. Rev. Lett. 113, 132502 (2014) B. Maheshwari, A.K. Jain, P.C. Srivastava, Phys. Rev. C 91, 024321 (2015) T. Kibedi, T.W. Burrows, M.B. Trzhaskovskaya, P.M. Davidson, C.W. Nestor Jr., Nucl. Instr. Meth. A 589, 202–229 (2008) http://bricc.anu.edu.au/ J.H. Mcneill et al., Phys. Rev. Lett. 63, 860 (1989) A. Astier et al., Phys. Rev. C 85, 064316 (2012) B. Maheshwari, A.K. Jain, B. Singh, Nucl. Phys. A 952, 62 (2016) T. Togashi et al., Phys. Rev. Lett. 121, 062501 (2018) I.O. Morales, P. Van Isacker, I. Talmi, Phys. Lett. B 703, 606 (2011) B. Pritychenko, M. Birch, B. Singh, M. Horoi, At. Data Nucl. Data Tables 107, 1 (2016) B. Maheshwari, H.A. Kassim, N. Yusof, A.K. Jain, Nucl. Phys. A 992, 121619 (2019) T. Kibedi, R.H. Spear, At. Data Nucl. Data Tables 80, 35 (2002)

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49. B. Maheshwari, S. Garg, A.K. Jain, Pramana- J Phys. (Rapid Commun.) 89, 75 (2017) 50. N.J. Stone, www-nds.iaea.org/publications, INDC(NDS)-0658, Feb. (2014) 51. D.T. Yordanov, D.L. Balabanski, M.L. Bissell, K. Blaum, A. Blazhev, I. Budinˇcevi´c, N. Frömmgen, Ch. Geppert, H. Grawe, M. Hammen, K. Kreim, R. Neugart, G. Neyens, W. Nörtershäuser, Phys. Rev. C 98, 011303(R) (2018) 52. B. Maheshwari, A.K. Jain, Nucl. Phys. A 986, 232 (2019) 53. D.T. Yordanov et al., Nature Comm. Phys. 3, 107 (2020)

Chapter 5

K -Isomers in Deformed Nuclei

Keywords K -quantum number · K -isomers · Multi-quasi particle states · Theoretical approaches · K -isomeric features · K -mixing · Examples

5.1 The K -Quantum Number Recent developments in the experimental techniques have enabled the discovery of a number of K -isomers in a broad range of mass regions. The variety of K -isomers offer new possibilities in both fundamental research as well as new applications. Their studies offer unique insight into the interplay between intrinsic excitation and collective rotation in nuclei. They have, therefore, been in the lime-light of nuclear physics. Many reviews/data tabulations have entirely focused on the K -isomers; we can mention the 2001 review of Walker and Dracoulis [1], a comprehensive collection of K -isomer data and configurations by Kondev, Dracoulis and Kibedi [2], and a more general review by Dracoulis, Walker and Kondev [3]. Besides this, several papers have been published, which also highlight various properties of K -isomers and their systematic features [4–6]. In order to understand the K -isomers, let us first understand the K quantum number. The schematic diagram in Fig. 5.1 depicts a deformed nucleus with axial symmetry along the z-axis. The axially-symmetric nuclear core is coupled to two nucleons, whose angular momentum construction is also shown. Each nucleon contributes an angular momentum j1 and j2 , which in turn arise from their respective spin and orbital angular momenta. The total angular momenta of the two nucleons j = j1 + j2 . The projection of j on the symmetry axis is generally denoted by  = 1 + 2 , where 1 and 2 are the projections of j1 and j2 , respectively. The nucleus rotates about an axis perpendicular to the symmetry axis with a rotational  which is perpendicular to the symmetry axis. The total anguangular momentum R, lar momentum of the nucleus I = R + j, and projection of I on the symmetry axis is denoted by K . We can see from the geometry of the system that K = . The © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. K. Jain et al., Nuclear Isomers, https://doi.org/10.1007/978-3-030-78675-5_5

79

80

5 K -Isomers in Deformed Nuclei

Fig. 5.1 Schematic representation of K -quantum number, which is the projection of total angular momentum j = j1 + j2 + ... on the the z-axis, the symmetry axis. The spin of nucleons s1 , s2 , ... and the orbital angular momenta l1 , l2 , ... are also shown with their respective projection quantum numbers as 1 , 2 ..... and 1 , 2 ...

quantum number K , thus, represents the inclination of the total angular momentum I with respect to the z-axis. If K is large, I makes a small angle with the z-axis. If K is small, then I makes a large angle with the z-axis, and becomes closer to the rotation axis. This is the situation of rotation alignment. K -isomers arise due to a large change in the K -value in going from the isomeric state to the final state, which entails a significant change in the orientation of the angular momentum during the decay. This implies a significant change in the wave function, while transiting to the final state, something not liked by the nucleus. At times, the change in the value of K , K , may be larger than the multi-polarity λ of the gamma-transition. Such a transition (K > λ) is forbidden, or hindered. The degree of forbiddenness ν is generally defined as ν = K − λ. We may again recall the historic example of the 8− isomer in 180 Hf, with a half-life of 5.5 h , which has been understood as a K -isomer. The isomer is the band-head of the K = 8 rotational band. It requires a change in K , K = 8, for the decay to the 8+ state of K = 0 band, while the multipolarity for this decay becomes λ = 1. This results in a high degree of hindrance as ν = 7, and gives rise to a long half-life of the isomeric state, as shown in Fig. 1.2. The K = 8 isomer in 180 Hf and its decay were of particular value in establishing the rotational motion in nuclei [7]. Quite often, the spectroscopic properties of isomers like energy, spin-parity, and electromagnetic moments etc. are more accessible to experiments in comparison to other nuclear states due to their long-lived nature. The 2015 collection of data by Kondev et al. [2] presents a detailed spectroscopic information of the decay properties

5.1 The K -Quantum Number

81

of high K -isomers in deformed nuclei with A > 100, and has proved to be very useful in probing the structure of deformed nuclei.

5.2 Deformed Nilsson Model and High-K States Just as the spherical shell model plays a crucial role in interpreting the spin isomers and the seniority isomers, the deformed shell model (or, the Nilsson model) plays a crucial role in interpreting the K -isomers. K remains a good quantum number as long as the axial symmetry in nuclear deformation holds. Therefore, observation of K -isomers may also serve the purpose of defining the applicability of axial symmetry in nuclei. It is, therefore, relevant to consider the appropriate Nilsson schemes and the high− Nilsson levels that lie in close proximity to the Fermi energy in specific mass regions. A large number of K -isomers have been identified, particularly in A ≈ 130 and 180, and more recently the A ≈ 250 region, since the high-j orbitals lie near the Fermi surface in these mass regions resulting in large angular-momentum projections. The i 13/2 neutron and h 11/2 proton orbitals get active in A ≈ 180 region, while j15/2 neutron and i 13/2 proton orbitals are occupied in the A ≈ 250 region. Here, we show as an example the Nilsson single-particle level schemes relevant for the mass region A = 150 − 190 only, both for protons and neutrons, as given in Figs. 5.2 and 5.3, respectively. It is useful at this point to recollect the quantum numbers used in labeling the single-particle levels of the Nilsson model. The single-particle levels plotted in the Figs. 5.2 and 5.3, are commonly labelled by the asymptotic quantum numbers π [N n z ]. These quantum numbers arise from an axially-deformed oscillator potential. Here, N is the principal quantum number, n z is the number of oscillator quanta along the symmetry axis,  is the projection of single-particle orbital angular momentum along the symmetry axis, and  = K is the projection of the total angular momentum along the symmetry axis. Incidentally, only  and, therefore, K and , both are the good quantum numbers. Rest of the quantities are nearly good quantum numbers in the limit of large deformation, hence termed as the asymptotic quantum numbers. Here,  =  + . The quantum numbers  = +1/2 or −1/2, denote the intrinsic spin projection of the particle. Parity is given by the symbol π, which is either + or −, depending on N being even or odd. High-j single-particle orbitals, which are generally the unique-parity orbitals, have special advantage as these have different parity from rest of the orbitals in the shell. High− orbitals generally emerge from the high- j orbitals after the shell is more than half-full. We will consider  ≥ 5/2 orbitals as high− orbitals. Since most of the high-K states have contributions from such high- orbitals, these high-K states may often remain quite pure. From the Figs. 5.2 and 5.3, we can see that the high− proton orbitals of interest in the rare-earth region are: 5/2[532], 5/2[413], 7/2[523], 7/2[404], 5/2[402], 9/2[514], and 11/2[505]. Similarly, the high− neutron orbitals in the

82

5 K -Isomers in Deformed Nuclei

Fig. 5.2 Single-particle proton-energy levels in the rare-earth region by using the modified harmonic-oscillator (MHO) potential for 2 = 0 to 0.28 with 4 = 0, for 4 = 0 to 0.08 with 2 = 0.28, and for 2 = 0.28 to 0.10 with 4 = 0.08. The parameters used are κ = 0.0620 and μ = 0.614. Reprinted with permission from Jain et al. [8], copyright (1990) by American Physical Society

rare-earth region are: 11/2[505], 5/2[642], 5/2[523], 7/2[633], 5/2[512], 7/2[514], 9/2[624], 11/2[615], 9/2[505], and 13/2[606]. Single-particle high− states alone, such as those in odd-A nuclei, may not be so important for creating high-K isomers. When we combine two- or more singleparticle states emerging from a broken pair, the K value rises and the possibility of K -isomer formation becomes larger. We may thus have two-neutron and two-proton configurations giving rise to high-K states in even-even nuclei. One neutron-one proton configurations in odd-odd nuclei constitute the lowest lying 2−qp configurations, and do not require any broken pair. An additional broken pair will lead to 4−qp configurations in odd-odd nuclei. In odd-A nuclei, we may have 3−qp configurations emerging from one broken pair to create high-K states. Since the high− single-particle states are close to Fermi energy once the shell is more than half-full, the resulting high-K states lie low in excitation energy. Higher number of particle configurations will lie further high in energy.

5.2 Deformed Nilsson Model and High-K States

83

Fig. 5.3 Single-particle neutron-energy levels in the rare-earth region by using the modified harmonic-oscillator (MHO) potential for 2 = 0 to 0.28 with 4 = 0, for 4 = 0 to 0.08 with 2 = 0.28, and for 2 = 0.28 to 0.10 with 4 = 0.08. The parameters used are κ = 0.0636 and μ = 0.393. Reprinted with permission from Jain et al. [8], copyright (1990) by American Physical Society

As discussed in Chap. 2, various isomers arising from 2−qp and 3−qp configurations in the high-j shell model levels, give rise to various isomers like 8+ , 10+ , 12+ , 21/2+ , etc. Some of these isomers lie away from the closed shells and are probably deformed K -isomers. As pointed out by Kondev et al. [2], a few combinations of Nilsson orbitals are quite common to spot in various MQP states. For example, it is common to see the combinations 6+ π(5/2+ [402] ⊗ 7/2+ [404]), 8− ν(7/2+ [404] ⊗ 9/2− [514]) etc. appearing in the MQP configurations as the basic building blocks of many isomeric states.

5.2.1 Quasi-particles Since we use the term quasi-particle quite frequently, it is best to introduce it more formally. The pairing interaction and its binding energy (commonly known as the

84

5 K -Isomers in Deformed Nuclei

pairing energy or, the pairing gap) plays a very important role in the excitation energies of the multi-particle excitation. The like-particle pairing energy and its variation with mass number A was briefly mentioned and used in the Figs. 1.7 and 1.8 to show the effect of broken-pairs on the excitation energy of the isomers. One usually treats the pairing interaction in a model [9], which is based on the BCS theory of Superconductivity [10]. The resulting phase transition involves only few nucleons near the Fermi energy [11]. The pairing interaction scatters the particles among the states near the Fermi energy and introduces a diffusion in the occupancy of states. As a result, the sharp cut-off of the Fermi-Dirac type, separating the occupied and unoccupied states, gives way to a smoothly changing diffused occupancy. Therefore, the single-particle states near the Fermi energy are partially occupied and partially unoccupied. As a result, it is the single-particle level density near the Fermi energy which mainly affects the pairing correlations. Thus, a given nucleon may occupy several single-particle states near the Fermi energy in such a way that the sum of the occupancy is unity. Due to the pairing interaction, the simplicity of the single-particle description is lost. This simplicity is restored in the independent quasi-particle description. The quasi-particles so defined in the new scheme do not interact through the selfconsistent field and the pairing  interaction [12, 13]. The BCS approach gives the quasi-particle energy as E i = (i − λ F )2 + 2 , where λ F is the Fermi energy and  is the pairing-gap energy, approximately given by  ≈ √12A MeV. Specific values of  may be calculated from the masses of neighboring odd and even nuclei [14].

5.2.2 High-K MQP States and Isomers Both protons and neutrons can play an important role in the formation of high-K isomers. As an example,  = 7/2 and 9/2 orbitals lie near the proton Fermi-energy in Hf (Z = 72) isotopes for a deformation β2 ≈ 0.25, and may lead to a low-lying K = 7/2 + 9/2 = 8 two-quasiproton states. This leads to the systematic existence of K = 8 isomers in all the even-even Hf-isotopes from 170 Hf to 184 Hf [15, 16]. Similarly, the  = 7/2 and 9/2 orbitals also lie close to the neutron Fermi-level for N = 106 and β2 ≈ 0.25. This can result in another K = 8 two-quasineutron states. Extending these arguments further, the K = 8 from the two-quasiprotons and K = 8 from the two-quasineutrons combine together to form a K = 16 four-quasiparticle state in 178 Hf, which shows up as a 31−year half-life isomer at 2.4 MeV energy, as shown in Fig. 5.4. This isomer exists in a region of well-deformed nuclei where many MQP isomers have been observed and is known to be the longest-lived, MQP isomer known so far. The deformed shell-model calculations can be used to predict many new isomers not seen so far. For example, Walker et al. [1, 17] and Xu et al. [18] highlighted the N = 114 − 116 and β2 ≈ 0.20 region in 186−188 Hf isotopes having  = 9/2 and

5.2 Deformed Nilsson Model and High-K States

85

Fig. 5.4 Partial level scheme of 178 Hf. The experimental data are taken from ENSDF [36]

11/2 neutron orbitals near the Fermi energy. This may lead to the K = 10 isomers in this region, as observed.

5.3 Calculation of MQP States Precise calculation of MQP energies remains an elusive and tedious task as so many factors are involved here. Such calculations require a good knowledge of the nuclear shape, the single-particle energies, the pairing energy, the residual nucleon-nucleon interaction energy to say the least. There could be frequent perturbations from Coriolis interaction, and phase transition/ rotation alignments could be marring the simple picture. Two-quasiparticle (2−qp) energy calculations represent the first step in this direction. 2−qp intrinsic excitations are observed in both the odd-odd and the even-even nuclei. The ground state and other low lying excitations in odd-odd nuclei are of 2−qp nature. However, 2−qp excitations occur above the pairing gap (about 1 MeV) in the even-even nuclei.

86

5 K -Isomers in Deformed Nuclei

Fig. 5.5 Schematic coupling scheme of K + = 1 + 2

Since the Nilsson single-particle states are doubly degenerate in spin projections, the projection of the single-particle angular momenta of the two particles on the symmetry axis, denoted by 1 and 2 , may couple in two ways such that their intrinsic spins are either parallel or, antiparallel. This results in the formation of two states with total projection of the system K + = (1 + 2 ) and K − = (1 − 2 ), as shown in Figs. 5.5 and 5.6. The residual neutron-proton interaction lifts the degeneracy of the K + and K − states in odd-odd nuclei. The residual neutron-neutron or, proton-proton interaction lifts the degeneracy in the case of 2−qp couplings in even-even nuclei. The well known Gallagher-Moszkowski rule [19] decides whether K + or K − will lie lower in energy for odd-odd deformed nuclei. For a neutron-proton pair, the configuration (either K + or K − ) with triplet spin projection lies lower in energy and this is known as the GM rule. It is one of the most successful rules in nuclear physics, with no known violations so far. As an example, there are two possible couplings of ν9/2[624] and π7/2[404] Nilsson orbitals, resulting in K − = |1 − 2 | = 1 and K + = 1 + 2 = 8. K − = 1 will be the lower-lying state because it is the triplet coupling state where both neutron and proton have their spins parallel to each other. One may infer this from the Nilsson quantum numbers. The K + = 8 state lies higher in energy at 176 keV in 180 Ta as it is the singlet coupling state. However, in the case of like-nucleon pairs, as in the 2−qp excitations in even-even nuclei, the reverse is true and the residual interaction favours the state with singlet spin projection configuration, as pointed out by Gallagher [20] in 1962. It is known as the Gallagher rule. In the special case of the K = 0 bands, the n − p residual interaction gives rise to an additional contribution, which is diagonal in nature. This term, commonly known

5.3 Calculation of MQP States

87

Fig. 5.6 Schematic coupling scheme of K − = |1 − 2 |. K − is more rotation aligned than K +

as the Newby shift, has an angular-momentum dependence and causes a displacement of the odd−I levels with respect to the even−I levels within the members of the rotational band based on K = 0 [21]. This term is still not fully understood and is known to have important consequences in the rare-earth [22] as well as the actinide regions [23]. Prediction of 2-qp Isomers in Odd-Odd Nuclei Calculations of neutron-proton 2−qp energies in odd-odd nuclei has been carried out for decades and has met with reasonable success by using the simple approach of combining empirical inputs with theory by Jain et al. [24]. An empirical version of this approach, which combines the residual energy calculations with empirical singleparticle energies, has been used by Sood and collaborators for several decades with good success [25–28]. A theoretical approach to calculate the 2−qp excitations in odd-odd nuclei uses various forms of residual neutron-proton interactions and often leads to reasonably precise information of the 2−qp isomers in odd-odd nuclei [24, 29]. A simple theoretical approach based on a zero-range delta interaction was first used by R. N. Singh [30] and applied very successfully in a number of cases to obtain the level structure of many odd-odd nuclei [25, 26]. An even more simple empirical version of the same approach has been successfully used by P. C. Sood to predict isomers in various odd-odd nuclei. For example, Sood and Gowrishankar recently provided a detailed understanding of the 2−qp isomers in 186 Ta [27] and 254 Md [28]. The 2−qp energy calculations involve three steps: First step is to identify the available single-particle configurations by using the experimental excitation-spectra of the nearest neighbour odd-A isotope/isotone. Odd-A isotope provides the proton level energies and odd-A iso-

88

5 K -Isomers in Deformed Nuclei

tone provides the neutron level energies. These are combined to obtain the 2−qp configurations and their energies, placing the pair of GM levels according to the rule. This allows an easy evaluation of the 2−qp energies by using the formula: E(K :  p , n ) = E 0 + E( p ) + E(n ) + Er ot + E pn Here, E() are the measured level energies of the corresponding Nilsson states in the neighbouring odd-A nuclei. The rotational contribution Er ot and the pninteraction energy contribution E pn leading to GM splitting, are also extracted from the measured values in odd-odd nuclei [27, 28]. Using this approach, they could recently make firm configuration assignments to the two long-lived 10 min and 28 min isomers in 254 Md as J π K = 1− 0 : 1/2− [521]π ⊗ 1/2+ [620]ν and J π K = 3− 3 : 7/2− [514]π ⊗ 1/2+ [620]ν , respectively. Such calculations could be very useful, for example, in the newly observed neutron deficient 244 Md isotope [31, 32], where the existence of a tentative isomer with a half-life of ∼5 ms has been suggested.

5.3.1 Three-Quasiparticle States This picture for 2−qp states was expanded by Kiran Jain and A. K. Jain [33] to obtain a very successful description of 3−qp intrinsic excitation in odd-A nuclei. A major difference from the 2−qp model was the inclusion of pairing energy, which becomes of critical importance as it is necessary to break a pair of protons or neutrons to form a 3−qp excitation. Two kinds of 3-qp states are possible: neutr on − neutr on − neutr on or pr oton − pr oton − pr oton type having identical nucleons, and neutr on − pr oton − pr oton, neutr on − neutr on − pr oton type having two types of nucleons. Coupling of three nucleons leads to four possible intrinsic states with allowed K = 1 ± 2 ± 3 . These four states are split by the residual neutr on − neutr on/ pr oton − pr oton/neutr on − pr oton interaction. Data for 3-qp states was first collected and discussed by Jain et al. and Singh et al. [8, 34]. As it turned out, not a single complete 3-qp quadruplet was known and this still remains a challenge to the experimentalists. Following the 2-qp scheme, the excitation energies of the members of quadruplet can be expressed as the sum of the energies of the three odd-nucleon excitation energies from neighboring nuclei plus the pairing energy required to create the broken pair (which was not present in the odd-odd 2-qp calculations) plus a rotational energy term coming from the band heads and the residual interaction energy term [33]. The model was reasonably successful in reproducing the correct orderings of the quadruplet and also the energies. Besides, it also gave rules like the GM rule for odd-odd nuclei. Jain and Jain [33] proposed generalized GM rules for the 3-qp states for the first time. They proposed two strong rules for pr oton − pr oton − neutr on and neutr on − neutr on − pr oton systems, according to which the state having all the spins as parallel, cannot lie lowest in energy in the quadruplet, and the state having spins of like particles parallel while that of unlike particle anti-parallel must lie the highest in energy. These rules were further expanded by Singh et al. [34] for fixing

5.3 Calculation of MQP States

89

the order of all the four members of a given 3-qp quadruplet. These rules come very handy in making the first guess of the ordering of levels in a given quadruplet.

5.3.2 MQP States An n-particle configuration in a nucleus gives rise to a multiplet having 2n−1 states. The residual n − n/ p − p/n − p interactions split the multiplet into several states, which may be labelled by the quantum number K . Here, K = (1 ± 2 ± 3 ± ...n ) is again a good quantum number for axially-deformed nuclei. However, broken pairs and the Coriolis mixing of rotational bands based on each intrinsic excitation causes a K -mixing, how-so-ever small. Very tedious yet simple calculations of MQP states were carried out by Kiran Jain, P. M. Walker and N. Rowley [35] following the generalization of the 2−qp/3−qp models. The residual interactions were shown to play a key role in this formulation as these caused shifts in the energy levels of the order of few hundred keV. This proved successful in reproducing the ordering of the three 6−qp states in 176 Hf, and also successfully obtained the MQP states in a series of Hf isotopes. These calculations firmly established the critical role of the residual interactions in MQP states. The Hf isotopes near the neutron numbers N = 104 to 108 lie in a region where high- orbitals occupy the space near Fermi energy. This is a region of low level density as N = 104 and N = 108, both appear as gaps in the single-particle spectrum (see Fig. 5.3). As a result high-K MQP states may occur at low excitation energies. When these states decay to lower-lying states, if these are connected by K -forbidden gamma decays only, observation of K -isomers becomes a distinct possibility. It is possible that the lower-lying states are connected by K -forbidden as well as highmultipole transitions. In such a situation, both the “spin” as well as “K ” changes drastically and very long half-lives may arise. Such is the case in the K = 16+ , 31 year isomer observed in 178 Hf (Fig. 5.4). Predictions of yrast traps were made by Aberg quite early [37] without taking into account the residual interaction contributions. However, inclusion of residual interactions proved very important for such calculations. Kiran Jain et al. [38] carried out the Nilsson model calculations with BCS pairing and residual interactions with great success. It enabled them to make predictions for high-K low-lying K = 8 and K = 16 isomers in 178 Hf already observed and K = 10 and K = 18 isomers in neutron-rich 188 Hf. We note that the nuclear shape plays an important role in such calculations and it may change from state to state. Xu et al. [18, 39] considered the effect of changing shapes in their calculations. Configuration dependent shape changes in the MQP states were used to calculate the 2−qp and 4−qp isomeric states in neutron-rich Hf isotopes with good results [40]. However, Hf isotopes, where a large number of K -isomers are concentrated, turn out to be very robust against changes in axial symmetry and shape variation. This explained the success of the calculations of Kiran Jain et al. [38]. Such changes do occur in higher−Z nuclei like Z = 76 Osmium

90

5 K -Isomers in Deformed Nuclei

isotopes [39, 41], where axial asymmetry sets in and triaxial deformation becomes important.

5.4 Some General Features of High-K States It is not necessary that all the high-K states are also isomers. However, an isomer renders some advantages in the measurements. Longer life-times enable many experimental techniques to be applied and measurement of quantities like the magnetic dipole moment, quadrupole moment, and charge radii become possible. Even then such measurements remain few only. Since the K -isomers occur in deformed nuclei, it is possible to have a rotational band built on each of these isomeric state. Such high-K rotational bands based on MQP states may be studied for the quenching of pairing correlations by observing the variation in the moment of inertia and rotation alignment. The phase transition from a BCS state to a broken-pair state while forming the MQP high-K state poses many challenges. It remains an open question, only partially answered that how many quasi-particles are required to reach the complete destruction of super-fluidity. The unpaired nucleons block the Nilsson states which were paired and diminish the pairing correlations as understood in the BCS approach [42]. The pair-blocking should lead to transition from paired to unpaired rigid body motion as the number of broken-pairs increases. Such a phase transition may also be evident as the angular momentum rises along a rotational band due to the Coriolis Antipairing effect at higher rotation. The Coriolis effect causes the high-j low- pairs to break first [43]. The broken pairs then align with the rotation axis and a transfer of angular-momentum from collective to single-particle mode takes place in steps. This also causes a sudden rise in the moment of inertia, which approaches the rigid body value. The rise in the angular-momentum of a given isomer may, therefore, be both collective and quasi-particle in nature. Approximately 400 MQP isomers are now known in the deformed regions of A ≈ 100 − 190, with a significant number residing in the A = 180 mass region [2, 44]. A significant number of these isomeric states are known to support a rotational band. Lizarazo et al. [45] has recently reported new information on the shape evolution of the very neutron-rich 92,94 Se nuclei from an isomer-decay spectroscopy experiment at RIKEN. The isomeric states are interpreted as high-K , oblate deformed quasineutron states in 94 Se. Following this, 94 Se has now become the lowest-mass neutron-rich K -isomeric nucleus and is also the very first observed oblate K -isomer. This opens up the possibility for a new region of K -isomers at low Z with oblate deformed neutron orbitals. Interestingly, the same neutron orbitals are also involved in the well known K -isomeric region around Hf isotopes but for prolate deformation. Before this recent observation, the lightest K -isomer was seen in 99 Y [46] at an excitation energy of 2.14 MeV, having spin-parity 17/2+ and half-life 8.0 µs. It is interesting that this 3−qp isomer decays to the levels of a K = 11/2+ rotational band with an

5.4 Some General Features of High-K States

91

M1 transition. But most of the properties of this isomer are known only tentatively and it needs to be explored further. A confirmed case of highest MQP high-K isomer is known in 175 Hf. The 9−qp K = 57/2− isomer has a half-life of 22 ns and excitation energy of 7.45 MeV. Kondev et al. [2] lists additional 9−qp and 10−qp isomers in 175 Hf (K = 61/2) and 178 W (K = 29 and 34) respectively, but their spins and half-lives (less than 1-2 ns) are tentatively known. As noted already, higher number of broken pairs should lead to depleted pairing correlations and take the nucleus closer to a rigid body situation. Since the Coriolis forces operate in the rotating frame, the Coriolis anti-pairing (CAP) effect becomes active and is responsible for breaking the paired nucleons and aligning them along the rotation axis much like a rotating top. The CAP effect is most intense in the pair of nucleons occupying high-j and low- orbitals. For example, the 1/2[541] proton orbital emanating from h9/2 shell model orbital has j = 9/2 and  = 1/2, and lies close to the Fermi level for Z ≈ 74, i.e. tungsten nuclei. MQP states having this orbital in their configuration are likely to experience highest CAP effect and a very large perturbation in the associated rotational bands. The highly perturbed rotational bands cannot be used directly to extract the moment of inertia values for a direct comparison with the rigid body value. It is necessary to correct for the Coriolis perturbations first. As pointed out by Walker and Dracoulis [1], the K = 30, 8−qp rotational band in 178 W has a configuration devoid of this orbital and, therefore, expected to experience least Coriolis perturbation. It is, therefore, possible to compare the moment of inertia extracted from the ground band and the K = 30 band with the rigid rotor value. The ground band 2 → 0 transition yields a moment of inertia value of J = 282 MeV−1 , while the K = 30 band 32 → 30 transition yields a value of J = 562 MeV−1 . This is only 2/3 of the rigid-body value J = 852 MeV−1 . In other words, we have not reached the elusive rigid body value even after four broken pairs and high-spin value of 30. There could be so many reasons for it. Some of the reasons could be the shell structure, residual pairing due to the rest of the pairs, tilted axis rotation due to large K -value at the band head having equally large angular momentum. Frauendorf et al. [47, 48] explored the last reason and were able to reproduce the full empirical value with no pairing correlations in the tilted axis cranking model. Dynamic pairing correlations may also play an important role in the high-K bands as shown in the case of 178 W [49].

5.4.1

K -Isomer in 250 No

In a very interesting study, a rare example of an isomer having half-life longer than the respective ground state was found in the heavy mass nucleus, 250 No. This isomer is now identified as a K -isomer [50]. While the ground state lives for only 3.8 µs, the isomeric state lives for 34.9 µs. There are only a few cases like this in the transFermium region (Z > 100). Other cases where the isomeric half-life is more than

92

5 K -Isomers in Deformed Nuclei

that of the ground state are: 254 Rf, 256 Es, and 270 Ds. This phenomenon of half-life inversion gives a new hope of finding a K -isomer which lives much longer than that of the ground state in the super-heavy nuclei. In fact the spontaneous fission half-life of the K -isomer in 250 No is estimated to fall by a factor of five in the ground state due to a lower barrier height. This is an open problem for both the theorists as well as experimentalists.

5.4.2

K -Mixing

If K is a good quantum number and there is no K -mixing, then a K -isomer should not decay to the lower-lying states having different K -values as it would violate the goodness of the K quantum number. However, a weakening of the K -hindrance is observed in all cases. It is a clear signature of some K -mixing among the initial and final states. We have already defined the degree of forbiddenness as ν = K − λ, where λ is the multipolarity of the transition. Goodness of the K quantum number is measured by defining f ν , the hindrance per degree of K forbiddenness or, the γ γ W W ), where T1/2 and T1/2 reduced hindrance. Hindrance itself is defined a F = (T1/2 /T1/2 are γ-decay and Weisskopf half-lives, respectively. The reduced hindrance is now defined in terms of F as f ν = F 1/ν . The hindrance factor F carries the dependence of half-life on the energy and multi-polarity of the transition. The reduced hindrancefactor carries the nuclear structure information of the hindrance on the degree of forbiddenness. Typical values of f ν could be around 100 according to Lobner [51]. It implies that the transition should be hindered by an additional factor of 100 for each additional unit rise of K . The tabulation of K -isomers [2] lists the f ν values of the K -isomers in the mass region A>100. Lower value of f ν implies a faster transition and larger value implies a slower transition. We can see from the tabulation that the value of f ν vary in a very broad range and most of the values fall in the range of 30–200; however, a significant variation in f ν is not uncommon. Very small f ν values clearly signal the presence of larger K -mixing in the decays. In fact only 26 K -isomers have half-life values in ms and above. Nearly all the longest-lived isomers (11 of them) fall in the island of Hf (Z = 72) and Ta (Z = 73) isotopes. Similarly, about six longer-lived isomers are observed in the island of Z = 94 − 102 nuclei. However, no K -isomer has been observed so far in the Cf (Z = 98) isotopes. All the longer lived K -isomers in the trans-actinides have a 2−qp configuration in contrast to the island of K -isomers in the rare-earth region which are mostly 4−qp states. Exploration of MQP K -isomers in the actinides and super-heavy region is, therefore, a very attractive and challenging option for the future experimental program. Kondev et al. [2] have provided a detailed discussion of the various factors that affect the hindrance factors. Some of the most prominent factors which lead to a weakening of K -hindrance are: Coriolis mixing of bands, γ-softness and tunneling, and mixing due to high level density. An early study [52] observed that there is an

5.4 Some General Features of High-K States

93

overall trend of decreasing hindrance with rising excitation energy. This could be due to the rising level density at high excitation-energy but other factors cannot be ruled out. Walker et al. [52] studied the 4−qp, K π = 12+ states in 172 Hf and 178 W nuclei and which decay to their respective ground state bands having K = 0. A correlation between the decreasing values of reduced hindrance and rising excitation energy in Hf, W, and Os isotopes, was established [3]. By a calculation based on statistical estimates of level density, this was largely attributed to higher statistical mixing at higher excitation energies due to rising level density. Coriolis interaction may also cause a mixing of levels having same spin but belonging to different bands having K values differing by one i.e. K = ±1. Closer the level spacing, larger is the mixing. The effect of Coriolis mixing on the reduced hindrance was demonstrated by Dracoulis et al. [53] by a particle-rotor model calculation to obtain the mixing amplitudes of the high-j components in the wave-function. It was noticed that the observed reduced hindrances (≈100) of the decay of 8− isomers in the N = 106 isotones of Er to Pb nuclei are quite low, considering the 7-times forbidden nature of the E1 decays. The E1 transitions are known to be hindered already when compared with the Weisskopf single-particle estimates. The 9/2+ [624] Nilsson orbital belonging to the i 13/2 shell model orbital can interact with other members of the i 13/2 orbital having  = 1/2 to  = 13/2. Mixing amplitudes of all the active components may be extracted and used to obtain the reduced hindrance-value which fits the observed hindrances. This exercise yielded the so called “unperturbed” values of the reduced hindrance, which are plotted in the Fig. 7 of Kondev et al. [2]. The corrected reduced hindrances rise by a factor of almost 100 in line with the degree of forbiddenness of the E1 decays. Nuclei in the mass region A ≈ 180, such as the isotopes of W, Os and Pt isotopes, are known to display gamma-softness which involves a deviation from the axial symmetry in deformation. The triaxial shapes introduce a K = 2 mixing among the states of such nuclei and K does not remain a good quantum number. The K mixing caused by the gamma-fluctuations entails a reduction in the K -hindrances resulting in low f ν values. There are many examples of such low hindrance factors in nuclei such as 188,190 W and 190,191,192 Os isotopes [2]. Another possibility is a statistical mixing of high-K isomeric states with lowK states lying in the region above the yrast states where the level density is high. The high-K isomers in the non-yrast region have a MQP structure and are observed to display a small reduced hindrance factor f ν < 5 for direct decays from the isomeric states to the K ≈ 0 levels bypassing the intermediate-K levels [52]. Even in a nucleus like 164 Er, a non-yrast 4−qp K π = 12+ K -isomer was observed which decays directly to the ground band with a hindrance-factor as low as those found in heavier nuclei [54]. It was concluded that the higher level-density is a key variable responsible for the K -mixing which causes the fall in hindrance. Swan et al. [54] had also shown (in Fig. 5 of their paper) the fall in reduced hindrance as a function of the increasing net energy above the yrast line, which is directly linked to the increasing level density.

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5 K -Isomers in Deformed Nuclei

However, there is another aspect, which is random in nature, i.e. the effect of chance near-degeneracies between specific pairs of states with the same spin and parity. 174 Lu is a good example in this regard [55]. The K π = 13+ , 280 ns and 4−qp isomer in the doubly odd 174 Lu decays by a K -forbidden transition to the 0+ band. This decay was explained by the occurrence of an extrapolated 13+ member of the 0+ band, whose energy nearly coincides with the 13+ isomer. In the two-level mixing scenario, the mixing matrix elements (∼eV) and wave functions (∼99.999% purity) can be determined for the two states, and it is evident that small reduced hindrances can arise from very tiny amounts of K -mixing, i.e. the half-lives are extremely sensitive to small amounts of K -mixing. It would be interesting to search for the 13+ member of the 0+ band and verify the assumption.

5.4.3

K -Isomeric Rotational Band

A high-K rotational band is a regularly spaced sequence of levels with energies approximately proportional to I (I + 1), connected by I = 1 and I = 2 γ-ray transitions, and with an isomeric band-head. The K -value is usually taken to be the spin of the band-head, and the Nilsson-model configuration assignment is required to have a consistent K -value and g-factor. These model calculated values are then compared with the measured values as obtained from the in-band γ-ray branching ratios [56], or from the band-head magnetic moment [57]. Not all the rotational K band heads are isomeric because of the availability of lower lying states with similar K -value and the resulting frequent decay. Matching K -values, excitation energies, g-factors and alignments with model calculations can provide confidence in the overall structure interpretation and can provide a unique configuration assignment. One also tries to keep the degree of rotation-alignment consistent with the assigned MQP configuration. In real life, the situation is not that simple leading to some ambiguities which may arise due to bandmixing, rotational-alignment effects, or deviation from the I (I + 1) energy spacing structure. For example, in 176 Hf, Iπ = 15+ and 16+ states were initially given I = K band-head status and separate configuration assignments [58], due to the lowestenergy 186 keV I = 1 transition. However, it was subsequently shown that the K = 16 assignment was inappropriate [59] and the 186 keV transition is of low energy due to band mixing. In spite of the difficulties, the level of understanding on comparing the theory and experiment, can further provide a good basis to determine K -isomer decay rates.

5.5 Theoretical Treatments Used for K -Isomers We have already pointed out that the Nilsson model [60] continues to be widely used for single-particle energy levels with deformation for the Z ≈ 72 and N ≈ 108 regions; however, Wood-Saxon model [61, 62] is more realistic. These model solu-

5.5 Theoretical Treatments Used for K -Isomers

95

tions are then corrected for pairing correlations using the well known BCS pairing [9, 42], or Lipkin-Nogami treatment [63–65]. In addition to this, one needs to take care of the blocking effect due to the block in the scattering of pairs into the singly-occupied quasiparticle orbitals [38, 66], and residual interactions between quasiparticles due to relative orientations of spins, so that the obtained energies start to agree with the known experimental numbers [19, 20]. The Nilsson model along with pairing, blocking and residual interactions [38] serve as a great tool in understanding the energies and configurations of MQPs in the well deformed nuclei. Such calculations are done at a fixed shape for a given nuclide, which are usually adopted from ADNDT 1995 mass table of Moller et al. [67]. Since the nuclear shape is configuration dependent, the configuration-constrained MQP energy surface calculations are needed. One of the famous prescription is to extend the Nilsson-Strutinsky approach [68] where the liquid-drop energy is modified as per the level density at the Fermi surface so that the minimum in the total energy can be obtained. Using the configuration-constraints, Nilsson quantum numbers at different deformations can be traced through the level crossings. Pairing correlation using Lipkin-Nogami approach not only avoids the spurious states of BCS pairing but also takes care of the blocking effect by simply removing the singly-occupied orbitals from the calculations. The potential energy surface (PES) calculations are usually done in the quadrupole (β2 , γ) space with the energy further minimised at each point with respect to the hexadecapole (β4 ) deformation, since these shape parameters are proved to be the most important ones. The realistic pairing strength as given by parameter G also influences the quasiparticle energies in a significant way [39]. The tuning of pairing strength can also be done by fitting the experimental odd-even mass differences which brings the calculated energies quite closer to the experimental data in various mass regions [69]. On a similar ground, the Wood-Saxon-Strutinksy PES calculations are also performed and explained the understanding of MQP states in the well deformed nuclei [18, 70]. These calculations could also handle the soft and shape-transitional nuclei by calculating the energies, shapes, charge-radii and configurations [18, 40, 70–79]. Furthermore, high-K states are also sensitive to the way they respond to the rotation, which is usually taken care by cranking approximation such as totalRouthian-surface (TRS) calculations [62]. TRS method is based on the energy calculations in deformation space at each given cranking frequency to determine the deformation of the rotational states. The pairing-deformation self-consistent TRS method [80] is quite successful in explaining the rotational-alignment and pairing correlations. An exact particle-number conserving (PNC) method [81–83] is then used to obtain the converged solutions in pairing correlations which overcomes the nonconvergence problem faced because of non-conservation of particle number in Lipkin-Nogami approach. This method was first developed by Zeng et al. [83] at fixed deformation with axial symmetry, using the Nilsson potential. The generalisation to variable deformation with triaxially deformed shapes was then achieved by Fu et al. [84] using the Woods-Saxon potential.

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5 K -Isomers in Deformed Nuclei

These configuration-constrained PES (or TRS) calculations require adiabatic blocking of the configuration orbitals in the deformation space (and the cranking frequency range), where the specific single-particle orbitals are kept singly-occupied while changing the deformation (and rotational frequency). The orbitals are usually traced by using the average Nilsson quantum numbers. The deformation evolution with changing configuration and rotational frequency plays a crucial role in explaining the experimental observables [84–86]. TRS approach is hence constructed within a rotational frequency basis whereas the real nuclei do conserve angular momentum. To avoid this ambiguity, one can rely on the projected shell model of Hara and Sun [87] with Nilsson potential and angular momentum projection. Projected Shell Model Calculations The half-lives of K -isomers critically depend upon the degree of K -violation due to K -mixing. Yang Sun [88] proposed a method based on the Projected Shell Model (PSM) [87] to describe the K -isomers. In general, the PSM starts with the deformed Nilsson single-particle states with BCS pairing, which defines a set of deformed MQP states [89, 90]. The shell model basis is constructed by considering a few quasiparticle orbitals near the Fermi energy and performing angular momentum projection for the chosen configuration. With such projected MQP basis states, the PSM can be used to describe the rotational bands built upon MQP excitations up to high-spin states. Sun [88] extended this model by transforming these states from the body-fixed frame to the laboratory frame and mixing them in the laboratory frame through the two-body residual interactions. The angular momentum projected MQP states may be written as  I,σ >= f κI,σ PˆMI K |φκ > (5.1) | M κ,K ≤I

=



f κI,σ PˆMI K κ |φκ >

(5.2)

κ

where the index σ labels the states with same angular momentum and κ defines the basis states. PˆMI K is the angular-momentum projection operator and the coefficients f κI,σ denote the weights of the basis states. These weights can be determined by diagonalization of the Hamiltonian leading to the eigenvalue equation for a given I ,



(Hκκ − E σ Nκκ ) f κσ = 0

(5.3)

κ

where the Hamiltonian and norm matrix elements are given by Hκκ = < φκ | Hˆ PˆKI κ K |φ κ > κ

Nκκ =


(5.4) (5.5)

5.5 Theoretical Treatments Used for K -Isomers

97

The angular-momentum projection operator on a MQP state |φκ > generates a band having a sequence of angular momenta. The corresponding energies of the band members may be defined as E κI = =

Hκκ Nκκ

(5.6)

< φκ | Hˆ PˆKI κ K κ |φκ > < φκ | PˆKI K |φκ > κ

(5.7)

κ

As defined in the previous sections, the dominant term in the energy of a MQP state may be taken as the sum of the component single-QP states. Y. Sun [88] modified this quantity in two ways. Firstly, the band energy is actually the correction due to angular momentum projection and two-body interactions. This takes care of the quantum mechanical couplings between the rotating body and the quasi-particles. Second, the consequent rotational states are mixed in K -values while solving the eigen-value equation for weight factors. The energies are hence further modified by such configuration mixing. In principle, the Hamiltonian in PSM uses the two-body interactions like pairing plus quadrupole-quadrupole with inclusion of quadrupole-pairing term,  1  ˆ† ˆ Q μ Q μ − G M Pˆ † Pˆ − G Q Pˆμ† Pˆμ Hˆ = Hˆ 0 − χ 2 μ μ

(5.8)

The strength of the quadrupole-quadrupole force χ is determined in a self consistent way with  the quarupole deformation 2 . The monopole-pairing force constants are G M = G 1 ∓ G 2 N −Z A−1 with “−” for neutrons and “+” for protons. This corA

rectly reproduces the odd-even mass differences in a given mass region if G 1 and G 2 are properly chosen. Commonly, the strength for quadrupole pairing G Q is assumed to be proportional to 0.16G M . Y.Sun [88] (and references therein) applied this model to study the well known K -isomer in 178 Hf with a half-life of 31 y. He found a decent agreement between the calculated and experimental energy level schemes. Using this model, he could not only generate the mixed wave functions but also calculate the electromagnetic transition probabilities between various states. He and his group then applied the model in various regions to study the K -isomers and reported a reasonable agreement between model results and experimental observations. Particularly, many nuclei of A ≈ 180 mass region support axially-symmetric shapes (where K behaves as an approximately good quantum number) and follow the extended PSM quite well. However, this model needs further improvements to take care of the highly K -forbidden transitions.

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5 K -Isomers in Deformed Nuclei

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35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48.

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Chapter 6

Shape and Fission Isomers

Keywords Double-hump barrier · Shell corrections · Fission isomers · Shape isomers · 0+ isomers · E0 → E0 transitions · Examples It is now well known that a nucleus may have various kinds of shapes, from simple spherical to complex deformed, depending upon the N , Z configuration, excitation energy, angular momentum etc. These distinct shapes arise because of the additional minima in the potential energy of the nucleus besides the ground state minimum, when plotted against the various degrees of freedom. When a nucleus gets trapped in a secondary minimum at a distinct shape different than its ground state, then a gamma decay from the higher lying state in the secondary minimum to the ground state involves a significant shape change. This hinders the decay and may give rise to a longer half-life of the state in the secondary minimum. Such isomeric states are known as shape isomers, as schematically shown in Fig. 6.1. A very well known example of shape isomer is the 0+ isomer in 72 Kr, a self conjugate nucleus (N = Z ), with a half-life of 26.3 ns. This isomer exists due to the shape hindered 0+ → 0+ transition, as shown in Fig. 1.2. Another important class of shape isomers, known as fission isomers, is found in heavier trans-actinides, which can also decay by spontaneous fission. The fission isomers exist at super-deformed shapes and also represent the first examples of superdeformation at very low spins. The collection of data on super-deformed bands also lists the fission isomers [1]. The Atlas of Isomers also contains these isomers but without an explicit classification [2].

6.1 Double-Hump Barrier and the Shell Corrections Exploration of the limits of nuclear stability, in terms of Z and N , has been an important area of research for decades, both theoretically as well as experimentally. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. K. Jain et al., Nuclear Isomers, https://doi.org/10.1007/978-3-030-78675-5_6

101

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Fig. 6.1 A schematic representation of shape and fission isomers in the nuclear potential well. Note that the liquid drop model cannot explain the origin of these isomers. The second minimum in the potential well is usually attributed to the shell corrections and so is the existence of shape/fission isomers. The second minimum is surrounded by two saddle points (the maxima). The scission point occurs at extreme right

Many theoretical approaches have been developed for predicting the highest possible atomic number (Z ), which may exist in nature or laboratory. The Potential Energy Surface (PES) calculations by using various approaches have been instrumental in these studies. Nearly all such calculations predict at least two shape minima in the potential energy landscape of trans-actinides. However, more than two minima are not ruled out. The existence of a pronounced secondary minimum in the actinide region besides the ground state minimum may be considered as a landmark finding as it explained the discovery of the fission isomers and a large number of features associated with them [3]. The quantal shell fluctuations culminating in strong shell effects, which provide extra binding, have been extensively studied by using the Strutinsky method of extracting the shell effects [4, 5]. In brief, the shell effects represent the quantum contribution to the nuclear potential energy that is missed in the potential energy of the classical liquid drop model (LDM) [6]. In the Strutinsky approach [4, 5], it is assumed that the general trend followed by the binding energy is that of the liquid drop. Shell effects and other quantum effects give rise to small correction to the LDM energy. The shell correction δU is defined as the difference, δU = U − U˜ , where U is the sum of the single-particle energies in an average potential and U˜ is the energy of an averaged out uniform level distribution, obtained by using one of the several prescriptions available in the literature [3]. The total potential energy is the sum of the LDM energy, the shell correction energy, and the residual interaction contributions like the pairing. All the contributions depend on deformation. The minima of this sum define the equilibrium shapes of the nucleus. For spherical shape, δU has a deep minimum for the magic number of nucleons and is maximum in the middle of the shell. We may conclude that the

6.1 Double-Hump Barrier and the Shell Corrections

103

magic nuclei lie in the regions having the smallest single-particle level density in the neighbourhood. This concept of magicity may be extended to the deformed nuclei also. It leads to the concepts of deformed magic numbers and stability for non-spherical shapes [7– 9]. It has been found that for near-magic nuclei, the ground state is spherical ( = 0.0) and the secondary minimum lies near  ≈ 0.4, where  is the axial deformation. For nucleon numbers away from the spherical magic numbers, the ground state is found at  ≈ 0.2 − 0.25 and the secondary minimum at  ≈ 0.4 − 0.7 depending on the nucleon number configuration. As we move along the line of stability from low- A to high-A nuclei, the LDM saddle region moves from large- to small-. For the beta stable rare-earth region, the LDM saddle points are located beyond  ≈ 1.0 and for the actinide region these are located near  ≈ 0.5 − 0.9. It is, therefore, obvious that the double hump structure in the fission barrier becomes prominent in the actinide region. A similar favourable region is that of neutron-deficient nuclei near Z ≈ 82. It is, therefore, expected that fission isomers will be found in these nuclei. The experimental observations coincide with these expectations. In 1990s, Moller et al. used FRLDM macroscopic-microscopic model [10] for calculating potential-energy surfaces of these nuclei. In this model, the macroscopic part was obtained by finite-range liquid drop model whose parameters were fixed by consideration of fission-barrier heights in addition to the nuclear masses. The microscopic corrections were obtained by single-particle energy levels using a foldedYukawa single-particle potential [11] following the Strutinsky method [4, 5]. The pairing corrections were calculated by the Lipkin-Nogami approximation [12–15]. They plotted the nuclear potential-energy surfaces as a function of spheroidal deformations 2 , axial asymmetric deformation γ, and hexadecapole deformations 4 . In a further refinement to these calculations, Moller et al. [16] reported the highlights of the global calculation of nuclear shape isomers in the full nuclear landscape. The theoretical details and full results for a total 7206 nuclei have been presented in 2012 by Moller et al. [17], where they portray the global potential-energy surface calculations from A = 31 to A = 290. They catalogued the deformations and energies of all minima deeper than 0.2 MeV and of all the saddle configurations between all pairs of minima. The respective results for the shape isomer calculations were given in Table 1 of their 2012 paper [17] along with more than 100 graphs for 1224 nuclei. However, the page-size graphs of each of the 7206 nuclei is available for download from their web site [18]. A very typical example of such calculations is that of 154 Sm, where only one minimum exists at the prolate ground state 2 = 0.25 and γ = 0.0◦ with energy E = 0.021 MeV. However, there are two minima obtained for 98 Sr; first one is the prolate minimum at 2 = 0.325 and γ = 0.0◦ with energy E = 2.225 MeV, and second one is located at 2 = 0.300 and γ = 60.0◦ with energy E = 4.205 MeV. Among these, the first one is the deeper minimum, which means this one belongs to the ground-state. The second one is, therefore, the shape isomeric state. The saddle point, which is separating these two minima, is located at 2 = 0.275 and γ = 40.0◦ with E = 4.738 MeV. It is also possible to have the potential energy surface with three minima. A surprising experimental observation of three co-existing shapes in 186 Pb has been

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reported along with the calculated potential-energy surface supporting three minima by Andreyev et al. [19]. The results from Moller’s table [17] for 186 Pb are found to be in similar agreement. However, many limitations in assigning the nuclear configurations still exist and require more efforts in this direction. Despite the limited information, it is clear to an extent that the excited states in second well may have longer half-lives than the respective second-well minima [20, 21]. This extra stability can be attributed to the high-K quantum number, which may be of great importance in the studies of superheavy nuclei. The extended half-lives of such high-K , second-well isomers can be understood by the configuration-constrained, potential energy surface calculations [21]. For a more first principle description, one may go beyond the mean-field to incorporate the pure microscopic description and account the mixing between wave functions localized near shape configurations. Several efforts [22–29] have been made in this direction, where the two-body effective interaction or density functional theory has been used for a microscopic description of the nuclei. Unfortunately, such models do not lead to any improvements and the results deviate significantly from the experimental data. We, therefore, still rely on the methods similar to the one used by Moller et al. [17] in the global calculation of nuclear shape isomers.

6.2 Discovery of the Fission Isomers The first spontaneously fissioning isomer was observed rather accidentally in 242 Am by Polikanov et al. [30]. An attempt was being made to study the properties of short lived heavy nuclei. A beam of heavy ions (16 O or, 22 Ne) from the Dubna Cyclotron was bombarded on a thin target of 238 U. The reaction products fled at high momentum from the target and were collected by a collector, which was rotating at a high speed, passing through two ionization chambers capable of detecting fission fragments very efficiently. It was ensured that only those pulses are recorded which represent the fragments from 238 U. An analysis of the pulse-spectrum and the range of the fragments confirmed that delayed fission was taking place. The half-life of the delayed activity was estimated to be of the order of (1 − 2) × 10−2 sec. This half-life is now measured to be 14 ms [31] with a 100 percent fission branching. 242 Am is known to exist in two states: the ground state and an isomeric state at an excitation of 48 keV. The ground state decays by β-decay or, K electron-capture and has a half-life of 16 h. The isomeric state has a half-life of 141 y and spontaneous fission (SF) partial half-life of 9.5 × 1011 y. On the other hand, the newly observed isomer is now known to have an excitation energy of 2200 keV, a SF half-life of 14 ms, and a 100% SF decay. This implies a decrease in the half-life of the order of 1020 . If a gamma decay hindrance is imagined to cause hindrance of this order, it would require a spin of tens of . The spin is now measured to be (2+ , 3− ). It was, therefore, a big puzzle to explain the short-lived fission activity. Even large excitation energy, implying a shorter barrier, could not explain it.

6.2 Discovery of the Fission Isomers

105

100

Fission isomers

Proton Number (Z)

98

96

94

92

90

140

142

144

146

148

150

152

Neutron Number (N) Fig. 6.2 The occurrence of fission isomers in nuclear landscape. All cases belong to the actinide region. The detailed properties can be found in Table 6.1

It is now known that such nuclei may get captured in the second minimum at a super-deformed shape. The captured nucleus may further decay via either spontaneous fission or gamma decay to the first minimum. The size and shape of the barrier decides the half-life of the fission isomeric state. B. Singh et al. has listed 47 fission isomers, along with their spectroscopic information in 2002 [1]. Since then, there is no new discovery along this line. We have summarized the most probable cases of fission isomers in the Table 6.1. The longest-lived fission isomer is of course 242 Am with a half-life of 14 ms. In comparison to the 2002 fission isomer table, the present list has an additional case in 246 Cf with 45 ns half-life and %S F ≤ 100 decay, reasonably fitting into the characteristics of fission isomers [2]. Also, the half-lives in the table have been updated in most of the cases and range from ps to ms, and adopted from the Atlas [2] and ENSDF [32] data base. Figure 6.2 represents the occurrence of fission isomers in the nuclear landscape. The (N , Z ) region of interest is very tightly located in the range 90 ≤ Z ≤ 98 and 141 ≤ N ≤ 151. There are a total of 48 fission isomers shown in Fig. 6.2, while there are two fission isomers in a single nucleus for 11 cases, resulting in 37 data points in the figure. The longest chain of fission isomers belong to the even Z = 94, Pu isotopes and the odd Z = 95, Am isotopes.

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Table 6.1 The list of isomers decaying via spontaneous fission (SF) decay mode. This has been prepared from our updated collection of isomers i.e. Atlas of Nuclear Isomers, to be published [2]. The data for 238 U and 244 Cm have been taken from [1] A S.N. Jπ T1/2 Decay Z XN 1. 2. 3.

233 Th 143 90 235 Pu 141 94 236 U 144 92

4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

236 Pu 142 94 236 Pu 142 94 237 Np 144 93 237 Pu 143 94 237 Pu 143 94 237 Am 142 95 238 U 146 92 238 U 146 92 238 Np 145 93 238 Pu 144 94 238 Pu 144 94 238 Am 143 95 239 Pu 145 94 239 Pu 145 94

(0+ )

18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28.

239 Am 144 95 240 Pu 146 94 240 Am 145 95 240 Cm 144 96 241 Pu 147 94 241 Pu 147 94 241 Am 146 95 241 Cm 145 96 242 Pu 148 94 242 Pu 148 94 242 Am 147 95

(7/2+ ) (0+ )

29. 30. 31. 32. 33. 34. 35.

242 Cm 146 96 242 Cm 146 96 242 Bk 145 97 242 Bk 145 97 243 Pu 149 94 243 Am 148 95 243 Cm 147 96

(0+ )

0+

(0+ ) (5/2+ ) (9/2− )

(2+ , 3− )

1–100 ns 25(5) ns 120(2) ns 40(15) ps 34(8) ns 45(5) ns 97(4) ns 1.1(1) µs 5(2) ns 280(6) ns >1 ns 112(39) ns 0.6(2) ns 6.0(15) ns 35 µs 7.5(10) µs 2.6(+40 − 12) ns 163(12) ns 3.7(3) ns 0.94(4) ms 55(12) ns 21(3) µs 32(5) ns 1.2(3) µs 15.3(10) ns 3.5(6) ns 28 ns 14.0(10) ms 40(15) ps 180(70) ns 9.5(20) ns 600(100) ns 46(13) ns 5.5(5) µs 42(6) ns

%S F ≈ 100 %S F ≤ 100 %I T = 87(6), %S F = 13(6), %α < 10 %S F ≤ 100 %S F ≤ 100 %S F ≤ 100 %S F > 0 %S F > 0 %S F > 0 %I T = 97.4(4), %S F = 2.6(4) %S F ≤ 100 %S F ≤ 100 %S F ≤ 100 %S F ≤ 100 %S F ≤ 100 %S F ≤ 100 %S F ≤ 100 %S F ≤ 100 %S F > 0 %S F ≤ 100 %S F ≈ 100 %S F = 100 %S F = 100 %S F = 100 %S F = 100 %S F ≤ 100 %S F ≤ 100 %S F ≈ 100 %I T > 0.0?, %α > 0.0? %S F ≤ 100 %S F =?, %I T =? %S F ≤ 100 %S F ≤ 100 %S F = 100 %S F ≤ 100 %S F ≤ 100 (Continued)

6.3 Additional Features of Fission Isomers Table 6.1 (Continued) A S.N. Z XN 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48.

243 Bk 146 97 244 Pu 150 94 244 Am 149 95 244 Am 149 95 244 Cm 148 96 244 Cm 148 96 244 Bk 147 97 245 Pu 151 94 245 Am 150 95 245 Cm 149 96 245 Bk 148 97 246 Am 151 95 246 Cf 148 98

Jπ

107

T1/2

Decay

5 ns (?) 0.40(10) ns 0.90(15) ms ≈6.5 µs 500 ns 820(60) ns 90(30) ns 640(60) ns 13.2(18) ns 2(1) ns 73(10) µs 45(10) ns

%S F %S F %S F %S F %S F %S F %S F %S F %S F %S F %S F %S F %S F

≤ 100 ≤ 100 ≤ 100 ≤ 100 ≤ 100 ≤ 100 ≤ 100 ≤ 100 ≤ 100 ≤ 100 = 100 ≤ 100 ≤ 100

6.3 Additional Features of Fission Isomers A few nuclei have two fission isomers known and few of them also have a rotational band built upon the isomeric state. Besides this, many new features related to the unique structure of the fission isomers have been observed [3]. These features arise because of the existence of the second well. One of such features is the existence of intermediate group structure, a very interesting phenomenon, which was observed in the fission cross-section with cold neutrons. The first such observation was reported in 1968 [33]. Interestingly, this experiment was carried out on 237 Np. Only two fission isomers have been observed in Np isotopes after many efforts. This experiment gave the first indication that a second well exists in Np isotopes also. In this experiment, Fubini et al. [33] bombarded slow neutrons on 237 Np and measured the total as well as the fission cross-sections. The behaviour of the total cross-section was similar to that in other odd-mass A = 237 nuclei. But the behaviour of the fission cross-section was very much different as it exhibited many high peaks, each peak consisting of several resonances, at definite energies. This phenomenon was readily understood in terms of the double-hump fission barrier. To understand the intermediate group structure, consider the nucleus to be located in the ground state of the first well. Spontaneous fission proceeds by barrier penetration of the fission barrier. If there was no second minimum, we would get resonances due to the energy levels of the compound nucleus formed in the neutron-capture. However, the existence of a second well brings in another possibility. The ground state of the second well lies at least 2–3 MeV higher than the ground state of the first well. Also, the captured neutron brings in at least 5 MeV of excitation energy. This leads to the shape isomeric state being formed at least 2–3 MeV above the ground

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6 Shape and Fission Isomers

state of the second well. Whenever the excitation levels in the second well and the first well coincide, there will be an enhanced barrier penetration. Since the second well is shallower than the first well, its levels are spaced further apart than those of the first well. Therefore, there will be a fine structure in the cross-section due to the first well, superimposed by the structure due to the larger spacing in the second well. Mignaco et al. [34] carried out subthreshold fission measurements of 240 Pu by using a high resolution time of flight spectrometer. A strong grouping of resonance fission cross-section was observed. The mean spacing between the various intermediate groups of resonances was observed to be 650 eV, while the compound level spacing was found to be 14.9 eV only. The ratio of the two spacing enabled the authors to estimate the energy of the isomeric minimum to be 2.1 MeV. This phenomenon, therefore, provides us an important tool to measure the energy of the second minimum.

6.4 The Low-Lying 0+ Shape Isomers Besides the fission isomers, the most common example of shape isomers in eveneven nuclei are the low lying 0+ states in addition to the ground 0+ states. In potential energy surface calculations, the two states will appear as the two minima with different shapes. However, all the low lying 0+ states are not shape isomers as we discuss now. There are 23 cases of low-lying 0+ isomeric states known throughout the nuclear landscape (See Fig. 6.3). These isomers are special as these are the lowest spin isomers. These can occur only in odd-odd or, even-even nuclei. It is easy to grasp the

Fig. 6.3 The occurrence of low lying 0+ isomers throughout the nuclear landscape

0+ isomers

Proton Number

80

40

0 0

40

80 120 Neutron Number

160

6.4 The Low-Lying 0+ Shape Isomers

109

occurrence of these isomers in odd-odd nuclei, where a multiplet of states emerge from the coupling of a proton and a neutron; the multiplet will always contain a higher spin state to which the 0+ can decay, giving rise to spin isomers. It is, however, not easy to comprehend their occurrence in even-even nuclei. As we shall see most such isomers in even-even nuclei are likely to be shape isomers. In even-even nuclei, an excited 0+ state may decay to the ground 0+ state via an electric monopole (E0) transition. For lower excitation energies, the E0 transition is usually very slow, and thus the excited 0+ state becomes a shape isomer. As we discuss now, each of the 0+ isomers has a different story to tell. Not only that, most of these stories are still incomplete, implying only partial understanding of the nature of these isomers. We list all the 0+ isomers, 23 in number, in Table 6.2, along with their basic properties. We discuss them, some in more detail, while others in passing remarks. 12 Be: The 0+ isomer in 12 Be is the lightest isomeric state in the Atlas of isomers [2]. 8 Be is generally accorded a two alpha-cluster structure. 12 Be has four extra neutrons beyond 8 Be, besides being a magic nucleus (N = 8). Various measured quantities are shown in Fig. 6.4. The B(E2; 21+ → 01+ ) = 14.2(22)e2 f m 4 (8.5 W.U.) is much larger to its value of 9.2(3)e2 f m 4 in 10 Be, suggesting large collectivity [35]. The N = 8 magic number also seems to disappear, and the B(E2; 02+ → 21+ ) = 7.0(6)e2 f m 4 is about 4.21 W.U. suggesting a shape change [36]. A back of the envelope calculation by Morse et al. [35] suggests that the distance between the two alpha clusters shrinks to 1.97 f m in 02+ isomer in comparison to 2.58 f m in 21+ state. Thus, the isomer appears to have shrunken considerably, and is suggested to have a near spherical structure. As shown by Chen et al. [37], calculations show that the percent intensities of s, d and p waves come out to be 19 ± 7, 57 ± 7, 24 ± 5 suggesting a prolate shape in 01+ ground state compared to 39 ± 2, 2 ± 2, 59 ± 5 percents in 02+ isomer, suggesting a more spherical structure. Thus, 12 Be 02+ isomer is an isomeric state with spherical configuration.

Fig. 6.4 Schematic level scheme of 12 Be. See text for details

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6 Shape and Fission Isomers

Table 6.2 The list of low lying 0+ isomers with a lower limit on half-life at 10 ns. Refer to the atlas [2] for more details, particularly for original references AX Energy (keV) Jπ T1/2 Multipolarity Decay 12 Be 26 Al 32 Mg 34 Si 38 K

40 K 44 S 68 Ni 72 Ga 72 Ge 72 Se 72 Kr 74 Ga 74 Kr 90 Zr 96 Zr 98 Sr 98 Zr 98 Mo 102 Pd 156 Tb 174 Tm 176 Ta

2251 (1) 228.305 (13) 1057 2718.4(1) 130.22 (16)

0+ 0+ 0+ 0+ 0+

233(8) ns 6.3460(8) s 7-26 ns 19.4(5) ns 924.4(3) ms

1643.638 (11) 1365.0 (8) 1603.5 (3) 119.66 (5) 691.43 (4) 937.22 (15) 671.0 (10) 59.571 (14)

0+ 0+ 0+ (0+ ) 0+ 0+ 0+ (0+ )

0.336(13) µs 2.619(26) µs 270(6) ns 39.68(13) ms 444.2(8) ns 17.5(17) ns 26.3(21) ns 9.5(10) s

E0 E0 (M2) E0 E0 E0 E0

509.25 (7) 1760.71 (14) 1581.64 (6) 215.35 (7) 854.02 (14) 734.75 (4) 1593.16 (22) 88.4 (2)

0+ 0+ 0+ 0+ 0+ 0+ 0+ (0+ )

14.2(10) ns 61.3(25) ns 38.0(7) ns 22.8(19) ns 64(7) ns 21.8(9) ns 14.5(4) ns 5.3(2) h

E2, E0 E0 E0 E2, E0 E0 E0 E0 E3

252.4 (5) 100.2 (10)

0+ (0+ )

2.29(1) s 30.5(10) ns

E3 E1

E2, E0

%I T = 100 % + %β + = 100

E0

%I T = 100 % + %β + = 99.9670(43) %I T = 0.0330(43) %I T = 100 %I T = 100 %I T = 100 %I T = 100 %I T = 100 %I T = 100 %I T = 100 %I T = 75(25) %β − = 25(25) %I T = 100 %I T = 100 %I T = 100 %I T = 100 %I T = 100 %I T = 100 %I T = 100 % + %β + > 0, %I T < 100 %I T = 100

Al: 26 Al is a self-conjugate N = Z = 13 nucleus and has an important isomer from astrophysical point of view. In a rare experiment, isomeric beam of 26 Al was used to carry out 26 Alm (d, p)27 Al reaction [38]. This provided, for the first time, an upper limit of the reaction rate for destruction of galactic 26 Al in stars and nova. The −1 odd-odd 0+ isomer is seen to arise from (1d5/2 )−1 π ⊗ (1d5/2 )ν configuration and all the members of the multiplet from I = 0 to 5 have been observed. The 0+ isomer lies just above the 5+ ground state, making it a spin isomer. It decays by beta decay only. 32 Mg: The 02+ isomer in 32 Mg was studied by using a novel in-beam spectroscopic technique, which involved the use of advanced γ-ray tracking array GRETINA, enabling the tracking of isomeric decay position [39]. This technique enables isomeric life-times of the order of 1–100 ns to be measured. The measured B(E2; 21+ → 26

6.4 The Low-Lying 0+ Shape Isomers

111

2 4 01+ ) = 94(16)e2 f m 4 , B(E2; 21+ → 02+ ) = 48+74 −20 e f m , and observed partial cross+ sections suggest that 02 isomeric state contains a strong admixture of 2 p − 2h and 4 p − 4h intruder configurations. It appears to be as collective in nature as 01+ state. It, therefore, appears to be a case of shape coexistence. 34 Si: 02+ isomer represents the first observation of a state with 2ω intruder character having a predicted oblate deformation, while the ground state 01+ state is spherical [40]. Measured B(E2; 21+ → 02+ ) = 47(19)e2 f m 4 or, 7.2(31)W.U. Calculations obtain a value B(E2; 21+ → 02+ ) = 78e2 f m 4 and B(E2; 21+ → 01+ ) = 12e2 f m 4 . Mixing ratio of (0 p − 0h): (2 p − 2h): (4 p − 4h) comes out to be 90:10:0 for 01+ and 4:89:7 for 02+ in percentage terms. This confirms the 2 p − 2h dominance with a trace of 4 p − 4h in 02+ isomer. 38 K: It is also a self-conjugate odd-odd nucleus like 26 Al. Bissell et al. [41] have measured the difference in the charge radii of ground and isomeric state, which is < rc2 >i.s. − < rc2 >g.s. = 0.100(6) f m 2 . The ground state is a 3+ state and 0+ lying next at 130 keV is clearly a spin isomer. The proton-neutron isovector pairing was treated on the same footing as proton-proton and neutron-neutron pairing, and was able to explain the isomer shift in charge radii. 40 K: It is also a spin isomer with lowest possible transition being M2. 44 S: It is a N = 28 shell closure neutron-rich nucleus. Chu et al. [42] report relativistic mean field calculations which obtain a central depression in charge density for the ground state, suggesting a neutron halo like structure. However, no specific results for 02+ isomeric state are given making it a blank case to decide on the nature of the isomer. We speculate it to be a shape isomer having different structure than 01+ ground state. It is worthwhile to mention that 4+ state at 2459 keV has been proposed as K -isomer having K = 4, B(E2; 4+ → 21+ ) = 0.61W.u. It is the first K -isomer to be seen in such a low mass nucleus. It is suggested that we look for more such cases. 68 Ni, 72 Ge, 72 Se, 72 Kr, 74 Kr, 90 Zr, 96 Ze, 98 Zr, 98 Sr, 98 Mo, 102 Pd are all most probably examples of shape-coexistence and shape isomers. What shape 0+ isomer may adopt, varies from nucleus to nucleus. This mass region appears to be quite prone to shape coexistence though. Nearly all the cases require more detailed investigations, both experimental as well as theoretical, to confirm the kind of shape coexistence. 156 Tb, 174 Tm, both are clear cases of spin isomers. However, 176 Ta, 0+ isomer decays to the 1− ground state by an E1 transition, both the spins estimated by using the G-M rule [43]. It, therefore, remains a doubtful case of spin isomer. We may mention here a highly interesting recent observation of a triplet of excited 0+ states observed in 66 Ni, which has been claimed as the lightest nucleus, which exhibits a different shape for each of the three 0+ states. The three 0+ levels exist at 2443, 2671, and 2974 keV and are oblate, spherical and prolate in nature according to the Monte-Carlo Shell Model (MCSM) calculations. The photon decay appears to be hindered and their respective half-lives are estimated to be 7.6, 134, and 20 ps only [44]. The half-lives of the 0+ states, which are less than 0.1 ns, do not fit into the definition of isomers adopted so far. This is the reason for not including this nucleus in our table of the 0+ states. Nevertheless, it is the most interesting example of shape co-existence at low spin in a very low mass nucleus.

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6 Shape and Fission Isomers

6.5 Half-Lives of the Shape Isomers Explaining the half-life of the shape isomers is one of the most crucial task of theoretical calculations and difficult too. The half-life of a shape isomer may depend upon: • the overlap of nuclear wave functions of the shape isomeric state and the ground state • the excitation energy of the shape isomeric state • the height of the saddle separating the shape isomer and the ground state, i.e. the barrier-heights Therefore, one needs to study the occurrence of minima in the potential-energy surfaces of such nuclei. Knowledge of such minima can be the starting point for estimating the isomeric half-lives. We provide brief details of some theoretical attempts in this direction.

6.5.1 Projected Shell Model Calculations Y. Sun [45] tried to explain the shape isomerism by using the projected shell model calculations and their competition with the K -isomers. For the K -isomer treatment, K -mixing is properly implemented in the projected shell model, as discussed in the previous chapter. To treat the shape isomers, the approach used is to allow a mixing of configurations belonging to different shapes. To calculate isomer life-time, decay probability is needed which involves transitions from the shape isomer to the ground state having a different shape or deformation minimum. If the energy barrier between the two minima is not very high, configuration mixing of the two shapes becomes possible. Sun [45] extended the PSM to take care of such mixing. The shapes can be of any kind of two deformed ones in a nucleus. For example, one of them can be a prolate-deformed and the other an oblate-deformed shape, or one of them can be a normally deformed and the other a super-deformed shape. Generalizing the method further, it can describe those transitional nuclei where energy surfaces are typically flat. The heart of the present consideration is the evaluation of overlapping matrix element in the angular-momentum projected basis. Let us start with the PSM wave function:  I,σ >= f κI,σ PˆMI K |φκ > (6.1) | M κ,K ≤I

=

 κ

f κI,σ PˆMI K κ |φκ >

(6.2)

6.5 Half-Lives of the Shape Isomers

113

For an overlapping matrix element, states in the left and right hand side must correspond to different deformed shapes. Therefore, two different sets of quasiparticle generated at different deformations are generally involved. Let us denote |φκ > explicitly as |φκ (a) > and |φκ (b) > , for which we define two sets of quasiparticle operators {a † } and {b† } associated with the quasi-particle vacua |a > and |b >, respectively. For the simple case of axial symmetry, the general three dimensional problem reduces to a one-dimensional problem with the following projection operator, PˆMI K = (I + 1/2)



π 0

I ˆ dβsinβd M K (β) R y (β)

(6.3)

ˆ

I where Rˆ y (β) = e−iβ Jy , d M K (β) is the small d-function and β is one of the Euler angles. In this way, the problem of calculating actual overlapping matrix element reduces to < φκ (b)| Oˆ Rˆ y (β)|φκ (a) >, which actually shows the matrix element of Oˆ operator sandwiched by a MQP state |φκ (b) > and a rotated MQP state Rˆ y (β)|φκ (a) >. With the overlapping matrix elements that connect configurations belonging to different shapes, one obtains the wave functions containing configuration mixing. Using these wave functions, one can further calculate decay probabilities from a shape isomer to the ground state.

6.5.2 Other Microscopic Calculations The existence of shape isomers in lighter systems continues to be debated for long. The Monte Carlo Shell Model (MCSM) is a type of configuration-interaction approach for atomic nuclei that employs the advantages of quantum Monte Carlo, variational, and matrix-diagonalization methods. The MCSM [46] has been proposed to solve the nuclear many-body problem even when the computational scale becomes too large for the conventional ‘large-scale’ shell model calculations. Recently, MCSM calculations have been performed for heavy nuclei like the Hg isotopes for the first time [47] by using the K -supercomputer facility. The calculations could reproduce the magnetic and quadrupole moments for various states in the Hg isotopes. Another example of triaxial shape isomer has been established in 73 Zn by Yang et al. [48], which was studied after the nuclear spin and moment measurements of neutron-rich Zn isotopes measured at ISOLDE-CERN by Wraith et al. [49], showing an uncommon behavior of the 5/2+ isomeric state in 73 Zn. Yang et al. [48] have addressed the detailed measurements and analysis of the 73m Zn hyperfine structure to further support its spin-parity assignment 5/2+ and to estimate its half-life. The systematic investigation suggested that the collectivity appears significantly for this 5/2+ isomer due to proton/neutron E2 excitations across the proton Z = 28 and neutron N = 50 shell gaps, which was confirmed by the good agreement of the

114

6 Shape and Fission Isomers

observed quadrupole moments with large-scale MCSM calculations. They further claimed it to be of a triaxial shape using the potential energy surface calculations. On the other hand, Bonche et al. [50] have tried to search for the shape isomers in regions of nuclear chart other than the actinides by using Microscopic HartreeFock plus BCS calculations in three-dimensional coordinate space. They have tried to locate the states having an energy structure similar to that of the well known shape isomers in actinides, but with suppressed fission channel by calculating the potential energy surfaces for quadrupole deformation in a Hartree-Fock plus BCS approximation for many nuclei. Many isotopes of platinum, mercury and osmium do exhibit a second minimum with a large deformation and are candidates for shape isomerism. The same feature also occurs around the 68 Ni nucleus. However, no experimental confirmation of these findings have been reported so far.

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Chapter 7

Unusual Isomers

Keywords Extreme isomers · Very high-spin/energy/half-life/multipolarity isomers · Extremely Low Energy (ELE) isomers · 229 Th · Proton decaying isomers · Beta-decaying isomers In this chapter, we briefly describe those examples of isomers which are rather unusual in any of their specific characteristics or properties. These isomers, which carry a property like half-life, spin, excitation energy at the extreme, either very low or very high, are being termed as extreme isomers. Such isomers generate a tremendous sense of excitement, and reflect very special circumstances of their formation. Probing of such isomers may offer insight into many fundamental issues related to nuclear stability at the extremes, and also potential novel applications. While a brief description is presented here, we may discuss a couple of these cases in more detail.

7.1 Examples of Unusual Isomers 7.1.1 High Energy Isomers Pb: The 28− isomer in 208 Pb with an excitation energy of 13.67 MeV and 60 ns half-life is the highest lying isomer known so far. • 178 Hf: The 31 year 16+ isomer in 178 Hf lies at a high excitation energy of 2.4 MeV, something very unusual for an isomer which is so long-lived. It supports an interesting band known up to 22+ , which has a constant moment of inertia. Interest in this isomer arises from its capacity as an energy storage device as well the possibility to obtain a high energy gamma ray laser. Due to its large half-life, it may be used as a target, or an isomer beam may be created to carry out some unusual reactions [1, 2].

•

208

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. K. Jain et al., Nuclear Isomers, https://doi.org/10.1007/978-3-030-78675-5_7

117

118

7 Unusual Isomers

7.1.2 Extremely Low Energy (ELE) Isomers Those nuclear excitations, which become isomer, only because of the extremely low excitation energy, fall in this category. It may be noted that none of these isomers have high-spin. Therefore, “no hindrance due to angular momentum” makes these isomers very unique. A few examples, which have an excitation energy less than 2 keV, are: • The 3/2+ isomer in 229 Th (Z = 90, N = 139) with an excitation energy of 8.12 eV is the lowest energy isomer known so far with a half-life of 7 µs. We will discuss this isomer in more detail in this chapter. • The 1/2+ isomer in 235 U (Z = 92, N = 143) lying at 0.076 keV is the second lowest isomer with 26.5 min half-life. • The 2− isomer in 110 Ag (Z = 47, N = 63) having an energy of 1.112 keV with 660 ns half-life. • The 1/2− isomer in 201 Hg (Z = 80, N = 121) having 1.56 keV energy with 81 ns half-life.

7.1.3 Very High-Spin Isomers We note that all the high-spin isomers generally have a high excitation energy also. • The (65/2− ) isomer in 213 Fr (Z = 87, N = 126) at 8.09 MeV energy with 3.1 µs half-life. • The (67/2− ) isomer in 151 Er (Z = 68, N = 83) at 10.28 MeV energy with 0.42 µs half-life. • The 34− isomer in 212 Rn (Z = 86, N = 126) at 10.61 MeV energy with ≈20 ns half-life. • The 30+ isomer in 212 Rn (Z = 86, N = 126) at 8.57 MeV energy with 154 ns half-life.

7.1.4 Very Long-Lived Isomers We note that most of these are high-spin and low excitation energy isomers, the ideal condition for spin isomer • The 9− isomer in 180 Ta (Z = 73, N = 107) with > 4.5 × 1016 y half-life at 77.2 keV energy. • The 9− isomer in 210 Bi (Z = 83, N = 127) with 3.04 × 106 y half-life at 271.31 keV energy.

7.1 Examples of Unusual Isomers

• The (8+ ) isomer in keV energy.

186

119

Re (Z = 75, N = 111) with 2 × 105 y half-life at 148.2

7.1.5 Highest Multipolarity Isomers Here we list only the E5 decaying isomers, which is the highest multipolarity observed in isomers so far. • The (8+ ) isomer in 186 Re (Z = 75, N = 111) with 2 × 105 y half-life at 148.2 keV energy. This is also a very long-lived isomer. • The 11− isomer in 192 Ir (Z = 77, N = 115) with 241 y half-life at 168.14 keV energy. • The 16+ isomer in 178 Hf (Z = 72, N = 106) with 31 y half-life at 2.44 MeV energy. • The 11/2− isomer in 113 Cd (Z = 48, N = 65) with 13.89 y half-life at 263.54 keV energy.

7.1.6 100% Proton Decaying Isomers • The 1/2+ isomer in 144 Ho (Z = 67, N = 74) with 7.3 µs half-life at 66 keV energy. • The 3/2+ isomer in 147 Tm (Z = 69, N = 78) with 0.36 ms half-life at 68 keV energy. • The (1− , 2− ) isomer in 150 Lu (Z = 71, N = 79) with 39 µs half-life at 22 keV energy. • The 3/2+ isomer in 151 Lu (Z = 71, N = 80) with 16 µs half-life at 61 keV energy.

7.1.7 Highest Quasi-particle Isomer The highest number of quasi-particles in an isomer is the nine-quasiparticle (57/2− ) isomer in 175 Hf lying at 7.455 MeV having a half-life 22 ns [3, 4].

7.2 ELE Isomers There have been many efforts in recent times to study and control the extremely low energy (ELE) isomer in 229 Th, which lies at close to 8 eV. This kind of excitation is very rare in nuclei. This has focused our attention on similar cases of ELE isomers,

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7 Unusual Isomers

Table 7.1 ELE isomers AX Energy (keV) 110 Ag 141 Sm 193 Pt 201 Hg 227 Ra 229 Th 235 U

1.112 1.58 (4) 1.642 (2) 1.56 1.733 (9) 0.00812 (12) 0.076

Multipolarity

Half-life

E1

660 ns

M1 M1 + E2

9.7(3) ns 81 ns

M1 E3

7(1) µs 26.5 min

which become isomers merely due to very low excitation energy. In other words, these isomers do not fall in any other category like spin isomers, or K -isomers etc. We have fixed a limit of less than 2 keV excitation energy to identify an isomer as an extremely low energy (ELE) isomer. There are seven examples of such ELE isomers in nuclei, which have the first excited state in the range of 2 keV or less. These are listed in the Table 7.1, and the data are taken from our Atlas. Of the seven cases, only 235 U isomer at 76 eV decays by a high multipolarity transition and may come under the category of spin isomers. Rest of the cases display isomeric behavior purely on the grounds of very low excitation energy. It may be noted that nearly all the cases belong to medium and heavy mass region. It is known that the energy levels begin to come closer as the mass number increases. It is, therefore, expected that as the mass number increases, many such cases may be found. This is an exciting possibility as it may also lead to the discovery of a relatively longer lived state in the much sought after super-heavy region. As discussed in Sect. 2.2, internal conversion is expected to play a crucial role in these low energy transitions. For transition energies close to the atomic shell binding energies, the kinetic energy of outgoing electrons is very low and the electron correlation effects may become significant. This is why the BrIcc calculates the ICC’s for transition energies 1 keV above the atomic shell binding energies of electron. In other words, the transition energy must be greater by at least 1 keV of the atomic shell binding energies in BrIcc calculations. It is based on the assumption, that the physical models and approximations may not be valid, when the kinetic energy of the outgoing electron is below 1 keV. However, Trzhaskovskaya et al. [5] have tested the code for very low energies, and concluded, that with some precaution the RAINE code can be used in this extremely low energy region also. In examining the resonances in the theoretical conversion coefficients, Trzhaskovskaya et al. [5] have done calculations using different atomic models for energies very-very close (within few eV) to the binding energy, and concluded that all models work with same accuracy, but it is hard to benchmark them. A study of the ELE isomers where ICC’s will play a crucial role is, therefore, an important problem not touched upon yet.

7.3 A Specific Case of ELE Isomer in 229 Th

121

7.3 A Specific Case of ELE Isomer in 229 Th The 229 Th ELE isomer, having an excitation energy of about 8 eV, is the lowest energy nuclear state in any nucleus seen so far, and its study has posed a challenge to the experimental nuclear physicists. This isomer opens up a window, which connects the atomic world to the nuclear world. Hence, 229 Th has emerged as the most important nuclear isomer of recent times, making it to the cover picture of Sept 12, 2019 issue of Nature [6]. It is being seen as a potential nuclear clock of utmost precision. A comprehensive review article, which covers the various aspects of atomic and nuclear clocks, and nearly definite possibility to achieve the goal of a nuclear clock in 229 Th, was recently published by Wense and Seiferle [7]. More complete listing of all the references related to this isotope may be found in this article. Such a low lying state in 229m Th has led to intense activities across several nuclear physics labs of the world, igniting the dreams of the first nuclear laser [8], the most precise nuclear clock [9, 10], emergence of a new area of nuclear quantum optics [11], possibility of testing the temporal variability of the fundamental constants of nature [12], and investigating the role of chemical environments on nuclear decay rates. Nearly all the studies on 229 Th have been carried out by gamma ray spectroscopy following the alpha decay of 233 U, gamma ray spectroscopy following the beta decay of 229 Ac, the particle transfer (d, t), (d, p) reactions, and also Coulomb excitation. The first ever reference to the 229 Th isomeric state appears in the Nuclear Data Sheets [13], where the existence of a 3/2+ state at nearly zero energy in the (d, p) reaction data was pointed out by Erskine in an unpublished report of 1971 [14]. The 1972 Nuclear Data Sheet also refers to a study of 229 Th by alpha decay of 233 U, using Ge(Li) and Si(Li) spectrometers, due to [15], again an unpublished report, where similar observations about a low lying state were made. Detailed analysis of this work was later on reported by Kroger and Reich [16], who observed the 5/2+ to 11/2+ members of a rotational band based on 3/2[631] Nilsson state in 229 Th. From indirect but convincing arguments, they concluded that the 3/2+ extrapolated band head of the 3/2[631] band would be very close, and less than 0.1 keV, to the 5/2+ , 5/2[633] ground state. A level having an energy of 1 ± 4 eV above the ground state of 229 Th was again proposed by Reich and Helmer [17] in 1990, from the difference in gamma ray energies observed in the alpha decay of 233 U populating the 229 Th nucleus (see the partial level scheme in Fig. 7.1). Dennis Burke and collaborators [18] measured the triton cross-section peaks from the 230 Th(d,t)229 Th pick-up reaction. Such reactions have been used in deformed nuclei because the various members of a rotational band have a characteristic pattern of cross sections, often called a ‘fingerprint’, a very helpful technique in identifying the Nilsson orbital on which the band is based [19, 20]. Burke et al. [18] reported firm evidence of a ≤ 5 eV level, member of the 3/2+ band, based on the characteristic finger-print cross-section data of 229 Th, confirming the 3/2[631] assignment. Eventually, Helmer and Reich [21] also refined their measurements, and proposed an excited state at 3.5 ± 1 eV in 229 Th. They carried out new measurements using

122

7 Unusual Isomers

Fig. 7.1 Partial level scheme of 229 Th. The two rotational bands are shown as shifted from each other for clarity, and assigned the Nilsson quantum numbers

more number of Germanium and Si(Li) detectors with better energy resolutions ranging from FWHM = 0.4 keV (for 29 keV γ rays) to approximately 1 keV (for 122 keV γ rays). Further experimental works were carried out by Barci et al. [22] using alpha decay of 233 U and by Gulda et al. [23] using beta decay of 229 Ac. Using the same γ-ray data but assuming slightly different decay patterns between low-energy levels, a reexamination of the two works led Guimaraes-Filho and Helene [24] to a value of 5.5(10) eV for the energy of the 3/2+ state. Several reports of observing an UV radiation from the decay turned out to be unsubstantiated. In 1997, Irwin et al. [25] reported the detection of ultraviolet (UV) light (2.5 and 3.5 eV) and visible photons from 233 U alpha decay. In 1998, Richardson et al. [26] also detected UV light with the same energies, but they were not able to confirm the origin of this radiation. In 1999, Utter et al. [27] and Shaw et al. [28] repeated these experiments but ‘in vacuo’, and no UV or visible light was detected. These new results confirmed that the light observed in earlier studies was likely to be caused by alpha-particle induced fluorescence of air. In 2005, no photon emission was observed by Kasamatsu et al. [29], who reported half-life limits of 60 h , and 400 days. A half-life range of 20–400 min was reported in an unpublished research work [30], which was considered unreliable in [31]. Therefore, this saga of UV light died out within a few years. In 2001, Eddie Browne et al. [31] tried to measure the half-life of the 3/2+ state in 229 Th by using a novel radiochemical technique which measured γ-rays as a function of time from the decay of freshly separated 229 Th after alpha decay from 233 U, seeking evidence of 229 Th (ground state) growth from the isomeric 229 Th (3/2+ ) decay. No

7.3 A Specific Case of ELE Isomer in 229 Th

123

evidence of such a growth was observed, and half-life limits of 20 days for 229 Th (3/2+ ) were deduced from this measurement. Gangrsky et al. [32] produced 229 Th (3/2+ ) using the 229 Th(γ, γ) reaction with an end-point energy of 8.2 MeV bremsstrahlung radiation. They did not observe any prompt or delayed light emission from the isomer. Mitsugashira et al. [33] produced 229 Th through the 230 Th(γ, n) reaction expecting a larger population of the 3/2+ state because of the low spin (0+ ) of the 230 Th target. They tried to detect alpha particles emitted from the 3/2+ , 3/2[631] state assuming a different decay pattern as that expected from the alpha decay of the ground state (5/2[633]). Very few alpha-particle events were detected within the expected 4.83 − 5.08 MeV energy range. The statistics was very low, and no spectral peak shape was observed. The half-life of decaying components was determined to be 13.9 ± 3 h. Beck et al. [34] obtained a more reliable energy for the first excited state in 229 Th. Using the NASA/electron beam ion trap x-ray micro calorimeter spectrometer with a resolution of 26 eV FWHM, they were able to resolve very close γ-ray doublets of 29 keV and 42 keV by observing the decay of 71.82 keV level, and thus obtained an energy of 7.6(5) eV for the 3/2+ state in 229 Th. This result confirmed that internalconversion decay is energetically allowed since the level energy is greater than the 6.3 eV ionization energy of an isolated Th atom. Later on, they improved the value to 7.8(5) eV by taking into account additional out-of-band transition [35]. This value is significantly higher than the earlier value of 3.5 eV and corresponds to photon wavelengths around 150–170 nm. These wavelengths fell outside the bandwidth of the detectors used in the previous searches, and also outside the transmission window of the quartz cell used in the samples, resulting in the failure of these experiments to detect the signals. Tremendous advancements have recently been made in trapping and cooling of single atom detection in specific states. This has enabled the detection of single photon transitions, for example the E3 atomic transition in single Yb+ ion by [36]. Campbell et al. [37] produced laser-cooled crystals of the triply charged Th ion, 232 Th3+ in a linear radio-frequency Paul trap. This is the first time that a multiply charged ion was laser cooled. This work has opened a new avenue for the excitation of the nuclear transition in a trapped, cold 229 Th3+ ion. Laser excitation of nuclear states can establish a new bridge between atomic and nuclear physics, with the promise of new levels of precision in metrology. Wense et al. [38] reported the first direct detection of 229 Th isomer, obtained from the alpha decay of 233 U in 2+ or, 3+ charge state, and funneled and guided by various magnetic arrangements, obtaining the isomeric Th as only 2% of the total Th ions. The isomer energy was measured to lie between 6.3 eV and 18.3 eV, with a half-life of less than 1 second in neutral state. The isomeric half-life in 2+ charge state was estimated to be longer than 60 sec. It may be noted that the isomeric state can decay by three methods (i) internal conversion, (ii) photon decay, and (iii) bound state internal conversion. Using the same experimental set-up, Seiferle et al. [39] reported a half-life of 7.0 ± 1.0 µs for the internal conversion decay of neutral 229m Th isomeric state lying at ultra-

124

7 Unusual Isomers

low energy. Finally, the same group measured [40] the most precise energy of the isomeric state to be 8.28 ± 0.17 eV using the ground state decay of the nucleus. The uncertainty includes a systematic error of 0.16 eV also. This translates into a wave-length of 149.7 ± 3.1 nanometer s, lying in the vacuum ultra-violet (VUV) region of the light spectrum. This precision measurement further simplifies the path to do nuclear spectroscopy by using lasers and to develop a precision nuclear optical clock. A new era of nuclear photonics may also be on the anvil. Half-life measurement of the 229m Th isomer in its neutral state poses a very big challenge to the experimentalists. Several novel approaches have already been tried. Inamura and Haba [41] used a technique from optical spectroscopy where a hollow electric discharge tube is used to excite the levels for measurements. A sample of pure 229 Th was loaded in the discharge tube and isomer excitation was attempted by nuclear excitation by electron transition (NEET). Electric discharge is expected to excite the Th atoms into higher excited states, some of which lie close to the isomer excitation energy. The electrons could then transfer this energy to the nuclear isomer excitation. Inamura and Haba [41] suggest that the isomer did get populated in this way and its alpha decay was observed after switching off the discharge. An estimate m < 3 min was obtained for the half-life. of 1 min ≤ T1/2 Kikunaga et al. [42], on the other hand, searched for a direct decay signal by using a large sample of 233 U, which decays by alpha decay. Alpha spectroscopy was expected to give alpha lines of isomer decay as well as ground state decay. No alpha events from isomer were seen and a limit of less than two h on the half-life was suggested. Masuda et al. [43] reported an active optical pumping of the isomeric state from the second excited state of 229 Th, lying at 29 keV. The 29 keV level was excited resonantly by using a narrow band 29 keV Synchroton radiation. The x-ray fluorescence emitted by the target after exposure was detected by an array of silicon avalanche-photo diode sensors. The energy and half-life of the 5/2+ excited state, which is a member of the 3/2+ isomeric rotational band, were measured accurately to be 29189.93 ± 0.07 eV and 82.2 ± 4 ps, respectively. The isomer energy was constrained in the range of 2.5 < E < 8.9 eV. Shigekawa et al. [44] produced 229 Ac by the 232 Th( p, α)229 Ac reaction, and successfully separated 229 Ac from the 232 Th target by chemical separation techniques. Using the high-purity 229 Ac source, coincidence measurement between high-energy electrons and all electrons from 229 Ac was performed. Signals, which correspond to the IC electrons of 229m Th produced from the beta decay of 229 Ac, were detected for the first time. The IC half-life of 229m Th was determined to be 10(8) µs. The method established in this study lays the foundations to study the IC-decay of 229m Th in various chemical environments, and measure their effect on the half-life. Yamaguchi et al. [45] determined the energy of the 229 Th isomeric state by measuring the absolute energy difference between the excitation energy required to populate the 29.2 keV state from the ground state and the energy emitted in its decay to the isomeric excited state. A single-pixel transition-edge sensor micro calorimeter was used to measure the absolute energy of the 29.2 keV γ ray following alpha decay of 233 U. Together with the cross-band transition energy (29.2 keV → ground) and the

7.3 A Specific Case of ELE Isomer in 229 Th

125

branching ratio of the 29.2 keV state measured in an earlier study, the isomer energy was determined to be 8.30 ± 0.92 eV. Wense and Seiferle [7] have taken the weighted average of best 8 measurements and give a value of 8.12 ± 0.12 eV for the isomeric excitation energy. Its radiative life-time has been estimated to be about 104 sec. An IC decay half-life has been measured to be 7 ± 1 µs as already noted. In view of the very large ICC of the order of 109 , the total half-life may be taken to be the same. A 229 Th+ ion, however, does not undergo IC as the energetics do not allow it. This inspiring story of investigating 229 Th provides an insight into the tremendous interest in this isomer as well the variety of experimental techniques that may be employed to study isomer properties. The possibility of a level so close to the ground state in 229 Th has also produced a significant amount of theoretical research [46–62].

7.4 β-Decaying Isomers Isomers which entirely decay by beta decay processes may be of special interest in nuclear astrophysics. It is known that beta decay is a slow process and an excited state will always try to find a gamma decay path which is much faster than beta decay. In case of 100% beta decay, such a path is suppressed and, therefore, only beta decay occurs. We feel that this is an interesting situation and needs further attention. It is possible that the gamma decay of some of these isomers is spin forbidden and they need to be identified. Still, the remaining cases would present an interesting situation to study. We list in the Tables 7.2 and 7.3, only those cases which have 100% β + and β − − decaying branch. Those beta decaying isomers which have even a small gamma branching or, their excitation energy is not known, have not been included; such cases are quite a few. All the data have been collected from the Atlas of Nuclear Isomers and its update. As an example of a recent study, three beta decaying isomeric states in 128 In and 130 In have been studied with the JYFLTRAP Penning trap at the IGISOL facility by Nesterenko et al. [63]. A new beta-decaying high-spin isomer has been discovered in 128 In at 1797.6 (20) keV, which is suggested to be a 16+ spin-trap using shell-model calculations. For the first time, the lowest-lying (10− ) isomeric state at 58.6(82) keV was resolved in 130 In using the phase-imaging ion cyclotron resonance technique. Such precise measurements on the energies of the excited states are quite crucial for improving the shell-model effective interactions near 132 Sn.

126

7 Unusual Isomers

Table 7.2 The list of 100% β + -decaying isomers AX

Energy (keV)

Jπ

T1/2

Decay

42 Sc

616.762 (46)

7+

61.7(4) s

% + %β + = 100

44 V

286 (28)

(6)+

150(3) ms

% + %β + = 100

50 Mn

225.28 (9)

5+

1.75(3) min

% + %β + = 100

52 Co

381 (13)

2+

102(6) ms

% + %β + = 100

54 Co

197.1 (4)

7+

1.48(2) min

% + %β + = 100

70 Br

2292.3 (8)

9+

2157(53) ms

% + %β + = 100

74 Br

13.58 (21)

4(+)

46(2) min

% + %β + = 100

84 Y

67.0 (2)

1+

4.6(2) s

% + %β + = 100

87 Nb

3.9 (1)

(9/2)+

2.6(1) min

% + %β + = 100

90 Tc

144.1 (17)

(1+ )

8.7(2) s

% + %β + = 100

92 Nb

135.5 (4)

(2)+

10.15(2) d

% + %β + = 100

106 Ag

89.66 (7)

6+

8.28(2) d

% + %β + = 100

106 In

28.6 (3)

(2)+

5.2(1) min

% + %β + = 100

108 In

29.75 (5)

2+

39.6(7) min

% + %β + = 100

110 In

62.08 (4)

2+

69.1(5) min

% + %β + = 100

116 Sb

383 (40)

8−

60.3(6) min

% + %β + = 100

118 Sb

250 (6)

8−

5.00(2) h

% + %β + = 100

120 I

132 (30)

(7− )

53(4) min

% + %β + = 100

135 (15)

8(−)

3.70(11) min

% + %β + = 100

14.2 (4)

(3/2+ )

3.7(4) min

% + %β + = 100

122 Cs 127 La 127 Ce 134 Pr 138 Pr

+

7.3 (11)

(5/2 )

28.6(7) s

% + %β + = 100

67.7 (4)

(6− )

≈11 min

% + %β + = 100

364 (23)

7−

2.03(2) h

% + %β + = 100

152.6 (5)

(11/2− )

110.0(14) s

% + %β + = 100

143 Dy

310.7 (6)

(11/2− )

3.0(3) s

% + %β + = 100, % p =?

147 Tb

50.6 (9)

(11/2− )

1.87(6) min

% + %β + = 100

90.1 (3)

+

(9)

2.20(5) min

% + %β + = 100

48.8 (2)

(1/2+ )

56(3) s

% + %β + = 100

461 (27)

9+

5.8(2) min

% + %β + ≈ 100

247.23 (16)

11/2−

4.59(11) min

% + %β + = 100

89 (12)

7(+)

3.7(3) min

% + %β + = 100

128 (8)

13/2(+)

50.8(15) min

% + %β + = 100

138 (45)

(7+ )

10.8(2) min

% + %β + = 100

95 (28)

(13/2+ )

5.8(2) min

% + %β + = 100

260 (15)

7+

32.8(2) min

% + %β + = 100

202.9 (7)

13/2+

15.0(12) min

% + %β + = 100

280 (40)

(7+ )

11.6(3) min

% + %β + = 100

143 Gd

148 Tb 149 Ho 150 Tb 189 Au 190 Tl 191 Hg 192 Tl 193 Pb 194 Tl 195 Pb 198 Bi

References

127

Table 7.3 The list of 100% β − -decaying isomers AX E(keV) Jπ 65 Fe 69 Ni 71 Ni 82 As 83 Se 84 Br 96 Y 98 Rb 100 Y 100 Nb 102 Nb 106 Rh 115 Cd 116 In 117 Cd 119 Cd 121 Cd 123 In 123 Sn 125 In 125 Sn 126 In 127 In 127 Sn 130 Sn 130 Sb 136 I 152 Pm 166 Ho 170 Ho 178 Lu

393.7 (2) 321 (2) 498.5 (7) 131.6 (5) 228.92 (7) 3.2E+2 (10) 1541 (10) ∼270 145 (15) 314 (23) 93 (23) ? 137 (13) 181.0 (5) 127.267 (6) 136.4 (2) 146.54 (11) 214.86 (15) 327.21 (4) 24.6 (4) 360.12 (9) 27.50 (14) 102 (64) 1697 (49) 5.07 (6) 1946.88 (10) 4.8 (2) 201 (26) 1.5E+2 (9) 5.969 (12) 120 (70) 123.8 (26)

(9/2+ ) (1/2− ) (1/2− ) (5− ) 1/2− (6)− 8+ (3+ , 4+ ) 4+ (5+ ) 1+ (6)+ 11/2− 5+ 11/2− 11/2(−) 11/2(−) (1/2)− 3/2+ 1/2(−) 3/2+ (8− ) (21/2− ) 3/2+ (7− ) (4, 5)+ (6− ) 4− 7− (1+ ) (9− )

T1/2

Decay

1.12(15) s 3.5(4) s 2.3(3) s 13.6(4) s 70.1(4) s 6.0(2) min 9.6(2) s 96(3) ms 0.94(3) s 2.99(11) s 1.31(20) s 131(2) min 44.56(24) d 54.29(17) min 3.441(9) h 2.20(2) min 8.3(8) s 47.4(4) s 40.06(1) min 12.2(2) s 9.52(5) min 1.64(5) s 1.04(10) s 4.13(3) min 1.7(1) min 6.3(2) min 46.6(11) s 7.52(8) min 1132.6 (39) y 43(2) s 23.1 (3) min

%β − %β − %β − %β − %β − %β − %β − %β − %β − %β − %β − %β − %β − %β − %β − %β − %β − %β − %β − %β − %β − %β − %β − %β − %β − %β − %β − %β − %β − %β − %β −

References 1. 2. 3. 4. 5. 6.

G.D. Dracoulis, P.M. Walker, F.G. Kondev, Rep. Prog. Phys. 79, 076301 (2016) P.M. Walker, Phys. Scr. 92, 054001 (2017) F.G. Kondev et al., Bull. Am. Phys. Soc. 45(5), 92 (2000) D. Ward, P. Fallon, Adv. Nucl. Phys. 26, 167 (2001) M.B. Trzhaskovskaya, T. Kibedi, V.K. Nikulin, Phys. Rev. C 81, 024326 (2010) J.T. Burke, Nature 573, 202 (2019)

≈ 100 ≈ 100 = 100 = 100 = 100 = 100 = 100 = 100 = 100 = 100 = 100 = 100 = 100 = 100 = 100 = 100 = 100 = 100 = 100 = 100 = 100 = 100 = 100 = 100 = 100 = 100 = 100 = 100 = 100 = 100 = 100

128

7 Unusual Isomers

7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.

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23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49.

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Chapter 8

Experimental Methods, Applications, Future Prospects

Keywords Experimental methods · Life-time measurements · Isomeric moment measurements · Fundamental and industrial applications of isomers · Isomeric targets and beams · Isomers in medicine and energy · Nuclear clocks · Gamma ray laser

8.1 Experimental Methods and Applications Nuclear isomers have turned out to be very useful tool in both the fundamental researches, as well as practical applications. Their use in uncovering the nuclear structure of nuclei is now well recognized and will be discussed with a few selected examples, where novel experimental techniques are being continuously devised and used. With the advent of RIB accelerators, isomer targets and isomer beams can play an important role in revealing the nuclear structure of drip-line nuclei as well as producing nuclei in the farthest regions of the nuclear chart. In the same spirit, isomers are also playing an important role in deciphering the r-process path of producing heavy neutron-rich nuclei. Isomers have also found a large number of applications in the medical field. Since isomers are excited states of a nucleus, they carry significant amount of energy depending on their excitation. This energy is mostly released in the form of nuclear radiation. In most of the cases, it is released in the form of gamma radiation. If we could collect 1 gm. sample of the 31 y isomer in 178 Hf, it would store approximately 1.3 Giga − J oules of energy. Isomers are, therefore, tiny store-houses of huge energy. They have the potential of being used as energy storage devices like nuclear batteries, realising the dream of gamma ray lasers triggering an era of controlled release of energy, possibility of constructing the most precise nuclear clock, and ushering in a new area of nuclear quantum optics. We will briefly look at these new developments. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. K. Jain et al., Nuclear Isomers, https://doi.org/10.1007/978-3-030-78675-5_8

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8.1.1 Gamma Ray Spectroscopy Gamma ray spectroscopy has emerged as the most powerful experimental tool to study nuclei. It is, therefore, natural that gamma ray spectroscopy is being used to study isomers also in a variety of ways. Although the list would be quite long, we cite here a few examples only. For example, Kanjilal et al. [1] found the first confirmed example of a 729 keV, 9− isomer in 210 Fr, from the time difference γγ-coincidence spectrum by using the INGA-Clover detector array at IUAC, New Delhi. In another example, Patel et al. [2] found the lightest four-qp K -isomer in the nucleus 160 Sm at an excitation of 2757 keV having spin 11+ , from the gamma-ray spectroscopy performed at RIBF, RIKEN, Japan. It becomes quite difficult to identify the gamma rays lying above the isomers. A prompt-delayed coincidence technique becomes very useful in such cases [3]. For example, in 130 Te, a gate was applied to the 935 keV (9− to 7− ) isomeric prompt gamma-transition and the 182, 331, 793, 839 keV gamma rays were observed in the delayed spectrum [4]. In another example of the prompt-delayed gamma ray technique, Costel Petrache et al. [5] identified a six-quasiparticle 20+ isomer in 140 Nd, which represents a fully aligned six-hole structure in the 146 Gd core. It is common to come across fission fragments produced in a reaction, which have long-lived isomers. In a latest example, Gerst et al. [6] applied the prompt and delayed γ-ray spectroscopy to the neutron-rich 94 Kr produced in fast neutron-induced fission of 238 U in conjunction with a neutron-ball array. A novel hybrid gamma spectrometer was used for both energy and life-time measurements. A 32 ns isomer was discovered at 3444 keV energy with 9− possible spin. Similarly, an exact knowledge of the level structure around the isomer is very important to determine its possible decay paths and hence the depletion rate at which isomers will decay and release energy. For example, the long-lived 438 y, 6+ isomer in 108 Ag at a low energy (110 keV) has been of considerable interest due to its high production possible through the (n, γ) reaction using the stable 107 Ag isotope. The odd-odd nature of 108 Ag implies the presence of a number of 2-qp structures, making the estimation of depletion pathways of the isomer very challenging. The high resolution gamma ray spectroscopy by using the Indian National Gamma Array (INGA) was carried out by the group at TIFR, Mumbai to decipher the low-lying structure, giving a clearer picture of the various possibilities [7].

8.1.2 Recoil/Fragment Mass Analyzers Reaction product mass spectrometers having sufficiently high mass resolution can be used to study longer lived isomers. Use of recoil mass analyzers along with gamma decay tagging techniques can give an additional edge to the discovery of newer isomers which are produced in small numbers. As an example, the Heavy Ion

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Reaction Analyzer (HIRA) at IUAC, New Delhi [8], was effectively used to measure the life-times of the high-spin isomer in 90 Nb. The INGA clover array facility was coupled to HIRA and recoil isomer tagging was employed, where the gamma ray measurements were made at both the target as well as at the focal plane of HIRA. This led to a new measurement for the 1880 keV, 11− isomer, which was found to have a life-time of 0.47 µs. A more advanced gas filled Hybrid Recoil Mass analyzer (HYRA) [9] was used in conjunction with a MWPC detector, Si-PAD detector, and one HPGe clover detector, to identify the evaporation residues from a fusion-fission reaction and a new 1.6 µs isomer having high-spin 31/2− and excitation 3336 keV was identified in 195 Bi. The residues were identified using the signals from the MWPC and the Si-PAD detectors and the gamma rays decaying from the isomer were detected in the clover detector [10]. A more detailed investigation was later on carried out by A. Herzan et al. [11], who could identify more isomeric states and rotational bands.

8.1.3 Mass Measurement Techniques In 2008, a new method to discover an isomer in the neutron-rich 65 Fe was reported [12]. High precision Penning trap mass spectrometry was used by the researchers at Michigan State University’s National Superconducting Laboratory (NSCL) to resolve the masses of the ground state and the isomeric state. In short, the low energy beam and ion trap (LEBIT) facility was used to extract rare isotopes from fast beam fragmentation, which were then thermalized, converted into low-emittance pulsed beams, and used for high-precision mass measurements [13]. Besides LEBIT at MSU, we can mention the Penning traps in operation like TITAN at TRIUMF, ISOLTRAP at CERN, JYFLTRAP at Jyvaskyla, and CPT at ANL at Argonne. A more detailed description of these facilities and their applications may be found in [14]. The projectile fragment separator (FRS) is one of the most versatile mass separators in use [15]. The FRS Ion catcher [16–22] consists of four parts: (i) the FRS including a mono-energetic degrader system, (ii) the cryogenic gas-filled stopping cell, (iii) the RF quadrupole-based beam transport and diagnostics unit and (iv) the multiple-reflection time-of-flight mass spectrometer. The final focal plane of FRS mainly consists of the cryogenic stopping cell, RFQ beamline and a MultipleReflection Time-of-Flight Mass Spectrometer (MR-TOF-MS). The MR-TOF-MS has played an important role in the separation of isomers and discovery of new isomers. The first spatial separation of longer lived isomers by using the MR-TOF-MS was reported by Timo Dickel et al. in 2015 [17]. This kind of separation sets the stage for pure isomeric beam experiments. Ions created in the FRS facility were transported to the MR-TOF-MS and the 210 Po, 210m Po ions were mass separated, where the spectrum obtained by using the Bradbury-Nielson-Gate (BNG) on, is able to select only the isomeric ions.

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Due to its high sensitivity and non-scanning measurement technique, MR-TOFMS is an ideal device for new isomers search as shown in the recent work of Hornung et al. [23]. A new low-lying isomer of nearly 100 ms half-life was discovered in 97 Ag in the vicinity of the doubly magic exotic nucleus 100 Sn, and isomers in odd-A 101−109 Ag isotopes were also observed in the same experiment. Besides the FRS facility at GSI, the TITAN at TRIUMF also has a MR-TOF-MS set up with a high resolution, sufficient to detect an isomer. RIKEN, ISOLTRAP, ANL are the other places with a MR-TOF-MS in operation, but they seem to have a bit lower mass resolving power.

8.1.4 Highly Charged Ion Storage Rings Direct observation of highly charged ions stored in a ring has been shown to be powerful tool for studying isomers. An excellent technical review of the area has recently been published by Steck and Litvinov [24]. The storage rings at GSI, Lanzhou, and RIKEN are now being used in nuclear physics experiments in general and nuclear isomer studies in particular. As an example, the GSI storage ring was used in identifying long-lived isomers in 212 Bi by Chen et al. [25]. Fragmented products from the 238 U projectile fragmentation at 670 MeV per nucleon were injected as highly charged ions (81+ charge state for Bi) into the GSI storage ring. The excitation energy of the first isomer of 212 Bi was confirmed and the second isomer was observed at 1478(30) keV, in contrast to the previously measured limiting value of >1910 keV. The fully stripped ions were found to have an extended Lorentz-corrected in-ring half-life >30 min, compared to 7.0 min for the neutral atom, which supports the presence of a significant IT component in the decay of the neutral atom.

8.1.5 Isomeric Targets and Isomeric Beams Isomeric targets and isomeric beams offer unique opportunities to study the nuclear structure properties of nuclei located near the drip-lines and otherwise also. Such isomers exist in excited states and also at high-spins promising a unique window in their structure, an example being 178m Hf, which has 2.4 MeV excitation and 16 angular momentum. Long-lived isomers (of the order of several months) are most suitable candidates to make an isomeric target [26]. Maunoury et al. present in step by step fashion [26], a method to produce an isomeric target of 177m Lu, having 1015 atoms with 45% purity. Similar isomeric targets have also been made for 178 Hf [27] and 242 Am [28]. Inelastic deutron scattering [27], two-neutron transfer [29] and neutron-capture reactions [30] have been carried out in 178 Hf, in addition to the photo-activation and photo-destruction. Inelastic neutron acceleration has also been observed in 178 Hf target [31], 160 d isomer target in 177 Lu [32, 33], and 5.5 h isomer target in 180 Hf [34].

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Short life-time isomers are suitable for isomeric beams provided it is possible to produce them in good number. A discussion and exploratory experimentation on isomeric beams has already begun in the nineties at the laboratories like the NSCL, GANIL, and RIKEN in the US, France and Japan, respectively [35]. Walker [36] has discussed the various possibilities of experiments with isomer beams. More references may be found in these works and the review [37]. For example, in an early experiment, Coulomb excitation of 174 Hf was studied by Morikawa et al. [38]. First experiment with post-accelerated isomer beam was reported by Stefanescu et al. [39] in Coulomb excitation of 68,70 Cu at the ISOLDE facility of CERN. The 6− radioactive beam of neutron rich odd-odd nuclei was used to study the multiplet of states (3− , 4− , 5− , 6− ) arising from the configuration (π2 p3/2 ⊗ ν1g9/2 ). Isomeric beams of heavier nuclei present an interesting option of producing heaviest nuclei. New possibilities are expected to emerge from the new high power laser based nuclear physics facility at ELI-NP, Romania. Isomers can be produced by using Bremsstrahlung photons from the high power laser driven electron acceleration [40]. For example, it has been estimated from the TALYS calculations that the (γ, n) and (γ, 2n) channels can give sufficient cross-section to produce the 178m Hf isomeric state.

8.1.6 Medical Applications of Isomers Isomers are being widely used for diagnostic as well as treatment purposes in the medical field [41]. Most commonly used diagnostic tool is the Single Photon Emission Computer Tomography (SPECT) for which isomers are quite useful. The positron emitting isomers are widely used in the Positron Emission Tomography (PET) imaging. Auger Electrons emitted after internal conversion decays of isomer have been used to treat cancer. We list in Table 8.1, the commonly used isomers in medical applications [37]. There are several pairs of isotopes listed in the Table 8.1, where the first member of the pair is the generator of the isomer listed along with it. The 99m Tc isomer, a daughter of 99 Mo is the most commonly used SPECT isomer, which emits a 142 keV gamma photon, easily detectable with high efficiency. The longest-lived Tc isotope is 98 Tc; and 99 Tc was the first artificial isotope produced in 1937. It binds to bio-molecules easily. 99m Tc is, therefore, used in over 40 million procedures every year in diagnosing heart disease and cancer. However, production of 99m Tc is limited to few old reactors, which use highly enriched uranium. Methods have been tried to produce 99m Tc by using non-HEU sources. However, it has been shown by model calculations that the proposed method of producing 99m Tc by using a neutron generator driven sub-critical assembly of natural or depleted uranium does not give enough of 99m Tc [42]. It is, therefore, becoming a challenge to sustain the global supply chain of 99m Tc. A novel method to produce this isomer is the need of the hour.

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Table 8.1 Isomers commonly used in medical applications. Table prepared on the basis of a similar table from [37]. Modifications made on the basis of newer data from the updated Atlas of Isomers (to be published) Generator/Nuclide T1/2 E*(keV) Decay mode Application 34m Cl 52 Fe/ 52m Mn 80m Br 81 Rb/ 81m Kr 82m Rb 94m Tc 99 Mo/ 99m Tc 103 Pd/ 103m Rh 113 Sn/ 113m In 115m In 117m Sn 167 Tm/ 167m Er 178 Ta /178m Hf 191 Os/ 191m Ir 193m Pt 195m Pt

31.99 min 8.275 h/ 21.1 min 4.4205 h 4.572 h/ 13.10 sec 6.472 h 52.0 min 65.976 h / 6.0060 h 16.991 d/ 56.114 min 115.09 d/ 99.476 min 4.486 h 14.0 d 9.25 d/ 2.269 sec 2.36 h/ 4.0 sec 14.99 d/ 4.899 sec 4.33 d 4.01 d

146.360 377.748

β + ; IT EC/β +

PET PET

85.843 190.46

IT IT

Auger SPECT; PET

69.0 76.0 142.6836

β+ β+ IT; β −

PET PET SPECT

39.753

IT

Auger

391.699

IT

SPECT; Auger

336.244 314.58 207.801 1147.416 171.29

IT; β − IT IT EC/ IT IT

SPECT SPECT; Auger SPECT; Auger SPECT; Auger SPECT; Auger

149.78 259.07

IT IT

Auger Auger

8.1.7 Isomer Depletion by External Triggers Internal conversion process is known to play an active role in the situations where decay by gamma emission is either not possible or is hindered due to various reasons such as small decay energy, and angular momentum restrictions. The nucleus then de-excites by transferring its energy to an atomic electron, which is ejected as an internal conversion electron. However, an inverse process was hypothesized in 1976, wherein a free electron is captured in a vacant atomic orbit, and the energy so released is used to excite the nucleus provided the kinetic energy of the free electron plus the binding energy of electron in the atomic orbital is sufficient for the excitation of the nucleus [43]. Since the free electron kinetic energy plus its binding energy may not be more than few keV, the nuclear excitation must be of the same order. This requires the two levels involved in the nucleus to be close together. This process is known as Nuclear Excitation by Electron Capture (NEEC). In the event of a nucleus being formed in an isomeric state, its excitation to a nearby level by NEEC becomes a possibility. It may be noted that isomer depletion may become of great importance

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in stellar environments and may lead to reduced abundance of an isotope after its population. Recently, a claim has been made of the first observation of NEEC in 93 Mo, by Chiara et al. [44]. In this work, 93 Mo was produced in its 21/2+ isomeric state at 2425 keV and half-life 6.85 h. Another level (17/2+ ) lies very close in energy at 2430 keV and a half-life of 3.5 ns. Only 5 keV is required for exciting the isomer to the 2430 keV level, which may then decay very fast, leading to what is called “isomer depletion". Once the isomer has been populated in a nuclear reaction, the recoiling nucleus may meet the right condition (resonance in energy as well as charge), and NEEC may occur. The observed isomer depletion was attributed to the extremely rare process of NEEC. However, this claim has been disputed in a theoretical calculation of NEEC probability, carried out in the beam-based set up by Wu et al. [45], who obtained a NEEC probability which is 9 orders of magnitude smaller than the probability claimed by Chiara et al. [44]. Isomer depletion by other means has also been investigated. For example, Belic et al. [46] used intense Bremsstrahlung radiation to excite the rarest isomer in nature i.e. 180m Ta by resonant photo-absorption. They claim to have obtained an enhanced isomeric decay under conditions similar to the s-process in nucleo-synthesis. Similarly, Kirischuk et al. [47] irradiated a 30 keV electron beam on the 31 year 178m2 Hf isomer target embedded in a Ta matrix. They observed enhanced counting rates for the ground-state band (with the γ-ray energies of 213, 325 and 426 keV) and K = 8− band (with the γ-ray energies of 216, 495 and 574 keV) from the 178m2 Hf isomer decay, suggesting a faster triggering of isomer decay. In another experiment, Roig et al. [33] has claimed to have obtained inelastic neutron acceleration by the long-lived isomer in 177 Lu.

8.1.8 Moments and g-Factor Measurements Nuclear moments and the g-factor measurements of isomers enable the determination of nucleon configuration either of single-qp or multi-qp isomers. Application of the additive rule for g-factor plays a very useful role in deciphering the configuration. A knowledge of moments and g-factors is, therefore, quite useful. Many techniques are used to measure the magnetic moments of isomers depending on their half-life values. An excellent exposition of these techniques has been presented by Hubel [48] and Neyens [49]. The additivity rule in obtaining the total magnetic moment by summing the moment contributions from individual particles in the configuration provides a simplified method to make configuration assignments. A commonly used method to measure the nuclear moments is based on the directional properties of the decaying gamma rays. The technique of Time-Dependent Perturbed Angular Distributions (TDPAD) may be applied to the initially oriented nuclei formed in heavy-ion and fusion-evaporation reactions for half-lives in the range of ns to µs. The nuclear dipole vector precesses in an external magnetic field, which in turn modulates the radiation intensity at any given angle, allowing a mea-

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surement of the precession frequency, and hence the dipole moment. For example, the magnetic dipole moments of K = 14, 41 ns isomer in 176 W [50] and K = 16, 88 ns isomer in 182 Re [51], K = 35/2, 750 ns isomer in 179 W [52] and the K = 25, 130 ns isomer in 182 Os [53] were measured by using the TDPAD technique. In a recent application, the g-factor of the 2738 keV, 25 ns isomer was measured by using the TDPAD technique with a superconducting magnet [54]. A precise knowledge of the magnetic field enabled a more precise value of the g-factor −0.049 to be obtained along with the assignment of major shell model configuration as π(d5/2 ) ⊗ ν(h 11/2 )−2 . An interesting case involved the measurement of the electric-quadrupole moment for 182 Os isomer by implantation into the osmium crystal [55]. The anomalously fast decay of isomer to the assumed K ≈ 0 band based on the ground state required a large K -mixing. The measured quadrupole moment was low and in line with the theoretical predictions supporting a triaxial shape [56]. A reduction in the axial symmetry results in the anomalous decay rate. Leuven group developed an alternative method of measuring the quadrupole moments of high-spin isomers [57] by extending their “level mixing resonances” technique [58]. As an example, the LEvel Mixing Spectroscopy (LEMS) technique was applied to the K = 35/2 isomer in 179 W to obtain the ratio of dipole and quadrupole moments by using a combination of magnetic and electric fields [59]. Knowledge of magnetic moment from TDPAD [52] allowed the quadrupole moment to be deduced [59]. The additivity rule, which works so well in magnetic moments, was, however, shown to fail in quadrupole moments [60]. The spectroscopic quadrupole moment of the 11− isomer in 196 Pb, was measured to be −3.41b, which gives a deformation of β2 = −0.156, a slightly oblate shape. Using this value and the previously measured value of quadrupole moment of 12+ isomer, the parameters of TAC model could be constrained very severely. These were used to check the additivity rule for quadrupole moment in the 16− band head. However, on coupling the measured quadrupole moments of the 11− proton and the 12+ neutron component of the 16− band head in an additive way results in a smaller value for the quadrupole moment than the one obtained from the TAC calculations which used the same parameter values. This result, however, still needs to be confirmed by a direct measurement of the quadrupole moment of the 16− band. For the half-life range of milliseconds or more, laser hyperfine spectroscopy could be used for the measurement of both the moments and mean-square charge radii. As an example, both quadrupole moment and charge radius of the well known K = 16, 31 y isomer in 178 Hf were studied by using the laser spectroscopy[61].

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8.2 Future Applications of Isomers 8.2.1 Isomer Battery and Gamma Ray Laser As pointed out, nuclear isomers store lots of energy. In fact any radioactive decay involves a release of stored energy. If one could find a way to release this energy in a gradual manner, it would lead to a number of useful applications. One of them would be nuclear batteries with a life-span of the order of hundreds of years. At present, 238 Pu has proven to be a very useful source to provide dependable power in long range space missions and is the material used in nuclear batteries. The heat from the alpha decay of 88 y half-life 238 Pu, is converted into electricity, which is then used to power the instruments on board the spacecraft or, the lander on Mars and Moon. The natural radioactive sources have a predetermined half-life, which cannot be changed. In such a scenario, isomers have the potential to provide alternative nuclear power sources. In a recent report by Feng et al. [62], an accelerated pumping of an isomer in 83 Kr has been claimed by Coulomb excitation of ions by quivering electrons induced by laser fields from a table top hundreds of TW laser system. If such a pumping is proven to be successful, it will pave the way to several new developments like isomer batteries, nuclear lasers, nuclear clocks etc.

8.2.2 Nuclear Clock The global time standards are mostly based on what are known as the atomic clocks such as Cesium, or the Strontium clocks. These clocks utilize the atomic transition as the frequency standard, timing the number of cycles in a second. The energy of the isomer in 229 Th is amazingly small, now measured to be 8.12 eV [63], and within the reach of laser nuclear spectroscopy. It is, therefore, a prime candidate for creating the first nuclear laser and the first nuclear clock [64]. We have already discussed this isomer in some detail in the previous chapter. It was shown quite early by Campbell et al. [65] that a Th ion based nuclear clock may allow an accuracy of one part in 10−19 . A more precise value of the isomer excitation energy and its life-time, therefore, remains a prime agenda for the experimental groups involved in this exciting area [66]. A nuclear clock of this high accuracy will open up many new possibilities like a more precise global positioning system, test of variation in the fundamental constants with time, trace the tiny fluctuations in the Earth’s gravitational field, to name a few.

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Summary

In Chap. 1, we have presented an overview of nuclear isomers, starting from the basic definition and early history, going up to the their classification on the basis of various hindrance mechanisms. We also discuss systematics of various isomeric properties and point out a number of interesting features. In Chap. 2, we have specifically focused on the origin of spin isomers and the associated isomeric transitions. The impact of internal conversion on isomeric halflives has been pointed out. A discussion of the island of isomers in terms of the high-j intruder orbitals has been presented and various exceptions pointed out. Few interesting examples of spin isomers are discussed in detail. In Chap. 3, we have highlighted the role of seniority in giving rise to the isomers. The physics origin of seniority isomers are discussed along with the quasispin scheme, The isomeric decay properties and moments are also presented in great detail using various examples from the semi-magic chains. Role of seniority mixing has also been pointed out. In Chap. 4, extension of seniority from single-j to generalized seniority in multi-j environment has been presented by using the multi-j quasi-spin scheme. A new kind of seniority isomer has been pointed out. As examples, various high-spin isomers in Sn, Pb isotopes and N = 82 isotones have been discussed. In addition, other excited states which are important for nuclear structure studies, such as the first-excited 2+ and 3− states have also been analyzed in terms of the generalized seniority arguments. Isomeric moments, both Q-moments and g-factors, have been discussed by using examples. The long standing puzzle of the asymmetric double parabolic behavior of B(E2) values in Sn isotopes has been explained. Similar behavior in the Cd and Te isotopes, which are away from the semi-magicity, could also be explained. In Chap. 5, we introduce the concept of K -isomers by using deformed Nilsson model and the calculations of high-K MQP states. Some general features of K -isomers including K -mixing and high-K rotational bands are discussed. A brief presentation of theoretical approaches used for K -isomers has also been made. K-

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isomers present a very interesting option of observing long-lived super-heavy elements. In Chap. 6, we have discussed the origins of shape and fission isomers and their characteristic features. The origin of shape isomers has been discussed with the specific examples of E0 transitions for the 0+ isomeric states. The competition of shape and fission isomers in heavy nuclei has been pointed out. We start from the double-hump potential barrier due to the shell corrections and explain the discovery of fission isomers. A very brief overview of theoretical treatment for these isomers has also been presented. In Chap. 7, we list some unusual type of isomers, such as high energy isomers, ELE isomers, very long-lived isomers etc. The ELE isomers, separated out for the first time, are discussed in detail with a specific and important example of 229 Th due to its possible application in the ultra-precise nuclear clock. A table of 100% β-decaying isomers is also presented which may be of use in nuclear astrophysical calculations. In Chap. 8, we briefly discuss the various experimental methods used for life-time and isomeric moment measurements, along with their applications in the fundamental understanding of nuclear structure using isomeric targets and beams. Several industrial applications of isomers in medicine and energy have been discussed. Future prospects of isomers in nuclear clocks, gamma ray laser, etc. are also covered very briefly.