Wobbling Motion in Nuclei: Transverse, Longitudinal, and Chiral (Springer Theses) 3031171497, 9783031171499

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Springer Theses Recognizing Outstanding Ph.D. Research

Nirupama Sensharma

Wobbling Motion in Nuclei: Transverse, Longitudinal, and Chiral

Springer Theses Recognizing Outstanding Ph.D. Research

Aims and Scope The series “Springer Theses” brings together a selection of the very best Ph.D. theses from around the world and across the physical sciences. Nominated and endorsed by two recognized specialists, each published volume has been selected for its scientific excellence and the high impact of its contents for the pertinent field of research. For greater accessibility to non-specialists, the published versions include an extended introduction, as well as a foreword by the student’s supervisor explaining the special relevance of the work for the field. As a whole, the series will provide a valuable resource both for newcomers to the research fields described, and for other scientists seeking detailed background information on special questions. Finally, it provides an accredited documentation of the valuable contributions made by today’s younger generation of scientists. Theses may be nominated for publication in this series by heads of department at internationally leading universities or institutes and should fulfill all of the following criteria • They must be written in good English. • The topic should fall within the confines of Chemistry, Physics, Earth Sciences, Engineering and related interdisciplinary fields such as Materials, Nanoscience, Chemical Engineering, Complex Systems and Biophysics. • The work reported in the thesis must represent a significant scientific advance. • If the thesis includes previously published material, permission to reproduce this must be gained from the respective copyright holder (a maximum 30% of the thesis should be a verbatim reproduction from the author’s previous publications). • They must have been examined and passed during the 12 months prior to nomination. • Each thesis should include a foreword by the supervisor outlining the significance of its content. • The theses should have a clearly defined structure including an introduction accessible to new PhD students and scientists not expert in the relevant field. Indexed by zbMATH.

Nirupama Sensharma

Wobbling Motion in Nuclei: Transverse, Longitudinal, and Chiral Doctoral Thesis accepted by University of Notre Dame, USA

Nirupama Sensharma University of North Carolina at Chapel Hill Department of Physics and Astronomy Chapel Hill, NC, USA

ISSN 2190-5053 ISSN 2190-5061 (electronic) Springer Theses ISBN 978-3-031-17149-9 ISBN 978-3-031-17150-5 (eBook) https://doi.org/10.1007/978-3-031-17150-5 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 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

To my parents, Aparna Sensharma and Supriyo Sensharma for always being there for me and To my husband, Josh Marzic for his unconditional love, support, and his ability to keep me sane even in the most trying times.

Supervisors’ Foreword

Although one intuitively thinks of atomic nuclei being “spherical,” they come in many shapes: spherical, axially deformed (both prolate—akin to an American football, and oblate—like a doorknob), pear-like (“octupole deformed”), and other more complicated shapes corresponding to higher-order deformations. These shapes give rise to specific properties that may be observed in experiments, specifically in the form of energy levels characterized by regular spacings and the electromagnetic character of the cascades of γ rays that are emitted as an excited nucleus decay to its ground state. Experimenters gain insights into the structure of the nucleus via γ -ray spectroscopy—detailed measurements and analyses of the properties of the emitted γ rays. An interesting, but very rare, shape assumed by nuclei is what has been termed “triaxial,” i.e., an ellipsoid with all three axes unequal. When a nucleus is in this shape, it exhibits two unique properties: chirality and wobbling. Chirality refers to handedness and the nucleus, in specific states of excitation, can have orientations of its associated angular momenta that could be left- or right-handed. This phenomenon, first predicted in nuclei some 25 years ago, has been observed in many nuclei now, and detailed investigations have led to a deeper understanding of the interactions that gave rise to this aspect of nuclear structure. The second unique characteristic associated with triaxial nuclei is the phenomenon of wobbling—precession of the total angular momentum about one of the axes. First discussed by the Nobel Laureates Bohr and Mottelson some 50 years ago for even-even nuclei, this phenomenon has been observed in only a few nuclei so far, and in the process, two forms of wobbling motion have been identified: transverse wobbling (precession about the short axis) and longitudinal wobbling (precession about the medium axis, the one corresponding to the maximum moment of inertia); indeed, what Bohr and Mottelson had talked about was what we now understand as longitudinal wobbling. However, until this work, all evidence of wobbling observed in nuclei corresponded to the transverse case. In this work, Dr. Nirupama Sensharma has used techniques of γ -ray spectroscopy to provide the first clear evidence for longitudinal wobbling in the nucleus 187 Au. As well, she has obtained preliminary evidence for a transverse wobbling band, vii

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Supervisors’ Foreword

making it the first possible case where both types of wobbling have been observed in the same nucleus. In addition, she has carried out detailed investigations on wobbling phenomenon in the nucleus 135 Pr, the first nucleus out of the mass A~160 region where transverse wobbling was observed. In the process, she has identified wobbling bands associated with second-phonon excitation, as also a chiral band structure, making this nucleus the first case where both signatures of triaxial nuclei—chirality and wobbling—have been observed together. These results have put the phenomenon of wobbling on a solid experimental footing and have provided fertile grounds for testing and enhancement of the underlying theoretical interpretation. It is our hope that this work would inspire further high-statistics measurements in different regions of the nuclear chart to fully elucidate this very interesting, and exotic, aspect of the structure of the atomic nuclei, and we are delighted that this dissertation is being published in the Springer Theses series. Notre Dame, IN, USA September 5, 2022

Umesh Garg

Notre Dame, IN, USA September 5, 2022

Stefan Frauendorf

Acknowledgments

I came to the United States as a young graduate student back in 2015 and here I am six years later, writing my dissertation and on my way to becoming a nuclear physicist. This journey has been long and difficult on one hand but also enjoyable and enriching on the other. Throughout this period, I have had the chance to meet and work with numerous people who have been invaluable to my professional and personal lives. I cannot comprehend ending this journey without thanking them all. I would start by thanking my advisor, Prof. Umesh Garg, for his guidance and help through every step of my PhD at the University of Notre Dame. His commitment to the field of nuclear physics continues to inspire me. He has also been a great teacher and a mentor throughout my time at Notre Dame. I am thankful to him for constantly motivating me to work harder and improve myself as a researcher. His patience throughout the period of my thesis writing has been exceptional. Without his comments and advice, all my publications and this manuscript would have been incoherent and substandard. To Prof. Stefan Frauendorf for his help in establishing a strong theoretical foundation for my work. Having him as our theory collaborator was extremely valuable. I am thankful for the time he took to guide me through difficult theoretical concepts and help me gain better insight into the physics behind my experimental work. I am also grateful to Dr. Q. B. Chen for taking so much time to help me through our theory calculations and for answering all of my questions with a lot of patience. Through my communications with him, I have been able to better understand and communicate my research. A note of thanks also goes to Dr. G. H. Bhat and Prof. J. A. Sheikh for their help with the theoretical calculations. I am especially thankful to Dr. A. D. Ayangeakaa and Dr. Shaofei Zhu for their help with codes and data analysis. With loads and loads of data drives, I was totally lost and confused, and it was through their help during the initial days of my PhD, that I was able to make sense of my data and progress my research. I owe them everything I have learned so far about my experiments and data analysis. I would like to express my gratitude to Dr. Anna Simon for agreeing to be on my research committee. Having her on my committee really helped me make better decisions as a graduate student. She has been an amazing mentor throughout the ix

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Acknowledgments

duration of my program. I am especially thankful to her for all the academic and career-related advice I got from her. Her advice has been instrumental in helping me decide on my career path. To Dr. Daryl Hartley for being a wonderful teacher and a collaborator. I am grateful for the long hours he spent trying to help me understand the complicated data analysis procedures. His help during my experiments at the Argonne National Laboratory was also especially appreciated. Having him around in the data acquisition room helped graduate students like me stay motivated. To Prof. Don Howard for being my mentor. Working with him during the Social Responsibilities of Researchers program at the Reilly Center was really a life-changing experience for me. It was through all the training I received in ethical and socially motivated research during that program that I understood my responsibilities as a scientist toward our society. At the end of the year-long program, I came out as a better person and a better scientist. To Dr. Kevin Howard for being my closest friend and confidant. Without him by my side, I would have long been lost in the struggle to settle in this new country. Equally appreciated were our late-night trips to get ice cream every time I felt overwhelmed with work. Having him in the same research group was almost a blessing in disguise. Be it physics or life, if I had a problem, Kevin for sure had the solution! I would also like to thank the undergraduate students in our research group-Joseph Cozzi and Sierra Weyhmiller-for keeping our group meetings alive with their vivacious nature and interesting questions. I was fortunate to enjoy the company of Joe Arroyo and Orlando Gomez as my office mates. They have been great friends and colleagues. I have enjoyed all our academic and non-academic discussions. Working long hours in the office didn’t seem so hard with them around. I am also thankful to the entire administrative staff at the Department of Physics. I especially want to thank Shari Herman for being so helpful and caring. Whatever the crisis, she is always there to help graduate students. A token of thanks also goes to Janet Weikel who has been my (and every other nuclear physics graduate student’s) go-to person. It is incredible how meticulously she manages the entire Nuclear Science Laboratory. We would be totally lost without her organizational and administrative skills. To my friends Dr. Farheen Naqvi, Saurabh Bansal, Lailatul Fitriyah, and Ann Marie Thornburg for being my support system in the city of South Bend. I am thankful for all the long cold nights when we sat together and enjoyed long conversations. My life is so much better with all of them in it. To my cousin, Anupama Sensharma, for being a wonderful sister and friend. Without her editing of my various letters and articles, all my documents would be poorer and full of grammatical errors. To Dr. Rashi Talwar and Dr. Jasmine Sethi for being so helpful during the initial days of my PhD program. Their support and guidance during my visits to the Argonne National Laboratory have been immensely helpful. I also wish to extend my thanks to my collaborators at the Argonne National Laboratory-Dr. Michael P. Carpenter, Dr. Torben Lauritsen, Dr. Darek Seweryniak, and Dr. Filip Kondev. Without their expertise and guidance, none of my experiments would have been

Acknowledgments

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possible. I am also thankful to the staff of the Argonne Tandem Linear Accelerator System (ATLAS) facility for providing excellent quality beams that helped me perform all my experiments as needed. Finally, it would have been absolutely impossible to come this far without the love and support of my family-my mom, dad, and my brother, Aalok. They have always believed in me and have stood by me in the worst of times. When everything seemed to fall apart and I had lost my way, they held my hand and helped me see the light. It is because of them that I came out the other end as a stronger person. Everything I am and everything I would be, I owe it to my family. This work has been supported in part by the US National Science Foundation [Grants No. PHY-1713857 (UND) and No. PHY-1203100 (USNA)], and by the US Department of Energy, Office of Science, Office of Nuclear Physics [Contracts No. DE-AC02-06CH11357 (ANL), No. DE-FG02-95ER40934 (UND), No. DE-FG02-97ER41033 (UNC), No. DE-FG02-97ER41041 (TUNL), No. DE-FG0294ER40834 (Maryland), and No. DE-SC0009971 (CUSTIPEN)].

Contents

1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Nuclear Deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Nuclear Triaxiality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Nuclear Wobbling Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Chirality of Nuclear Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Motivation for Studying 135 Pr Nucleus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Motivation for Studying 187 Au Nucleus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 4 5 6 7 8 8

2

Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Nuclear Shell Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Harmonic Oscillator Potential. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Woods-Saxon Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Nuclear Shapes and the Deformed Shell Model . . . . . . . . . . . . . . . . . . . . . . . 2.3 Collective Rotation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Rotation About a Principal Axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Rotation About a Tilted Axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Rotation Out of the Principal Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Wobbling Modes in Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Wobbling in Even-Even Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Wobbling in Odd-A Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Chiral Rotation in Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Nuclear Models to Study Wobbling Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 Particle Rotor Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.2 Quasiparticle Triaxial Rotor Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.3 Triaxial Projected Shell Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Theory of Angular Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11 11 12 14 16 19 20 20 22 23 23 25 27 29 29 30 31 32 35

3

Experimental Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Interaction of Gamma Rays with Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Gamma-Ray Spectroscopy with Germanium Detectors. . . . . . . . . . . . . . .

37 37 38 xiii

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Contents

3.2.1 High Purity Germanium (HPGe) Detectors . . . . . . . . . . . . . . . . . . . 3.2.2 Response Function of HPGe Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Compton Suppression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 The Gammasphere Array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Angular Distributions and DCO-Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Reduced Transition Probability Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Heavy-Ion Fusion-Evaporation Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Beam and Target Specifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 ATLAS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Run Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.1 135 Pr Run . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.2 187 Au Runs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.3 Chiral Wobbler Run . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Pre-Calibration with Digital Gammasphere Data . . . . . . . . . . . . . . . . . . . . . 3.9 Energy and Efficiency Calibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39 41 43 44 48 50 51 54 55 55 55 56 58 58 60 61

4

Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Data Sorting Into Matrices and Cubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Background Subtraction Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Coincidence Relationships and Level Schemes . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 135 Pr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 187 Au . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Spin and Parity Assignments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Angular Distributions and DCO-Like Ratios . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63 63 64 65 66 71 77 78 94

5

Results and Theoretical Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Two-Phonon Wobbling in 135 Pr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Wobbling Motion in 187 Au . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Longitudinal Wobbling Bands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Transverse Wobbling Bands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Chiral Wobbling in 135 Pr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

97 97 101 101 104 110 114

6

Updates and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Updates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

117 117 117 118

A Gammasphere Detector and Ring Information. . . . . . . . . . . . . . . . . . . . . . . . . . . 119 B Digital Gammasphere Calibration Parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . 123 C Gammasphere Energy and Efficiency Calibration Parameters . . . . . . . . 135 Curriculam vitae of Nirupama Sensharma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

Parts of This Thesis Have Been Published in the Following Journal Articles

1. N. Sensharma, U. Garg, S. Zhu, A. D. Ayangeakaa, S. Frauendorf, W. Li, G. H. Bhat, J. A. Sheikh, M. P. Carpenter, Q. B. Chen, J. L. Cozzi, S. S. Ghugre, Y. K. Gupta, D. J. Hartley, K. B. Howard, R. V. F. Janssens, F. G. Kondev, T. C. McMaken, R. Palit, J. Sethi, D. Seweryniak, R. P. Singh. Two-phonon wobbling in 135 Pr. Phys. Lett. B, 792:170, 2019. https://doi.org/10.1016/j.physletb.2019. 03.038. 2. N. Sensharma, U. Garg, Q. B. Chen, S. Frauendorf, D. P. Burdette, J. L. Cozzi, K. B. Howard, S. Zhu, M. P. Carpenter, P. Copp, F. G. Kondev, T. Lauritsen, J. Li, D. Seweryniak, J. Wu, A. D. Ayangeakaa, D. J. Hartley, R. V. F. Janssens, A. M. Forney, W. B. Walters, S. S. Ghugre, and R. Palit. Longitudinal Wobbling Motion in 187 Au. Phys. Rev. Lett., 124:052501, Feb 2020. https://doi.org/10. 1103/PhysRevLett.124.052501.

xv

Symbols

c h¯ Z N A

Speed of light in vacuum Reduced Planck constant Number of protons in a nucleus Number of neutrons in a nucleus Number of nucleons in a nucleus

xvii

Chapter 1

Background and Motivation

Abstract Triaxial nuclear shapes are very rare and their experimental identification is usually done via two signatures—chiral rotation and wobbling motion. A study of these exotic rotational modes in two different regions of the nuclear chart viz. A ∼ 130 and 190 has been done. Prior to this work, nuclear wobbling motion was well established only in the A ∼ 160 region. With a detailed study of the level structures and the prevalent wobbling motion in the 135 Pr nucleus, A ∼ 130 region has been indicated as a new region of interest where wobbling motion and, hence, triaxiality can be observed. Beyond the A ∼ 130 region, the nucleus 187 Au (A ∼ 190) has also been studied to seek evidence of nuclear wobbling motion.

1.1 Nuclear Deformations Nuclei are usually characterized by spherical or very simple shapes (see Fig. 1.1). When considering the rotational motion of nuclei, quantum mechanics states that the wave function remains invariant for rotation about an axis of symmetry, and that rotation is not allowed for any spherical body. A deformation is, therefore, a prerequisite for the appearance of rotational spectra [1]. Hence, rotational motion is characteristic of non-spherical or deformed objects. Most deformed nuclei are axially symmetric and are predicted to have a prolate or an oblate shape. A closer inspection of the nuclear chart done according to the quadrupole deformation parameter, as shown in Fig. 1.2, indicates that there is a dominance of prolate ground-state shapes over most of the nuclear chart. Certain islands of oblate nuclei are also found for N ∼ Z. However, any deviation from this prediction is an extremely rare phenomenon. The breaking of axial symmetry leads to another shape of interest that is manifested as a triaxial deformation. A triaxial nucleus (shown in Fig. 1.3) is analogous to a triaxial ellipsoid and has three axes each with a different moment of inertia (MOI). The three axes are referred to as long (l), short (s), and medium (m) axes. Consider a 1-, 2-, 3-axis system with J1 , J2 and J3 being the three respective moments of inertia. In the body-fixed frame, the relative difference in the length of © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Sensharma, Wobbling Motion in Nuclei: Transverse, Longitudinal, and Chiral, Springer Theses, https://doi.org/10.1007/978-3-031-17150-5_1

1

2

1 Background and Motivation

Fig. 1.1 (a) Prolate, (b) spherical, and (c) oblate nuclear shapes. Images generated in Mathematica

the three axes is given by Ring and Schuck [2]:  δRx = R0

2π 5 β cos(γ − k), 4π 3

(1.1)

where R0 = 1.2A1/3 , β is the quadrupole deformation parameter, γ is the triaxiality parameter, and k = 1, 2 and 3 corresponding to the three axes. The upper panel of Fig. 1.4 shows the relative difference in the length of the three axes and the lower panel shows the irrotational flow type MOI (Eq. (2.35)) as a function of γ . It is seen that for γ ∈ [0,60], 1-, 2- and 3-axis corresponds to m-, s-, and l-axis, respectively. Furthermore, this figure also makes the following characteristics immediately clear: • γ = 0: J1 = J2 , J3 = 0 (indicates a prolate deformation) • γ ∈ (0,30) – The length of the m-axis is closer to that of the s-axis – s-axis has a larger MOI than the l-axis • γ = 30: J1 > J2 = J3 = 0 (indicates a triaxial deformation)

1.1 Nuclear Deformations

3

Fig. 1.2 The nuclear chart colored according to the quadrupole deformation parameter obtained from the B(E2) value (β2 = (4π /3ZR20 )[B(E2)]1/2 , where R0 = 1.2A1/3 ), and the B(E2) values are obtained from ENSDF. Image created with data from www.nndc.bnl.gov/nudat Fig. 1.3 Triaxial nuclear shape with three unequal axes, referred as long, medium and short axes. Image generated in Mathematica

• γ ∈ (30,60) – The length of m-axis is closer to that of the l-axis – s-axis has a smaller MOI than the l-axis • γ = 60: J1 = J3 , J2 = 0 (indicates an oblate deformation)

4

1 Background and Motivation

Fig. 1.4 Length of axes (relative difference) and the irrotational flow type MOI for the three principal axes as functions of γ from 0◦ to 60◦

1.2 Nuclear Triaxiality Nuclear triaxiality is a very rare phenomenon and is predicted for the ground state of only a few small regions of the nuclear chart. Reference [3] studied the effect of axial asymmetry on the ground-state mass of 5900 nuclei from A = 31 to N = 160. Their results are summarized in Fig. 1.5, which shows the lowering of the nuclear ground-state energy when axial symmetry is broken relative to that in axially symmetric shapes. The colder, bluer colors in the chart indicate regions of stronger well-established triaxiality. The experimental estimation of the magnitude of γ deformation of nuclei in their ground states is not very direct. Therefore, experiments often rely on the observation of γ bands in nuclei for regions of the nuclear chart where the ground-state shape is triaxial. Figure 1.5 shows the characteristic γ bands in the measured spectrum of the three regions (marked in blue) with stronger ground-state triaxiality. For triaxiality manifesting at excited states, two unique experimental signatures are relied on—wobbling motion and chiral rotation. These modes of rotation are extremely rare and usually originate when particles with high angular momentum couple with the collective motion of a triaxial core.

1.2 Nuclear Triaxiality

5

Fig. 1.5 Calculated lowering of the nuclear ground-state energy when axial symmetry is broken, relative to axially symmetric shapes. Reprinted figure with permission from Möller et al. [3]. Copyright (2022) by the American Physical Society

The dynamics of the coupled system of the odd nucleon and the rotating core are based on three physical mechanisms [4]: • The triaxial core which has three moments of inertia prefers rotation about the axis with the largest moment of inertia in order to minimize the rotational energy. • The coupling of the odd nucleon to the core prefers maximal mass overlap. • The interaction between the angular momentum of the odd nucleon and the rotating core due to centrifugal and Coriolis forces leads to the corresponding energy being minimal for a parallel alignment of the two.

1.2.1 Nuclear Wobbling Motion Nuclear wobbling motion is an exotic phenomenon that arises for a triaxial nuclear density distribution. It is described as the harmonic oscillation of one of the three principal axes of a triaxial rotor about its space-fixed total angular momentum vector. Bohr and Mottelson predicted this motion in even-even nuclei over 40 years ago [1]. The observed experimental evidence is however still rather limited. Prior to this work, wobbling had been observed only in five nuclei in the A ∼ 160 region: 161 Lu [5], 163 Lu [6, 7], 165 Lu [8], 167 Lu [9] and 167 Ta [10]; two in the A ∼ 130 region: 135 Pr [12] and 133 La [13]; and in 105 Pd [14]. Wobbling motion has since been observed in 130 Ba [15] and 183 Au [16] as well.

6

1 Background and Motivation

The experimental manifestation of this exotic motion is the appearance of multiple rotational bands, each corresponding to a particular wobbling phonon number (nω ) with the following properties: • Connecting I = 1 transitions are observed between successive wobbling bands (nω+1 → nω ), • Due to the collective nature of these excitations, the I = 1 connecting transitions are predominantly E2 in nature and, • High reduced E2 transition probability ratios, B(E2out )/B(E2in ), are observed for the connecting I = 1 transitions. Wobbling in odd-A nuclei is categorized into two types—longitudinal and transverse. This classification is based on the coupling of the odd nucleon with the triaxial core. If the odd nucleon aligns parallel (perpendicular) to the maximum moment of inertia axis of the triaxial core, it results in a longitudinal (transverse) wobbler. These two types of wobbling motion are described in further detail in the next chapter.

1.2.2 Chirality of Nuclear Rotation An object is said to possess chiral symmetry if it can be superimposed on its mirror image by virtue of rotation only. Objects that do not exhibit this property are said to be chiral in nature. The phenomenon of chirality has been studied extensively in molecules. Nuclei, on the other hand, had been assumed to not exhibit chiral properties owing to their relatively simple shapes as compared to molecules. However, Ref. [17] showed that the rotational motion of triaxial nuclei attains a chiral character if the angular momentum has substantial projections on all three principal axes of the triaxial density distribution. Experimentally, chiral bands appear as two I = 1 bands having the following characteristics: • • • •

same parity, close excitation energies, constant staggering parameter (relative position of energy levels) and, identical B(M1)/B(E2) ratios

So far, 62 candidate chiral doublet bands in 49 nuclei (including 6 nuclei with multiple chiral doublets) have been observed in odd-odd nuclei as well as in oddA and even-even nuclei, and these are spread over the A ∼ 80, 100, 130, and 190 regions (See Fig. 1.6) [18].

1.3 Motivation for Studying 135 Pr Nucleus

7

Fig. 1.6 The nuclei with wobbling bands (black triangles) and chiral bands (blue squares) observed in the nuclear chart so far

1.3 Motivation for Studying 135 Pr Nucleus The first experimental observation of nuclear wobbling motion was in the nucleus 163 Lu [6, 7], in the year 2001. Some other observed wobblers were 167,165,161 Lu [5, 8, 9] and 167 Ta [10]. All of these were in the A ∼ 160 region and searches outside of these isotopes, even in the same region of the nuclear chart, did not produce any fruitful results. For a long time, therefore, wobbling was known only in these five nuclei. The identification of 135 Pr as a wobbling nucleus in 2015 by Ref. [12] marked a turning point in the study of nuclear wobbling motion. For the first time, wobbling motion was observed outside of the A ∼ 160 region. This observation was important not only because it indicated the A ∼ 130 region to be a new region of interest where wobbling motion could be found, but also because wobbling in 135 Pr was found at a much lower deformation of  ∼ 0.16 and based on an odd h11/2 proton configuration while all the previously known wobblers were based on a much larger deformation of  ∼ 0.40 and the odd i13/2 proton configuration. Reference [19] also suggested the existence of a second-phonon (nω = 2) wobbling band in the nucleus 135 Pr but was unable to firmly establish it due to insufficient statistics. The present work, therefore, aimed to observe the nω = 2 band in this nucleus by performing a high-statistics experiment, and further affirm the existence of nuclear wobbling motion in the A ∼ 130 region. Another interesting feature observed in the 135 Pr nucleus was the possible coexistence of nuclear wobbling motion and chiral rotation. With increased statistics, the present work was able to identify two interconnected I = 1 bands, building over the nω = 1 and 2 bands. A closer examination of the two bands revealed possible chiral rotation built over wobbling excitations, thereby, identifying 135 Pr as the first observed chiral wobbler.

8

1 Background and Motivation

1.4 Motivation for Studying 187 Au Nucleus With the observation of wobbling motion in the A ∼ 130 ([11–13, 15]), A ∼ 160 ([5–10]) and A ∼ 100 ([14]) regions, it has been established that wobbling is not limited to any particular region of the nuclear chart, and can be observed with different neutron and proton configurations. Taking the next step, the present work has investigated the existence of nuclear wobbling motion in the A ∼ 190 region. A systematic study of potential energy surfaces has led to the prediction of significant triaxiality at low spins in nuclei in the A ∼ 190 region [20]. Moreover, chiral band pairs, which serve as another clear signature of triaxiality, have also been observed in a number of nuclei in this mass region (188 Ir [21], 194 Tl [22], and 198 Tl [23] to name a few). Within this landscape, the 186 Pt nucleus is known to exhibit strong triaxial + behavior as seen from the energy staggering of the 2+ 2 and 41 states [24]. Moreover, the presence of the π h9/2 orbital is expected to lead to the stabilization of triaxial shapes [25]. This suggested seeking wobbling adjacent to the even-even platinum cores as an attractive option, and the present work has therefore chosen 187 Au as its nucleus of interest.

References 1. A. Bohr, B.R. Mottelson, Nuclear Structure, vol.II, chap. 4 (W. A. Benjamin, New York, 1975). https://books.google.com/books?id=bDXgCO3Z4bIC 2. P. Ring, P. Schuck, The Nuclear Many-Body Problem, 1st edn. (Springer-Verlag, Berlin/Heidelberg, 1980). https://www.springer.com/gp/book/9783540212065 3. P. Möller, R. Bengtsson, B.G. Carlsson, P. Olivius, T. Ichikawa, Phys. Rev. Lett. 97, 162502 (2006). https://doi.org/10.1103/PhysRevLett.97.162502. https://link.aps.org/doi/10. 1103/PhysRevLett.97.162502 4. J. Meyer-Ter-Vehn, Nucl. Phys. A 249(1), 111–140 (1975). https://doi.org/10.1016/03759474(75)90095-0. http://www.sciencedirect.com/science/article/pii/0375947475900950 5. P. Bringel, G.B. Hagemann, H. Hübel, A. Al-khatib, P. Bednarczyk, A. Bürger, D. Curien, G. Gangopadhyay, B. Herskind, D.R. Jensen et al., Eur. Phys. J. A 24, 167–172 (2005) https:// doi.org/10.1140/epja/i2005-10005-7. https://doi.org/10.1140/epja/i2005-10005-7 6. S.W. Ødegård, G.B. Hagemann, D.R. Jensen, M. Bergström, B. Herskind, G. Sletten, S. Törmänen, J.N. Wilson, P.O. Tjøm, I. Hamamoto et al., Phys. Rev. Lett. 86(26), 5866– 5869 (2001) https://doi.org/10.1103/PhysRevLett.86.5866. https://link.aps.org/doi/10.1103/ PhysRevLett.86.5866 7. D.R. Jensen, G.B. Hagemann, I. Hamamoto, S.W. Ødegård, B. Herskind, G. Sletten, J.N. Wilson, K. Spohr, H. Hübel, P. Bringel et al., Phys. Rev. Lett. 89, 142503 (2002). https://doi.org/ 10.1103/PhysRevLett.89.142503. https://link.aps.org/doi/10.1103/PhysRevLett.89.142503 8. G. Schönwaßer, H. Hübel, G.B. Hagemann, P. Bednarczyk, G. Benzoni, A. Bracco, P. Bringel, R. Chapman, D. Curien, J. Domscheit et al., Phys. Lett. B 552, 9– 16 (2003). https://doi.org/10.1016/S0370-2693(02)03095-2. http://www.sciencedirect.com/ science/article/pii/S0370269302030952 9. H. Amro, W.C. Ma, G.B. Hagemann, R.M. Diamond, J. Domscheit, P. Fallon, A. Görgen, B. Herskind, H. Hübel, D.R. Jensen, Phys. Lett. B 553, 197–203 (2003). https://doi. org/10.1016/S0370-2693(02)03199-4. http://www.sciencedirect.com/science/article/pii/ S0370269302031994

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10. D.J. Hartley, R.V.F. Janssens, L.L. Riedinger, M.A. Riley, A. Aguilar, M.P. Carpenter, C.J. Chiara, P. Chowdhury, I.G. Darby, U. Garg et al., Phys. Rev. C 80, 041304 (2009). https://doi. org/10.1103/PhysRevC.80.041304. https://link.aps.org/doi/10.1103/PhysRevC.80.041304 11. N. Sensharma, U. Garg, S. Zhu, A.D. Ayangeakaa, S. Frauendorf, W. Li, G.H. Bhat, J.A. Sheikh, M.P. Carpenter, Q.B. Chen et al., Two-phonon wobbling in 135 Pr, Phys. Lett. B 792, 170–174 (2019). ISSN 0370-2693. https://doi.org/10.1016/j.physletb.2019.03.038. https:// www.sciencedirect.com/science/article/pii/S0370269319301959 12. J.T. Matta, U. Garg, W. Li, S. Frauendorf, A.D. Ayangeakaa, D. Patel, K.W. Schlax, R. Palit, S. Saha, J. Sethi et al., Phys. Rev. Lett. 114, 082501 (2015). https://doi.org/10.1103/ PhysRevLett.114.082501. https://link.aps.org/doi/10.1103/PhysRevLett.114.082501 13. S. Biswas, R. Palit, U. Garg, G.H. Bhat, S. Frauendorf, W. Li, J.A. Sheikh, J. Sethi, S. Saha, P. Singh et al., Eur. Phys. J. A 55, 159 (2019). https://doi.org/10.1140/epja/i2019-12856-5 14. J. Timár, Q.B. Chen, B. Kruzsicz, D. Sohler, I. Kuti, S.Q. Zhang, J. Meng, P. Joshi, R. Wadsworth, K. Starosta et al., Phys. Rev. Lett. 122, 062501 (2019). https://doi.org/10.1103/ PhysRevLett.122.062501 15. Y.K. Wang, F.Q. Chen, P.W. Zhao, Phys. Lett. B 802, 135246 (2020). ISSN 0370-2693. https://doi.org/10.1016/j.physletb.2020.135246. http://www.sciencedirect.com/science/article/ pii/S0370269320300502 16. S. Nandi, G. Mukherjee, Q.B. Chen, S. Frauendorf, R. Banik, S. Bhattacharya, S. Dar, S. Bhattacharyya, C. Bhattacharya, S. Chatterjee et al., Phys. Rev. Lett. 125, 132501 (2020). https://doi.org/10.1103/PhysRevLett.125.132501 17. S. Frauendorf, Rev. Mod. Phys. 73, 463–514 (2001). https://doi.org/10.1103/RevModPhys.73. 463 18. Q.B. Chen, J. Meng, Nucl. Phys. News 30(1), 11–15 (2020). https://doi.org/10.1080/10619127. 2019.1676119 19. J.T. Matta, Exotic Nuclear Excitations: The Transverse Wobbling Mode in 135 Pr (Springer International Publishing, New York, 2017). https://doi.org/10.1007/978-3-319-53240-0 20. T. Nikši´c, D. Vretenar, P. Ring, Prog. Part. Nucl. Phys. 66(3), 519–548 (2011). ISSN 0146-6410. https://doi.org/10.1016/j.ppnp.2011.01.055. http://www.sciencedirect.com/ science/article/pii/S0146641011000561 21. D.L. Balabanski, M. Danchev, D.J. Hartley, L.L. Riedinger, O. Zeidan, Jing-ye Zhang, C.J. Barton, C.W. Beausang, M.A. Caprio, R.F. Casten et al., Phys. Rev. C 70, 044305 (2004). https://doi.org/10.1103/PhysRevC.70.044305 22. P. Masiteng, A. Pasternak, E. Lawrie, O. Shirinda, J. Lawrie, R. Bark, S. Bvumbi, N. Kheswa, R. Lindsay, E. Lieder, R. Lieder et al., Dsam lifetime measurements for the chiral pair in 194tl. Eur. Phys. J. A 52, 02 (2016). https://doi.org/10.1140/epja/i2016-16028-y 23. E.A. Lawrie, P.A. Vymers, J.J. Lawrie, C. Vieu, R.A. Bark, R. Lindsay, G.K. Mabala, S.M. Maliage, P.L. Masiteng, S.M. Mullins, S.H.T. Murray, I. Ragnarsson, T.M. Ramashidzha, C. Schück, J.F. Sharpey-Schafer, O. Shirinda, Phys. Rev. C 78, 021305 (2008). https://doi. org/10.1103/PhysRevC.78.021305 24. G. Hebbinghaus, T. Kutsarova, W. Cast, A. Krämer-Flecken, R.M. Lieder, W. Urban, Nucl. Phys. A 514(2), 225–251 (1990). ISSN 0375-9474. https://doi.org/10.1016/03759474(90)90068-W. http://www.sciencedirect.com/science/article/pii/037594749090068W 25. V.I. Dimitrov, S. Frauendorf, F. Dönau, Phys. Rev. Lett. 84, 5732–5735 (2000). https://doi.org/ 10.1103/PhysRevLett.84.5732

Chapter 2

Theory

Abstract Stable nuclei with essentially spherical shapes can be described very well using spherically symmetric nuclear potentials. However, a transition to the deformed shell model is essential to correctly describe the appearance of rotational spectra in nuclei that take up well-deformed shapes. This chapter discusses in detail the phenomenon of collective rotation in a deformed nuclear system and the various rotational symmetries associated with it. Nuclear wobbling motion and Chiral rotation are two such collective rotation phenomena appearing only in triaxiallyshaped nuclei. These modes have been described in detail using different nuclear models that consider a triaxial rotor as the core. Finally, a detailed theory of γ -γ directional angular distributions has also been presented.

2.1 Nuclear Shell Model Developed in 1949 by several independently working physicists, the Nuclear shell model describes the arrangement of nucleons within a nucleus. There has been considerable experimental evidence that suggests a shell model structure for nucleons. For instance, when adding nucleons to a nucleus, there are certain points where the binding energy of the next nucleon is significantly less than the last one. Another example is when the proton and neutron separation energies are plotted as deviations from the predictions of the semi-empirical mass formula, sharp discontinuities corresponding to the filling of major shells are seen [1]. This sudden and discontinuous behavior occurs at specific proton or neutron numbers called the magic numbers. These magic numbers correspond to Z or N = 2, 8, 20, 28, 50, 82, and 126, and represent the effects of filled major shells. The basic assumption of the nuclear shell model is a mean-field potential i.e., the potential governing the motion of a single nucleon is caused by all the other nucleons. Under the influence of such a mean-field potential, the nucleons can occupy the energy levels of a series of subshells in accordance with the Pauli exclusion principle. The first step in developing such a model is therefore to choose a potential that can correctly reproduce all the magic numbers and, hence, explain the observed nuclear properties. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Sensharma, Wobbling Motion in Nuclei: Transverse, Longitudinal, and Chiral, Springer Theses, https://doi.org/10.1007/978-3-031-17150-5_2

11

12

2 Theory

Fig. 2.1 Schematic representation of the functional form of three shell-model potentials: Squarewell, Harmonic oscillator, and Woods-Saxon

The three generally used shell-model potentials include the square-well, harmonic oscillator, and the Woods-Saxon potentials. Figure 2.1 shows a schematic of the general form for these three potentials. Since the nuclear potential closely approximates the nuclear charge and matter distribution falling smoothly to zero beyond the mean radius, it cannot have a sharp edge as in a square-well [1]. Hence, the square-well potential was not found to be a good approximation. The next best approximation is the harmonic-oscillator potential that has a flat bottom but still requires infinite separation energies. A more realistic picture is that given by the Woods-Saxon potential. The following subsections discuss the latter two potentials in more detail.

2.1.1 Harmonic Oscillator Potential Considering a single-particle spectrum, a one-body hamiltonian can be written as [2]: H (r) = −

h¯ 2 2 ∇ + V (r), 2m

(2.1)

where r is the coordinate of the nucleon and m is its mass. Assuming a central potential that depends only on the magnitude of r but not on its direction, the harmonic oscillator well potential is given by Wong [2]:

2.1 Nuclear Shell Model

13

V (r) =

1 mω02 r 2 , 2

(2.2)

where ω0 is the frequency of the simple harmonic motion of the particle in a nucleus. The equation of motion for a simple harmonic oscillator is: −

h¯ 2 2 1 ∇ ψ(r) + mω02 r 2 ψ(r) = εψ(r), 2m 2

(2.3)

and the energy eigenvalues obtained from Eq. (2.3) are:   3 hω εN = N + ¯ 0, 2

(2.4)

where N is the number of oscillator quanta that characterizes each major shell for an isotropic, three-dimensional harmonic oscillator potential. All states belonging to a given shell (i.e. the same N) are then degenerate with energy given by Eq. (2.4). The allowed orbital angular momenta, l, for each shell takes the following values: l = N, N − 2, . . . , 1 or 0.

(2.5)

Since each nucleon has an intrinsic spin s = 1/2, the number of states, DN , representing the maximum number of neutrons or protons a harmonic oscillator shell can accommodate is given by: DN = 2



(2l + 1) = 2

allowed l

N +1 

k = (N + 1)(N + 2),

(2.6)

k=1

where the factor of 2 in front of the summations accounts for the two possible orientations of nucleon intrinsic spin. The total number of states, Dmax , up to Nmax , is given by a sum over all N -values to Nmax (Obtained using the identity  n 1 2 k=1 k = 6 n(n + 1)(2n + 1).) Dmax =

N max  N =0

DN =

1 (Nmax + 1)(Nmax + 2)(Nmax + 3). 3

(2.7)

Substituting Nmax = 0, 1, 2, 3, 4, . . . in Eq. (2.7), one obtains Dmax = 2, 8, 20, 40, 70, . . . . Using the harmonic oscillator potential, hence, reproduces the first three magic numbers—2, 8, and 20. However, deviations are found beyond Nmax = 2. In order to correct this, additional terms need to be introduced into the single-particle Hamiltonian as given by the harmonic-oscillator potential in Eq. (2.2). An addition is made to the single-particle Hamiltonian to take into consideration the short-range nature of the strong nuclear force. This term corrects for the steepness of the harmonic-oscillator potential and is achieved by including an

14

2 Theory

attractive term, l 2 , in the nuclear potential. This centrifugal term provides an angular momentum barrier that lifts the degeneracy between levels with the same N but having different values of l. The correct sequence of the magic numbers is produced by including a strong spin-orbit coupling interaction to the single-particle potential [3, 4]. Taking into consideration the spin dependence in the nucleon-nucleon interaction, a potential having a term depending on the intrinsic spin of a nucleon, s, and its orbital angular momentum, l, is invoked. Since j = s + l, two possible states can be formed for a given l. Depending on whether s is parallel (j = l +1/2) or anti-parallel (j = l −1/2) to l, the two states would have different energies. The single-particle Hamiltonian then takes the form: H (r) = −

h¯ 2 2 1  s, ∇ + mω02 r 2 + bl 2 + a l. 2m 2

(2.8)

where the parameters b and a depend on the nucleon number A and can be determined by fitting observed single-particle energies. With the added spin-orbit term, the single-particle energy given by Eq. (2.4) gets modified to the following:    3 + 12 al for j = l + 12 εN = N + hω ¯ 0= 2 − 12 a(l + 1) for j = l − 12 .

(2.9)

The energy splitting between the two levels given by Eq. (2.9) is a(2l + 1)/2. The sequence of magic numbers generated by including the spin-orbit term to the potential is schematically shown in Fig. 2.2.

2.1.2 Woods-Saxon Potential Named after Roger D. Woods and David S. Saxon, the Woods-Saxon potential is a mean-field potential used to describe the forces applied to each nucleon within the nuclear shell model. The potential takes the form [5]: V (r) = −

V0 1 + exp( r−R a )

,

(2.10)

where V0 represents the depth of the potential (typically ≈ 50 MeV), a is the surface thickness (≈ 0.5 fm), and R = r0 A1/3 is the nuclear radius with r0 = 1.2 fm.

2.1 Nuclear Shell Model

15

Fig. 2.2 Schematic showing the single-particle energy spectrum with the correct sequence of magic numbers marked in red

16

2 Theory

The Woods-Saxon potential has the following properties: • It increases monotonically with distance, i.e., attracting in nature, • For large A, it is approximately flat in the center of the nucleus, • Nucleons near the surface of the nucleus (r ≈ R) experience a large force towards the center and, • It rapidly approaches zero as r goes to infinity, thereby reflecting the short range of the strong nuclear force. Due to the aforementioned properties, the Woods-Saxon potential is considered a more realistic description of the mean-field potential.

2.2 Nuclear Shapes and the Deformed Shell Model For stable nuclei, the nuclear shape is essentially spherical [1], and their properties can be explained very well using the spherically symmetric nuclear potentials described in the previous section. However, as we depart from sphericity, the nuclear density distribution takes up a deformed shape that can be described in terms of a set of shape parameters [2]: ⎛ R(θ, φ; t) = R0 ⎝1 +



⎞ αλμ (t)Yλμ (θ, φ)⎠ ,

(2.11)

λμ

where R(θ, φ; t) is the distance from the center of the nucleus to the surface at angles (θ, φ) and time t, R0 is the equilibrium radius for a sphere having the same volume, αλμ (t) is the amplitude of nuclear deformation of the nuclear core, and Yλμ is a harmonic function for the spherical coordinates λ and μ. Each mode of order λ has 2λ + 1 parameters, corresponding to μ = -λ, -λ + 1, . . . , λ. However, in order to keep R(θ, φ; t) real, symmetry requirements impose the following conditions: ∗ Yλμ (θ, φ) = (−1)μ Yλ,−μ (θ, φ)

(2.12)

αλμ (t) = (−1)μ αλ,−μ (t).

The λ = 1 mode corresponds to an oscillation about a fixed point in the laboratory and is usually of no interest to studying the internal dynamics of a nucleus. The next mode, λ = 2 is however of prime importance as it represents quadrupole deformation in a nucleus. In this case, Eq. (2.11) reduces to: ⎛ R(θ, φ; t) = R0 ⎝1 +

2  μ=−2

⎞ α2μ (t)Y2μ (θ, φ)⎠ ,

(2.13)

2.2 Nuclear Shapes and the Deformed Shell Model

17

with five shape parameters, α2μ (t) for μ = -2, -1, 0, +1, +2. From the symmetry requirements in Eq. (2.12), we have α2,−1 = α2,1 = 0 and α2,−2 = α2,2 . The three non-zero parameters, α2,0 and α2,±2 are usually described in terms of β and γ such that, α2,0 = β cos γ 1 α2,2 = α2,−2 = √ β sin γ , 2

(2.14)

which indicates that the parameter β provides a measure of the extent of deformation while γ provides a measure of the departure from axial symmetry. In other words, β is the quadrupole deformation parameter while γ is the triaxiality parameter. For an axially symmetric case where γ = 0, a negative β indicates an oblate nucleus while a positive β indicates a prolate nucleus (see Fig. 1.1 for different nuclear shapes). Figure 2.3 shows the various nuclear shapes in the (β, γ )-plane as defined by the Lund convention. Within this convention, at γ = 0◦ and −60◦ , the rotational axis is perpendicular to the nuclear symmetry axis which indicates

Fig. 2.3 Schematic showing the nuclear deformations described in the Lund convention

18

2 Theory

collective motion (described in the next section). At γ = +60◦ , the shape is oblate while γ = −120◦ , the shape is prolate. However, for both these cases, the rotational axis is oriented along the symmetry axis which indicates a contribution from the intrinsic motion of individual nucleons i.e., a non-collective motion. The γ parameter with all other values describes various degrees of triaxiality with the maximum degree of triaxial deformation at γ = 30◦ . The next higher mode, λ = 3 corresponds to an octupole deformation which manifests as reflection-asymmetric pear-shaped nuclei [6]. The next higher order, λ = 4 corresponds to a hexadecapole deformation. λ = 4 and other higher order terms are sufficiently small and can be ignored. Since these are not the focus of the present work, the higher order terms are not discussed any further. For nuclei with a deformed shape, the spherical shell model (described in Sect. 2.1) cannot be used to explain the residual interactions between the valence nucleons i.e., nucleons outside the closed shell. A deformed shell model is, therefore, needed to describe the properties of such nuclei. The Nilsson potential, also known as the modified harmonic-oscillator potential [7] forms the basis of such a deformed shell model. This potential allows to take deformations into account, and the associated Hamiltonian can be written as: HN ilsson = −

  

h¯ 2 2 1  s − μ l2 − l2 ∇ + m(ωx2 x 2 + ωy2 y 2 + ωz2 z2 ) + 2κ hω , ¯ 0 l. N 2m 2

(2.15)    s is the spin-orbit term and (l 2 − l 2 ) term is added to simulate the where l. N flattening of the nuclear potential at the bottom of the well (similar to that obtained using the Woods-Saxon potential). The factor κ represents the strength of the spinorbit term while μ represents the strength of the l 2 term. The value of hω ¯ 0 is taken as 41A−1/3 MeV for spherical nuclei. The ωx,y,z terms are the one-dimensional oscillator frequencies that can be expressed as a function of the deformation, , which relates to the quadrupole deformation as  = 0.95β. For an axially symmetric case:   2 2 2 2 ωx = ωy ω0 1 +  3   4 ωz2 = ω02 1 −  (2.16) 3   4 2 16 3 −1/6 where ω0 () = ω00 1 −  −  . 3 27  1/3 The parameter ω00 ( = 0) = ωx ωy ωz , is the harmonic oscillator parameter which incorporates the principle of volume conservation for deformed nuclei. The energy eigenvalues, εi , can be obtained by using the Nilsson Hamiltonian (Eq. (2.15)), and solving the Schrödinger equation: HN ilsson ψi = εi ψi ,

(2.17)

2.3 Collective Rotation

19

Laboratory axis

Symmetry axis

Fig. 2.4 Asymptotic quantum numbers for the Nilsson model. R, J , and j are the collective, total, and particle angular momenta, respectively. M and K are the projections of J onto the laboratory and the symmetry axes, respectively

where i represents the set of asymptotic quantum numbers used to specify the Nilsson orbitals, π [Nnz ]. Here,  is the projection of the particle’s total angular momentum onto the symmetry axis, π is the parity defined as π = (−1)l = (−1)N , N is the oscillator quantum number, nz is the number of oscillator quanta, and  is the projection of the particle’s orbital angular momentum onto the symmetry axis. Figure 2.4 shows the coupling of angular momenta in a deformed nucleus, and the aforementioned asymptotic quantum numbers used in the Nilsson model. To show single-particle levels as a function of deformation, a Nilsson plot, also known as a Nilsson diagram can be drawn. Figures 5 and 6 in Ref. [8] show Nilsson diagrams for neutrons and protons in the range 50 ≤ Z ≤ 82. At  = 0, a large gap between energy levels is seen. These gaps indicate a particle number at which there is a shell closure i.e., reproduces the magic numbers.

2.3 Collective Rotation The collective model developed by Bohr and Mottelson [9] was derived from the success of the liquid-drop model formulated by Bohr [10], and the prediction of nuclear deformation by Ref. [11]. Within the collective model, the nuclear density distribution is pictured as a droplet of nuclear matter characterized by shape degrees of freedom, and capable of vibrations and rotations. For more information on

20

2 Theory

collective vibrations, the interested reader is referred to Refs. [2, 12, 13]. In the following, we discuss the collective rotational phenomenon in more detail. Bohr and Mottelson have shown that the wave function associated with a nucleus remains invariant for rotation about an axis of symmetry, and that rotation cannot occur for a spherical body. A deformation is therefore a prerequisite for the appearance of rotational spectra [9]. The orientation and the rotational degree of freedom for a deformed nucleus are specified by its anisotropic density distribution. For such a deformed nuclear system comprised of many particles, each of which contributes to the total angular momentum, the rotation is classified as being collective in nature. Figure 2.5a shows the schematic of a mean-field nuclear density distribution having three principal axes (1, 2, and 3) and the total angular momentum J pointing along the axis of rotation. The following subsections discuss the different rotational symmetries associated with such a deformed nuclear system.

2.3.1 Rotation About a Principal Axis This subsection deals with cases where the rotational axis coincides with one of the principal axes of the deformed density distribution. As an example, consider a system as shown in Fig. 2.5b where the rotational axis, J points along the 3-axis of the deformed nucleus. For such a system, Rz (π ) | = exp−iαπ | ,

(2.18)

where Rz (π ) corresponds to rotation about the z-axis by an angle of π , and α is a quantum number called signature. For rotation about a principal axis, signature is a good quantum number, and gives a selection rule for the total angular momentum, I = α + 2n, n = 0, ±1, ±2, . . . ,

(2.19)

where α takes the value of 0 or 1 for an even-A nucleus while for an odd-A nucleus, it takes the value of ±1/2. Therefore, for a given parity π , principal axis rotation leads to sequences of I = 2 bands having alternate signatures.

2.3.2 Rotation About a Tilted Axis This subsection deals with planar cases where the rotational axis, J, is tilted away from the principal axes (as shown in Fig. 2.5c) but still lies in one of the three principal planes. This kind of rotation is manifested in nuclei with axial deformation. For such a system,

2.3 Collective Rotation

21

Fig. 2.5 Schematic representation of (a) a mean-field nuclear density distribution with three principal axes 1, 2, and 3, and the axis of rotation, J with orientation angles θ and φ, (b) rotation about a principal axis, (c) rotation about a tilted axis, and (d)–(e) right- and left-handed solutions respectively arising from rotation out of the principal planes

22

2 Theory

Rz (π ) | = exp−iαπ |

(2.20)

The values that the total angular momentum I can take are, therefore, not limited as in Eq. (2.19). The resulting rotational bands correspond to sequences of states of all possible values of I and fixed π .

2.3.3 Rotation Out of the Principal Planes This subsection deals with planar cases where the rotational axis, J, does not lie in any of the three principal planes spanned by the density distribution (shown in Fig. 2.5d and e). This kind of rotation is manifested in nuclei with a non-axial deformation. For such a system, the symmetry corresponding to a rotation about the x-axis by an angle of π combined with the time reversal (T ), T Ry (π ), is broken. The breaking of the T Ry (π ) symmetry leads to a doubling of the rotational levels. Thus, there are two identical I = 1 sequences with the same parity, which are the even and odd linear combinations of the left-handed |l and right-handed |r mean-field solutions. The T Ry (π ) symmetry is broken if the angular momentum has components on all three principal axes. This solution is therefore manifested only in triaxial nuclei and is called chiral rotation. Figure 2.6a, b, and c show sample band structures resulting respectively from a principal axis rotation, a tilted axis rotation, and from the breaking of the T Ry (π ) symmetry.

Fig. 2.6 Example of band structure arising from (a) rotation about a principal axis leading to a I = 2 sequence with fixed parity, (b) rotation about a tilted axis leading to a I = 1 sequence with fixed parity, and (c) rotation out of the principal planes leading to two identical I = 1 sequences with the same parity

2.4 Wobbling Modes in Nuclei

23

2.4 Wobbling Modes in Nuclei Wobbling was first introduced by Bohr and Mottelson [9] as a collective mode that appears when the moments of inertia of all the three principal axes of the nuclear density distribution are unequal. The appearance of wobbling excitations, thereby, serves as a definite indication of the existence of triaxiality. In classical mechanics, uniform rotation about the axis with the largest moment of inertia corresponds to minimal energy for a given angular momentum. At slightly larger energy, the other two axes also contribute to the motion resulting in a nonuniform rotation. Reference [9] refers to this non-uniform rotation as wobbling motion. In the intrinsic body-fixed frame, wobbling motion is interpreted as the oscillation of the total angular momentum vector (J) about the axis with the largest moment of inertia. For a triaxial nucleus, the medium (m) axis corresponds to the axis with the largest moment of inertia, and so, it is this axis that J wobbles about.

2.4.1 Wobbling in Even-Even Nuclei The description of wobbling motion, as given in Ref. [9], was based on a triaxial even-even nucleus. This is called a simple or a standard wobbler. Wobbling excitations in a simple wobbler are described as small amplitude oscillations of the total angular momentum vector J about the axis with the largest moment of inertia (m-axis). Bohr and Mottelson considered the ideal case of harmonic oscillations, which appears as the small-amplitude limit of the exact eigenfunctions of the triaxial rotor Hamiltonian [9]. In accordance with the interpretation of nuclear vibrations, wobbling excitations are also classified by their phonon number when their structure deviates from the harmonic limit [14, 15]. Figure 2.7 gives the schematic of a standard wobbler. Fig. 2.7 Schematic representation of a standard wobbler. l, m and s represent the long, medium, and short axes, respectively. R and J are the rotor and total angular momentum, respectively

24

2 Theory

The Hamiltonian of a rigid triaxial rotor, in the body fixed frame, is given by: H = A3 Jˆ32 + A1 Jˆ12 + A2 Jˆ22 ,

(2.21)

where Jk are the angular momentum components, and Ak are the rotational parameters for the three principal axes (k = 1, 2, and 3). The Ak parameters are defined as: Ak =

h¯ 2 , 2Jk

(2.22)

where Jk are the three principal moments of inertia. From the conservation of angular momentum, we obtain: J 2 = J12 + J22 + J32 = I (I + 1),

(2.23)

and the energy is given by: E = A3 J32 + A1 J12 + A2 J22 .

(2.24)

Denoting the three axes of the triaxial rotor as 1, 2, and 3 with the 3-axis being the axis having the largest moment of inertia, Eq. 2.22 gives the inequalities: A1 > A2 > A3 and J1 < J2 < J3 .

(2.25)

The wobbling excitations are then defined as small amplitude oscillations of the angular momentum vector (J1 , J2 , J3 ) about the 3-axis of the largest moment of inertia. Their energy is given by a harmonic spectrum of wobbling quanta [9]   1 H = A3 I (I + 1) + n + hω ¯ w, 2

(2.26)

where n is the number of wobbling quanta, and hω ¯ w is the wobbling energy defined as the energy associated with wobbling excitations, and is given by Bohr and Mottelson [9]: 1/2 hω ¯ w = 2I ((A1 − A3 ) (A2 − A3 )) .

(2.27)

Using the inequality given by Eq. (2.25) in Eq. (2.27), it is seen that for a simple wobbler, the wobbling energy must increase with increasing spin.

2.4 Wobbling Modes in Nuclei

25

2.4.2 Wobbling in Odd-A Nuclei To describe wobbling in odd-A nuclei, Frauendorf and Dönau utilized the Quasiparticle Triaxial Rotor Model [14] (described later in Sect. 2.6.2). Within this model, an odd-mass triaxial nucleus is described as a triaxial even-even core coupled to a quasiparticle (an unpaired proton or neutron with a high angular momentum). This coupling provides the necessary alignment and results in a considerably modified wobbling motion. Depending on the triaxial core and quasiparticle coupling, wobbling in odd-A nuclei can be classified into two different types—longitudinal and transverse.

2.4.2.1

Longitudinal and Transverse Wobbling Bands

If the odd quasiparticle aligns parallel to the maximum moment of inertia axis (m-axis) of the triaxial core, the alignment gives rise to longitudinal wobbling. Figure 2.8a shows the schematic of a longitudinal wobbler where the odd particle with angular momentum j aligns with the m-axis. As a result of this alignment, the m-axis oscillates about the space-fixed total angular momentum vector J . For longitudinal wobbling, therefore, the following relations hold true: J3 = Jm ; J3 > J2 and J3 > J1 .

(2.28)

On the other hand, if the odd quasiparticle aligns perpendicular to the maximum moment of inertia axis (s- or l-axis) of the triaxial core, the alignment gives rise to transverse wobbling. Figure 2.8b shows the schematic of a transverse wobbler where an odd particle with angular momentum j aligns with the s-axis. As a result

Fig. 2.8 Schematic representation of (a) Longitudinal wobbling and (b) Transverse Wobbling. l, m, and s represent the long, medium, and short axes, respectively. R, j , and J are the rotor, odd particle, and total angular momentum, respectively

26

2 Theory

of this alignment, the s-axis oscillates about J . For transverse wobbling, therefore, the following relations hold true: J3 = Js ; J3 < J2 and J3 > J1 .

(2.29)

To account for the additional odd quasiparticle, the hamiltonian (Eq. (2.26)), and correspondingly the expression for wobbling energy (Eq. (2.27)) changes such that [14]: j hω ¯ w= J3

     1/2 J J3 J J3 1+ −1 1+ −1 . j J1 j J2

(2.30)

Using Eqs. (2.28) and (2.29) in Eq. (2.30), it is seen that the wobbling energy increases with J for longitudinal wobbling but follows a decreasing trend for transverse wobbling. It must be noted here that Eq. (2.30) is valid only for a harmonic frozen approximation where the odd particle is assumed to be rigidly aligned to one of the principal axes. The terminology of transverse and longitudinal wobbling can still be applied for strongly anharmonic motion, as long as the modes have the same topology as the harmonic limit from which they quantitatively deviate.

2.4.2.2

Yrast and Signature Partner Bands

In addition to the aforementioned wobbling bands, a triaxial rotor coupled to an odd quasiparticle also gives rise to yrast and signature partner bands. The yrast band is defined as the lowest energy band and is seen when the angular momentum of the triaxial rotor and the high-j quasiparticle is completely aligned with the m-axis. Figure 2.9 gives the geometry corresponding to the yrast band. Figure 2.10 shows the geometry for the signature partner band. The signature partner band arises due to the dealignment of the quasiparticle with respect to a principal axis of the rotor. As shown in Fig. 2.10, the odd quasiparticle with angular momentum j is not aligned with the s-axis. Instead, it precesses around the s-axis resulting in a significant contribution from the quasiparticle motion. For an axial nuclear core coupled with a quasiparticle, there is an yrast band with α = − 12 and a signature partner band with α = + 12 [16]. For a triaxial nucleus, however, in addition to the yrast and signature partner bands, a wobbling band arises due to the collective motion of the core. The E2 transition probability, B(E2) associated with a collective core is larger than the B(E2) associated with the quasiparticle motion. Hence, the wobbling and the signature partner bands can be differentiated by comparing their respective B(E2) values.

2.5 Chiral Rotation in Nuclei

27

Fig. 2.9 Schematic representation for the geometry of an yrast band. l, m, and s represent the long, medium, and short axes, respectively. R, j , and J are the rotor, odd particle, and total angular momentum, respectively

Fig. 2.10 Schematic representation for the geometry of a signature partner band. l, m, and s represent the long, medium, and short axes respectively. R, j , and J are the rotor, odd particle, and total angular momentum, respectively

2.5 Chiral Rotation in Nuclei The occurrence of chirality in triaxially deformed nuclei was first suggested by Frauendorf and Meng in 1997 [17] by applying the Tilted Axis Cranking (TAC) theory [18] to the model of two particles coupled to a triaxial rotor. As described in Sect. 2.3, chirality occurs when the axis of rotation does not lie in any of the three principal planes spanned by the density distribution. This occurs when high-j particles of one kind of nucleon align with the s-axis, high-j holes of the other kind align with the l-axis, and the triaxial core itself rotates about the maxis. This arrangement leads to the breaking of the time-reversal symmetry and

28

2 Theory

Fig. 2.11 Schematic representation of (a) left- and (b) right-handed chiral systems for a triaxial odd-odd nucleus. R, jπ , jν , and J are the rotor, proton, neutron, and total angular momentum, respectively

the total angular momentum pointing outside all of the principal planes. The soformed principal axes system is right-handed if the s-, m-, and l-axes are ordered counterclockwise with respect to the total angular momentum and left-handed otherwise. Figure 2.11 gives the schematic of a chiral system. To understand the arrangement of this system, consider a triaxial nucleus with one proton in an orbital just above the Fermi surface (particle) and one neutron just below the Fermi surface (hole). The proton particle aligns its angular momentum jπ along the s-axis of the density distribution to maximize the overlap of its orbital with the triaxial density. Since the core-particle interaction is attractive, the alignment of jπ along the saxis corresponds to minimum energy. On the other hand, the neutron hole aligns its angular momentum jν along the l-axis to minimize the overlap of its orbital with the triaxial density. Since the core-hole interaction is repulsive, the alignment of jν along the l-axis corresponds to minimum energy. The angular momentum of the core R is collective in nature and aligns itself along the axis having the largest moment of inertia i.e. the m-axis. As a result of these alignments, the total angular momentum J lies outside the three principal planes. With respect to J , the three axes form either a left-handed or a right-handed system as shown in Figs. 2.11a and b, respectively. The transformation from a left-handed to a right-handed system occurs by reversing the direction of the component of the angular momentum on the m-axis. Experimentally, chiral symmetry breaking is manifested as two identical I = 1 sequences that correspond to the left-handed |l and right-handed |r mean-field solutions, which follow the relation given by Frauendorf [18]: |l = T Ry (π ) |r .

(2.31)

2.6 Nuclear Models to Study Wobbling Motion

29

Reference [18] has shown that chirality in nuclei is based only on the symmetry of the rotating triaxial nucleus, and is independent of how the three components of the angular momentum are composed. Hence, chirality is expected for all configurations that have substantial angular momentum components along the three principal axes, no matter how the individual components are composed.

2.6 Nuclear Models to Study Wobbling Motion This section deals with three approaches that have been used in the present work to study nuclear wobbling motion occurring as a consequence of nuclear triaxiality.

2.6.1 Particle Rotor Model The Particle Rotor Model (PRM) considers a high-j quasiparticle coupled to the rotor, and is used to study the wobbling motion of triaxial nuclei. The total Hamiltonian of the PRM is given by Bohr and Mottelson [9] and Ring and Schuck [16]: Hˆ P RM = Hˆ coll + Hˆ intr ,

(2.32)

where Hˆ intr is the intrinsic Hamiltonian and Hˆ coll is the collective rotor Hamiltonian. Hˆ coll =

3 3   Rˆ k2 (Iˆk − jˆk )2 = , 2Jk 2Jk k=1

Hˆ intr =



(ν − λ) aν† aν +

ν

(2.33)

k=1



 δ (μ, ¯ ν) aμ† aν† + aν aμ . 2 μ,ν

(2.34)

The index, k = 1, 2, 3 denotes the three principal axes of the body-fixed frame and Rˆ k is the collective rotational angular momentum operator of the rotor. Iˆk and jˆk are the total and quasiparticle angular momentum operators respectively. Jk corresponds to the three principal moments of inertia,  is the pair-correlation parameter in the BCS approximation, and ν expresses the one-particle energies for a potential V . The PRM employs hydrodynamical moments of inertia given by: Jk =

  2 4 J0 sin2 γ + π k , 3 3

(2.35)

30

2 Theory

where γ is the triaxiality parameter and k = 1, 2, 3. Equation (2.35), therefore, restricts rotation to the region with −60◦ < γ < 0◦ . The medium axis of the triaxial rotor can, hence, be understood as the axis with the largest moment of inertia, and consequently, with the largest component of collective rotational angular momentum [15]. As a quantal model coupling the collective rotation to the single-particle motion, PRM has been widely used to successfully describe wobbling bands [14, 15, 19–24].

2.6.2 Quasiparticle Triaxial Rotor Model The Quasiparticle triaxial rotor (QTR) model considers the core as a triaxial rotor, which is coupled to a quasiparticle using the core-quasiparticle coupling (CQPC) model to obtain the properties of the odd-mass nucleus so formed [25]. As shown in Fig. 2.12, this model can be used to describe an odd-A nucleus either as a particle or a hole coupled to a triaxial rotor. To account for the presence of a high-j quasiparticle, the triaxial rotor Hamiltonian (Eq. (2.21)) must be replaced by the QTR Hamiltonian given by Frauendorf and Dönau [14]:

2

2

2 H = hdqp + A3 Jˆ3 − jˆ3 + A1 Jˆ1 − jˆ1 + A2 Jˆ2 − jˆ2 ,

(2.36)

where the operators jˆk and Jˆk are the angular momentum of the odd quasiparticle and the total angular momentum, respectively. The coupling of the odd quasiparticle to the triaxial core is reflected by the hdqp term. It is responsible for aligning a highj particle with the s-axis to maximize the overlap between the density distributions of the particle and the triaxial core, thereby, minimizing the attractive short-range core-particle interaction. Similarly, the hdqp term in Eq. (2.36) is responsible for aligning a high-j hole with the l-axis to minimize overlap between the density distributions of the hole and the triaxial core, thereby, minimizing the repulsive short-range core-hole interaction. The coupling further aligns a quasiparticle arising from a half-filled high-j orbital with the m-axis. In practice, the QTR calculations determine the energy of the first 2+ state of the triaxial core by making the level spacing of the yrast band in the odd-A nucleus agree with the experimental values [25]. Further, by adjusting the triaxial deformation parameter, γ , wobbling energy is obtained for the odd-A nucleus to match the experimental values.

2.6 Nuclear Models to Study Wobbling Motion

31

Fig. 2.12 Schematic of a quasiparticle coupling with a triaxial even-even core. This model serves as the basis of the Quasiparticle triaxial rotor model

2.6.3 Triaxial Projected Shell Model The Projected Shell Model (PSM) follows the basic philosophy of the standard shell model approach with the exception of employing a deformed basis instead of a spherical one. PSM has been successfully used to describe diverse nuclear structure phenomena such as backbending [26], superdeformation [27, 28], and signature dependence, [26] to name a few. To explain the aforementioned phenomena, PSM has historically assumed axial symmetry for the deformed system. However, for transitional nuclei in the rareearth region, it becomes imperative that the assumption be changed to incorporate a triaxial basis to describe the additional collective excitations generated by deviations from the axial shape. The simplest example includes the observation of gamma bands in triaxial nuclei. The model so developed has, hence, been referred to as the Triaxial Projected Shell Model (TPSM). To generate the deformed single-particle wave functions, TPSM utilizes the triaxial Nilsson potential specified by the deformation parameters  and   , and is given by Sheikh and Hara [29]:   ˆ −2 ˆ +2 + Q Q 2  ˆ0 +  Hˆ N = Hˆ 0 − h¯ ω  Q , √ 3 2

(2.37)

where Hˆ 0 is the spherical harmonic-oscillator single-particle Hamiltonian. The interested reader is referred to Ref. [29] for an in-depth description of the TPSM approach.

32

2 Theory

Fig. 2.13 Sample spin sequence of a gamma transition

2.7 Theory of Angular Distributions In order to establish nuclear wobbling motion and chiral rotation in triaxial nuclei, it is important to study the spectroscopic properties of the γ transitions connecting the bands of interest. Angular distributions provide a reliable means to extract multipolarities of the transitions and, hence, form a pivotal part of the present work. Here, we describe the theory behind γ -γ directional angular distributions. The probability of emission of a γ ray from the excited state of a radioactive nucleus depends on the angle between the direction of emission of the γ ray and the nuclear spin axis. Under ordinary circumstances, nuclear spins are randomly oriented in space and, therefore, the total radiation emitted is isotropic. An anisotropic pattern of radiation can only be observed from an ensemble of nuclei with spin orientations that are not randomly oriented [30]. Shown in Fig. 2.13 is a sample spin sequence for a γ transition. We consider a collection of identical nuclei in an excited initial state (Ji , Mi ) that decays to the final state (Jf , Mf ) by emission of a γ ray. Assuming all the Mi states in the initial level to be equally populated at the time of emission of the γ ray, a relative population PMf of each Mf sublevel in the final state is defined. PMf depends on the emitting γ radiation, and can be obtained by summing over all Mi → Mf transitions leading to the final Mf substate. P (Mf ) =



|Ji Mi |H |Jf Mf |2 FLM (θ ),

(2.38)

i

where H is the hamiltonian (See Ref. [31] for definition) which gives rise to the emission of γ radiation having angular momentum L and the angular momentum component along the axis of orientation M = Mi − Mf . Following this, Eq. (2.38) can be written as:  P (Mf ) ∝ |Ji Mi Jf Mf |LM |2 FLM (θ ), (2.39) i

2.7 Theory of Angular Distributions

33

where |Ji Mi Jf Mf |LM | are the Clebsch-Gordan coefficients representing the transition probability from Mi → Mf substates and FLM is the characteristic directional distribution for each Mi → Mf transition. θ is the angle between the direction of emission of γ and the axis of orientation. The directional angular distribution function, W (θ ), is the probability of emission of γ at an angle θ with respect to the orientation axis. It depends on the relative population of the Mi states, the transition probability, and the directional distribution for each component. W (θ ) ∝



P (Mi )|Ji Mi Jf Mf |LM |2 FLM (θ )

(2.40)

Mi Mf

It can be seen from Eq. (2.40) that except for the directional distribution function, all other terms reduce to coefficients. The FLM (θ ) terms can be expressed as Legendre Polynomials [32]. The angular distribution of γ transitions relative to the beam axis, therefore, takes the form: W (θ ) =

k max

(Ak Pk (cos θ ))

(2.41)

k

The Ak coefficients are the angular distribution parameters that depend on the degree of alignment, the spins of the initial and final states, the mixing ratio, and the multipolarity of the γ rays involved. We now consider the angular momentum of the emitted γ ray to have two components L1 and L2 (L2 = L1 + 1) with mixing ratio, δ. The Ak coefficients can then be expressed in terms of Fk coefficients [33]. Ak (Ji L1 L2 Jf ) =

 ρk Fk (Jf L1 L1 Ji ) + 2δFk (Jf L1 L2 Ji ) + δ 2 Fk (Jf L2 L2 Ji ) , 2 1+δ

(2.42)

where the Fk -coefficients are defined as: Fk (Jf L1 L2 Ji ) = (−1)Jf −Ji −1 [(2L1 + 1)(2L2 + 1)(2Ji + 1)]1/2 × L1 1L2 − 1|k0 W (Ji Ji L1 L2 ; kJf )

(2.43)

Here, L1 1L2 − 1|k0 is the Clebsch–Gordan coefficient and W is the Racah coefficient. The mixing ratio, δ is defined as the ratio of the matrix elements, δ=

Jf ||L2 ||Ji Jf ||L1 ||Ji

(2.44)

Under cases of complete alignment, a statistical tensor term (Bk (J )) for a state J is defined in Eq. (2.45). The values that this tensor takes for various spin values are listed in Ref. [33].

34

2 Theory

Bk (J ) = (2J + 1)1/2 (−1)J J 0 J 0|k 0 , 1

= (2J + 1)1/2 (−1)J − 2 J

for integral spin

1 −1 J |k 0 , 2 2

for half-integral spin (2.45) For the ideal case, i.e., for complete alignment, the Ak coefficients defined by Eq. (2.42) modify to the following: Amax k (Ji L1 L2 Jf ) =

 1 fk (Jf L1 L1 Ji ) + 2δfk (Jf L1 L2 Ji ) + δ 2 fk (Jf L2 L2 Ji ) , 2 1+δ

(2.46)

where fk (Jf L1 L2 Ji ) = Bk (Ji )Fk (Jf L1 L2 Ji )

(2.47)

In practical cases, however, the alignment is only partial. The definition of the Ak coefficients for partial alignment changes to, Ak (Ji L1 L2 Jf ) = αk (Ji )Amax k (Ji L1 L2 Jf ),

(2.48)

where αk (Ji ) is the attenuation coefficient of the alignment: αk (Ji ) =

ρk (Ji ) Bk (Ji )

(2.49)

and ρk is defined as: ρk (J ) = (2J + 1)1/2



(−1)J −M J MJ − M|k0 P (M)

(2.50)

M

For an aligned state, (P (M) = P (−M)), ρk (J ) vanishes unless k is even. Moreover, for fusion-evaporation reactions (as employed in the present work), the population parameters P (M) are assumed to follow a Gaussian probability distribution:

P (M) =  J

exp

m =−J

−m2 2σ 2



exp

−m2 2σ 2

,

(2.51)

where σ is the width of the distribution. The case of σ = 0 corresponds to complete alignment, thereby rendering the attenuation coefficient, αk (J ) = 1 (from Eq. (2.49)). σ is obtained experimentally by fitting the initial state’s spin Ji to the function σ = C × Ji . For typical fusion-evaporation reactions, the parameter C usually takes values in the range 0.2–0.4 [34].

References

35

References 1. K.S. Krane, Introductory Nuclear Physics, 3rd edn. (Wiley, Hoboken, 1987). https://www. wiley.com/en-us/Introductory+Nuclear+Physics%2C+3rd+Edition-p-9780471805533 2. S.S.M. Wong, Introductory Nuclear Physics, 2nd edn. (Wiley, Hoboken, 1998). https://doi.org/ 10.1002/9783527617906. https://www.wiley.com/en-us/Introductory+Nuclear+Physics%2C+ 2nd+Edition-p-9780471239734 3. M.G. Mayer, Nuclear configurations in the spin-orbit coupling model. ii. theoretical considerations. Phys. Rev. 78, 22–23 (1950). https://doi.org/10.1103/PhysRev.78.22 4. J.H.D. Jensen, M.G. Mayer, Elementary Theory of Nuclear Shell Structure. (Wiley, Hoboken, 1960). https://books.google.com/books?id=z24sAAAAYAAJ 5. R.D. Woods, D.S. Saxon, Phys. Rev. 95(2), 577–578 (1954). https://doi.org/10.1103/PhysRev. 95.577. https://ui.adsabs.harvard.edu/abs/1954PhRv...95..577W 6. P.A. Butler, W. Nazarewicz, Rev. Mod. Phys. 68(2), 4 (1996). https://doi.org/10.1103/ RevModPhys.68.349. https://www.osti.gov/biblio/286260 7. S.G. Nilsson, Binding States of Individual Nucleons in Strongly Deformed Nuclei (CERN Publications, I Kommission hos Munksgaard, 1955). https://books.google.com/books?id= R4L5PwAACAAJ 8. High Energy Physics Division, Appendix Nuclear Structure. Website, http://dbserv.pnpi.spb.ru/ elbib/tablisot/toi98/www/struct/struct.pdf. Accessed Jun 22, 2022 9. A. Bohr, B.R. Mottelson, Nuclear Structure, vol.II, chap. 4. (W. A. Benjamin, New York, 1975). https://books.google.com/books?id=bDXgCO3Z4bIC 10. N. Bohr, Nature 137, 344–348 (1936). ISSN 1476-4687. https://doi.org/10.1038/137344a0 11. J. Rainwater, Phys. Rev. 79, 432–434 (1950). https://doi.org/10.1103/PhysRev.79.432 12. C.A. Bertulani, Nuclear Physics in a Nutshell (Princeton University Press, Princeton, 2007). ISBN 9781400839322. https://books.google.com/books?id=n51yJr4b_oQC 13. E. Eichler, A survey of some properties of even-even nuclei. Rev. Mod. Phys. 36, 809–814 (1964). https://doi.org/10.1103/RevModPhys.36.809 14. S. Frauendorf, F. Dönau, Phys. Rev. C 89, 014322 (2014). https://doi.org/10.1103/PhysRevC. 89.014322 15. I. Hamamoto, Phys. Rev. C 65, 044305 (2002) https://doi.org/10.1103/PhysRevC.65.044305 16. P. Ring, P. Schuck, The Nuclear Many-Body Problem, 1st edn. (Springer-Verlag, Berlin/Heidelberg, 1980). https://www.springer.com/gp/book/9783540212065 17. S. Frauendorf, J. Meng, Nucl. Phys. A 617(2), 131–147 (1997). ISSN 0375-9474. https://doi. org/10.1016/S0375-9474(97)00004-3 18. S. Frauendorf, Rev. Mod. Phys. 73, 463–514 (2001). https://doi.org/10.1103/RevModPhys.73. 463 19. J.T. Matta, U. Garg, W. Li, S. Frauendorf, A.D. Ayangeakaa, D. Patel, K.W. Schlax, R. Palit, S. Saha, J. Sethi et al., Phys. Rev. Lett. 114, 082501 (2015). https://doi.org/10.1103/ PhysRevLett.114.082501 20. N. Sensharma, U. Garg, Q.B. Chen, S. Frauendorf, D.P. Burdette, J.L. Cozzi, K.B. Howard, S. Zhu, M.P. Carpenter, P. Copp et al., Phys. Rev. Lett. 124, 052501 (2020). https://doi.org/10. 1103/PhysRevLett.124.052501 21. J. Timár, Q.B. Chen, B. Kruzsicz, D. Sohler, I. Kuti, S.Q. Zhang, J. Meng, P. Joshi, R. Wadsworth, K. Starosta et al., Phys. Rev. Lett. 122, 062501 (2019). https://doi.org/10.1103/ PhysRevLett.122.062501 22. N. Sensharma, U. Garg, S. Zhu, A.D. Ayangeakaa, S. Frauendorf, W. Li, G.H. Bhat, J.A. Sheikh, M.P. Carpenter, Q.B. Chen et al., Phys. Lett. B 792, 170 (2019). https://doi.org/10.1016/j.physletb.2019.03.038. http://www.sciencedirect.com/science/article/ pii/S0370269319301959 23. S. Nandi, G. Mukherjee, Q.B. Chen, S. Frauendorf, R. Banik, S. Bhattacharya, S. Dar, S. Bhattacharyya, C. Bhattacharya, S. Chatterjee et al., Phys. Rev. Lett. 125, 132501 (2020). https://doi.org/10.1103/PhysRevLett.125.132501

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24. E. Streck, Q.B. Chen, N. Kaiser, U.-G. Meißner, Phys. Rev. C 98, 044314 (2018). https://doi. org/10.1103/PhysRevC.98.044314 25. W. Li, Algebraic Collective Model and Its Application to Core Quasiparticle Coupling. Ph.D. Thesis, University of Notre Dame, 2016 26. K. Hara, Y. Sun, Nucl. Phys. A 529(3), 445–466 (1991). ISSN 0375-9474. https://doi. org/10.1016/0375-9474(91)90580-Y. http://www.sciencedirect.com/science/article/pii/ 037594749190580Y 27. Y. Sun, M. Guidry, Phys. Rev. C 52, R2844–R2847 (1995). https://doi.org/10.1103/PhysRevC. 52.R2844 28. Y. Sun, J.-Y. Zhang, M. Guidry, Phys. Rev. Lett. 78, 2321–2324 (1997). https://doi.org/10. 1103/PhysRevLett.78.2321 29. J.A. Sheikh, K. Hara, Phys. Rev. Lett. 82, 3968–3971 (1999). https://doi.org/10.1103/ PhysRevLett.82.3968 30. K. Siegbahn, Beta- and Gamma-ray Spectroscopy (North-Holland Publishing Company, Amsterdam, 1955). https://books.google.com/books?id=Y2x4BYWdCiAC 31. S.R. DeGroot. Physica 18(12), 1201–1214 (1952). https://doi.org/10.1016/S00318914(52)80196-X. http://www.sciencedirect.com/science/article/pii/S003189145280196X 32. R.W. Carr, J.E.E. Baglin, At. Data Nucl. Data Tables 10(2), 143–204 (1971). https://doi. org/10.1016/S0092-640X(71)80042-6. http://www.sciencedirect.com/science/article/pii/ S0092640X71800426 33. T. Yamazaki, Nucl. Data Sheets A 3(1), 1–23 (1967). https://doi.org/10.1016/S0550306X(67)80002-8. http://www.sciencedirect.com/science/article/pii/S0550306X67800028 34. J.T. Matta, Exotic Nuclear Excitations: The Transverse Wobbling Mode in 135 Pr (Springer International Publishing, New York, 2017). https://doi.org/10.1007/978-3-319-53240-0

Chapter 3

Experimental Details

Abstract The experimental study of the nuclear structure largely depends on radiation detectors and associated electronics. With the advancement in detection techniques and the development of highly-efficient Gamma-ray detector arrays such as Gammasphere, it has become possible to identify some rare and exotic nuclear phenomena such as that exhibited by triaxial nuclei. This chapter deals with the experimental methods implemented to populate high-spin states in 135 Pr and 187 Au nuclei using fusion-evaporation reactions performed at the Argonne Tandem Linear Accelerator System and detected using the Gammasphere array. Also discussed is the digital data acquisition system and the energy and efficiency calibration methods employed prior to data analysis.

3.1 Interaction of Gamma Rays with Matter When interacting with matter, gamma rays (γ rays) undergo collision with atoms and lose energy either by ionization or excitation. This can occur via three general processes: Photoelectric Absorption The phenomenon of a γ ray interacting with an absorber atom leading to its complete absorption followed by the ejection of an energetic photoelectron by the atom from one of its bound shells is called photoelectric absorption. This process can only occur for electrons bound in an atom. The released photoelectron has kinetic energy hν − Eb , where, hν is the energy of the incident γ ray and Eb is the binding energy of the photoelectron in its original shell. Depending on its energy, the photoelectron can then further ionize and excite the absorber atoms or escape the medium. Photoelectric absorption is the primary interaction mode for low-energy γ rays and for absorber materials with a high atomic number, Z. The photoelectric absorption cross-section varies approximately as [1]: σph ∝

Zn , Eγ3.5

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Sensharma, Wobbling Motion in Nuclei: Transverse, Longitudinal, and Chiral, Springer Theses, https://doi.org/10.1007/978-3-031-17150-5_3

(3.1)

37

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3 Experimental Details

where n takes values between 4 and 5 over the region spanned by γ energies. Compton Scattering The interaction of a γ ray with a loosely bound electron of an absorber atom leads to Compton scattering. In this process, the γ ray imparts some of its energy to the target electron and then scatters away at an angle with reduced energy. The target electron recoils with kinetic energy equal to the difference in the energies of the incident and the scattered γ ray. The energy of the scattered photon in terms of the energy of the incident photon and the scattering angle θ is given by: hν  =

hν 1+

hν (1 − cos θ ) m0 c2

(3.2)

The γ ray can scatter at all possible angles, thereby, leading the recoil electron to have energies ranging from zero to a large fraction of the incident Eγ . Compton scattering primarily occurs for moderate γ -ray energies and for absorber atoms of low Z. The probability of Compton scattering per atom of the absorber depends on the number of electrons available as scattering targets and, therefore, increases linearly with Z [1]. Pair Production When the energy of a γ ray is greater than twice the rest mass energy of an electron (2 × me where me = 0.511 MeV), it can undergo pair production in the interacting medium i.e., completely disappear by giving rise to an electron-positron pair. All the excess energy of the γ ray is shared by the released electron-positron pair as kinetic energy. As the created positron slows through the interacting medium, it annihilates and gives rise to two annihilation photons which act as the secondary products of the interaction. Pair production is predominant for high-energy γ rays and varies as the square of the atomic number of the absorber [1].

3.2 Gamma-Ray Spectroscopy with Germanium Detectors Gamma-ray spectroscopy is a tool that is used to study the structure of a nucleus by identifying its energy levels via the measurement of their characteristic decay γ rays [1]. The tool of γ spectroscopy is largely dependent on the choice of the kind of radiation detector. The following detector properties are considered when selecting a γ ray spectrometer: • • • •

High energy resolution—helps distinguish between closely spaced γ energies. High detection efficiency—enables identification of the weaker channels. Granularity—uniquely defines the γ -ray scattering angle. Fast timing properties—ensures fast detector response.

3.2 Gamma-Ray Spectroscopy with Germanium Detectors

39

Fig. 3.1 Fractions of the photopeak contributed by the different γ interaction processes for photons incident upon the face of a 6 cm diameter, 6 cm length coaxial Ge detector

In light of the above-mentioned properties, Germanium (Ge) detectors serve as an ideal choice for γ -ray spectroscopy studies. The following subsections discuss in detail the working principle and properties of Germanium detectors used in the present study. Figure 3.1 shows the relative fractions of the photopeak contributed by different γ -interaction processes (as described in the previous section) for photons incident upon the face of a 6 cm diameter, 6 cm length coaxial Germanium detector.

3.2.1 High Purity Germanium (HPGe) Detectors Germanium detectors fall under the category of semiconductor detectors. When radiation interacts with a germanium detector (germanium crystal arranged between two electrodes), electron-hole pairs are created. The electron-hole pairs so created drift towards the respective electrodes under the influence of an applied voltage giving rise to a current. The current is proportional to the energy loss of the radiation in the germanium crystal. The bandgap of germanium is only 0.7 eV [1]. Therefore, electron-hole pairs can be created in a germanium crystal solely under the influence of temperature. The probability of thermal creation of electron-hole pairs, P (T ), can be described by a Boltzmann distribution,

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P (T ) = CT

3/2

  Eg exp − , 2kT

(3.3)

where Eg is the band gap in germanium, k is the Boltzmann’s constant, and C is a constant which is dependent on the material. Equation (3.3), thus, shows that for a small Eg , the probability for thermal excitation is large. This results in unwanted noise in the detector signals. In order to reduce thermally induced noise, germanium detectors are operated at very low temperatures. Liquid nitrogen (≈77 K) dewars are, therefore, kept in thermal contact with the germanium crystal to keep them cold and at operable temperatures. To further reduce noise generated from electronics, the preamplifier associated with the germanium detector is placed as close to the detector as possible and is also cooled along with the crystal. Charge pairs can only be created in a germanium detector when radiation interacts within the neutral depletion region of the crystal. As the width of the depletion region increases, so does the probability of interaction of the incoming radiation within the detector. The detector is, hence, operated in reverse bias (sometimes close to the breakdown voltage) to obtain depletion depths ranging around 2–3 mm. For γ -spectroscopic measurements, however, much larger depletion depths are required. The depth of the depletion region is given by Knoll [1]:  d=

2V eN

1/2 ,

(3.4)

where V is the reverse bias voltage, N is the net impurity concentration in the bulk germanium material,  and e are the dielectric constant and electronic charge, respectively. At a given applied voltage, larger depletion depths can be achieved by lowering the value of N by further reducing the net impurity concentration. To achieve this condition, High Purity Germanium (HPGe) detectors were introduced in the mid-1970s. These detectors utilize an ultrapure germanium crystal (with impurity concentration reduced to ≈1010 atoms/cm3 ) as the detector material. Using HPGe detectors, depletion widths of several centimeters can be achieved thereby increasing the effective active volume of the detectors making them ideal for γ spectroscopy. The fabrication of HPGe detectors starts with the process of zone refining [1], wherein large germanium crystals are treated to reduce their impurity concentration to the minimum possible. Depending on the remaining low-level impurities, the type of the crystal is specified. The presence of low-level acceptor impurities like aluminum leads to a high purity p-type crystal whereas if donor impurities are left behind, it leads to a high purity n-type crystal. Two configurations are available during the manufacturing of HPGe detectors— planar and coaxial. Each configuration involves a high-purity p- or n-type germanium crystal arranged between two electrical contacts to minimize leakage current. The electrical contacts are made with germanium heavily doped with acceptor (donor) impurities leading to a p+ (n+ ) material. Considering an active

3.2 Gamma-Ray Spectroscopy with Germanium Detectors

41

detector volume manufactured with a high-purity n-type germanium crystal (with the depletion region extending throughout the pure crystal), the junction is made with a very thin p+ electrical contact. Additional blocking contact is needed on the back surface of the crystal to avoid any leakage current (due to the presence of a non-negligible number of minority carriers). The blocking contact is always made of the same type of material as the crystal itself. For a high-purity n-type germanium crystal, an n+ blocking contact is added. Since the blocking contact is of the same type as the crystal, no junction exists at this end and, hence, any minority carrierinduced leakage current is suppressed. The coaxial configuration of an HPGe crystal is favored over the planar configuration due to its ability to be manufactured in large sizes with lower capacitance [1]. The coaxial configuration, hence, allows for larger active detector volumes. Figures 3.2a and b show the parallel and perpendicular cross-sectional views from the cylindrical axis of an HPGe crystal. As shown in Fig. 3.2a, a closed-ended coaxial crystal with rounded front ends is the adopted manufacturing design for HPGe detectors. This design helps in achieving a radial electric field throughout the crystal. Figure 3.2b shows a p+ − n − n+ configuration for an HPGe detector. Similarly, an n+ − p − p+ configuration can also be manufactured.

3.2.2 Response Function of HPGe Spectra When a γ ray passes through an HPGe detector, it interacts with the semiconductor material via one or more of the three processes described in Sect. 3.1. These result in an energy spectrum with characteristics of the type of γ interaction within the

Fig. 3.2 Schematic of a large-volume closed-ended coaxial Ge detector showing the crosssectional view (a) parallel and (b) perpendicular to its cylindrical axis. The Ge crystal shown here is n-type, sandwiched between heavily doped p + and n+ electrical contacts. The thick black arrows show the direction of motion of the electrons and holes created within the detector

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Fig. 3.3 Effect of interaction processes on the predicted detector response function for monoenergetic γ rays with hν >> 1.022 MeV. Reprinted figure with permission from Buchtela [2]. Copyright (2022) by the Academic Press

detector. A typical energy spectrum obtained from an HPGe detector (see Fig. 3.3 for the predicted detector response function for γ rays with hν >> 1.022 MeV) consists of the following different components: Photo Peak: Also known as the full-energy peak, this corresponds to the γ peak formed due to the complete absorption of the incident γ energy. This peak is usually the result of multiple Compton scatterings followed by photoelectric absorption. An ideal detector will have all the counts in its photo peak. Compton Background: As mentioned in Sect. 3.1, when a γ photon Compton scatters, the scattering angle, θ can range from 0◦ to 180◦ . This produces a continuum of energy that the scattered γ photon can take, giving rise to a Compton background with energies less than the photo peak. Compton Edge: When the γ photon Compton scatters at θ = 180◦ , Eq. (3.2) takes the form: hν  =

hν 1+

2hν m0 c2

.

(3.5)

This gives rise to the Compton continuum being cut off at hν—hν  . This cut-off is called the Compton edge. Escape Peaks: When Eγ > 1.02 MeV, it can undergo pair production. The positron so formed then annihilates to produce two 511-keV γ rays. One or both of these can escape the detecting medium and result in a single-escape and a double-escape peak

3.2 Gamma-Ray Spectroscopy with Germanium Detectors

43

at (hν—511) keV and (hν—1.02) MeV, respectively; where hν is the energy of the photo peak. Backscatter Peak: When a γ ray scatters backward in the material surrounding the detector, it gives rise to a Backscatter peak. The energy of this peak approximately lies between 200 and 250 keV which is the difference in the energies of the photo peak and the Compton edge. Annihilation Peak: If the initial radiation undergoes pair production in the material surrounding the detector giving rise to annihilation radiation, the released 511keV γ ray is then measured in the detector. This γ -energy peak at 511 keV is the annihilation peak. X-Ray Escape Peaks: When an incident particle strikes a bound electron in an atom, the inner-shell electron is ejected leading to an outer-shell electron falling into the inner shell to fill the vacant energy level. This results in the emission of an X-ray which is characteristic of the element. This low-energy X-ray is usually reabsorbed. However, if this X-ray escapes the detector, it gives rise to a peak at an energy which is the difference between the energies of the photo peak and the emitted X-ray. For γ -spectroscopic measurements, these characteristic X-rays are usually eliminated by putting absorber sheets (like Aluminum) in front of the detecting medium. Summation Peak: Isotopes that emit multiple γ rays can give rise to a summation peak at the sum of the respective γ energies. This occurs due to the coincident detection of multiple γ rays of different energies.

3.2.3 Compton Suppression The atomic number of germanium is Z = 32, which is low as compared to some other commonly used scintillation crystals like Sodium Iodide (NaI) or Bismuth Germanate (BGO). Equation (3.1) shows that a small Z leads to a low photoelectric absorption cross section which further renders a small peak-to-total ratio (P/T). The P/T ratio is defined as the ratio of the counts in a photo peak to that in the entire spectrum. In order to improve the P/T ratio, Compton suppression is required for γ -spectroscopic measurements with HPGe detectors. It is important that the detectors used for such measurements are large enough and highly efficient to detect any events that do not lead to complete photoelectric absorption. Scintillation detectors such as NaI(Tl) or BGO (Bi4 Ge3 O12 ) are used for this purpose. Due to higher density and greater Z, BGO detectors can be manufactured into more compact configurations and are, therefore, favored as Compton suppressors [1].

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3 Experimental Details

Fig. 3.4 Schematic showing the cross-sectional view perpendicular to the cylindrical axis of a Compton suppressed HPGe detector. Shown here are γ1 and γ2 . Both γ undergo multiple Compton scatterings with γ1 ending in full absorption while γ2 scatters into the surrounding BGO

Figure 3.4 shows an HPGe detector surrounded by an annular BGO detector; this arrangement serves as a Compton suppressor. Also shown in this figure are two incident γ rays—γ1 and γ2 . γ1 undergoes two scatterings followed by a photoelectric absorption. The detection of these three events will hence lead to counts in the photo peak. γ2 , however, scatters and escapes into the surrounding BGO detector. Events produced by the interaction of γ2 within the detector lead to a continuum in the energy spectrum. A coincident circuit is set up to detect such escaping γ photons and reject them. This is the anticoincidence measurement technique utilized to suppress Compton scattering events. Compton suppression is, hence, a method used to reject events that do not contribute to the full energy absorption peak (photo peak) in a γ -ray spectrum. The surrounding BGO detector is used in anticoincidence with the HPGe detector and events that lead to the Compton continuum are preferentially rejected, without affecting the full-energy events, thereby, suppressing the total background in the spectrum. Figure 3.5 shows an example of the effect of Compton suppression on a γ -energy spectrum obtained from an HPGe detector. Compton suppression becomes more critical for high-fold γ -coincidence measurements. By eliminating accidental coincidences caused by the coincident measurement of successive Compton scattered events, the technique of Compton suppression helps in achieving a good resolving power and, hence, a good overall peak-to-total ratio.

3.2.4 The Gammasphere Array Gammasphere is one of the largest γ spectrometers and was commissioned in December 1995 [3]. High energy resolution, granularity, and high detection

3.2 Gamma-Ray Spectroscopy with Germanium Detectors

45

Fig. 3.5 A sample superimposed spectrum showing the differences between an unsuppressed and a Compton suppressed spectrum

efficiency are some of the features that make Gammasphere the most powerful γ -detector array in the present times. In the last 25 years, Gammasphere has successfully operated at the 88-Inch Cyclotron of the Lawrence Berkeley National Laboratory and the Tandem-Linac Accelerator System (ATLAS) at the Argonne National Laboratory. Figure 3.6 shows a view of the Gammasphere array stationed at the Argonne National Laboratory. Gammasphere is an array of 110 Compton suppressed HPGe detectors arranged in spherical geometry with a solid angle coverage of 0.46 × 4π [4]. The 110 detectors are arranged symmetrically around the beamline in 17 rings with each ring at a particular angle with respect to the direction of the incoming beam. The details of all the detectors and rings are provided in Table A.1. As mentioned in Sect. 3.2.3, Compton suppression is achieved using large BGO scintillation detectors arranged in hexagonal shapes around each detector. This arrangement is called a honeycomb suppression scheme (discussed in detail in Sect. 3.2.4.1). This arrangement enables a high P/T ratio, making high-fold γ coincidence measurements more practical. Table 3.1 gives a list of the different properties of the Gammasphere. High energy resolution allows the distinction between various closely spaced γ -energy peaks and a high granularity enables the localization of individual γ rays, thereby, reducing the probability of two γ -ray hits in one detector from the same event. These properties further make the Gammasphere array suitable for high-multiplicity measurements.

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3 Experimental Details

Fig. 3.6 Gammasphere array at the Argonne National Laboratory Table 3.1 Summary of Gammasphere features Number of detectors HPGe crystal size Detector volume Relative efficiency of each HPGe detectora Target to detector distance Total solid angle Total peak efficiency Singles P/T (with honeycomb suppression) Energy resolution a w.r.t

110 7.1 cm (D) × 8.22 cm (L) 312.8 cm3 70–75% 24.6 cm 0.46 × 4π 0.094 at 1.3 MeV 0.6 at 1.3 MeV 2.1 keV at 1.3 MeV

a 3 × 3 NaI detector

3.2.4.1

Honeycomb Suppression Used in Gammasphere

The Compton suppression shield employed for Gammasphere consists of six BGO scintillators and a BGO back plug surrounding each HPGe detector. Figure 3.7 shows the schematic honeycomb arrangement of HPGe detectors and their BGO Compton suppression shields as used in the Gammasphere. Shown in Fig. 3.7 is an XX’ plane along which the BGO segments are split, with each half having its own photomultiplier tube (PMT). The signals from two neighboring HPGe detectors are investigated and if both of them fire simultaneously, the two BGO segments operate individually and provide their own respective suppressions. However, if only one of the HPGe detectors fires, the outputs produced

3.2 Gamma-Ray Spectroscopy with Germanium Detectors

47

Fig. 3.7 Schematic of the honeycomb suppression scheme used in the Gammasphere array

by the split BGO segments are combined electronically, thereby, producing a suppression that is similar to that for unsplit segments. In addition to Compton suppression, this arrangement helps in avoiding false vetoes that can occur if two adjacent HPGe detectors share the same BGO suppressor [5]. The BGO back plug is more sensitive to low energies leading to a single scattering event in the HPGe detector. It also helps in improving the P/T ratio for forward scattering γ rays. High energy γ rays, however, can undergo multiple scatterings and the BGO back plug enables only a slight improvement for them.

3.2.4.2

Digital Gammasphere Acquisition System

The VME eXtensions for Instrumentation (VXI) electronics associated with the Gammasphere were developed over 20 years ago and the aging of the electronic system demanded an upgrade. The analog data acquisition system had the following limitations [6]: • Count rates in excess of 40,000 for singles and 15,000 for high-fold events could not be processed. • The processing time for HPGe shaper is ∼10 μs. This led to a ∼6% pileup at a count rate of 10,000 per second.

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3 Experimental Details

Fig. 3.8 Block diagram of the digital Gammasphere data acquisition system

• The acquisition system was dead for at least ∼25 μs for triggered events. This saturated the rate capability for single and higher-fold events. In view of the above limitations, the analog data acquisition system was replaced by a digital pulse processing data acquisition system. Following this upgrade, the HPGe shaper time could be decreased from ∼10 μs to just ∼2.5 μs. This enabled the detectors to run at a count rate of 40,000 with the same ∼6% pileup as the analog system. The digital system also allows for an improved trigger model making it feasible for count rates to go up to 500,000 for singles and 50,000 for highfold events. Figure 3.8 shows the block diagram for the digital Gammasphere data acquisition system.

3.3 Angular Distributions and DCO-Ratios Angular distribution measurements and DCO-ratios constitute very powerful tools to help in the study of excited states in nuclei. These measurements rely on investigating the radiation emitted by an oriented nucleus. Through these measurements, one can determine the multipolarity of the γ rays involved along with their associated mixing ratios. These also help in the assignment of spin sequences of the nuclear energy levels. As mentioned in Sect. 2.7, an anisotropic spin orientation is required to obtain angular distributions of γ rays. A specific orientation of nuclear spins can be achieved using any of the following three processes: • Placing the radioactive sample in a very low-temperature environment with a strong magnetic field or electric field gradient. This aligns the ensemble of nuclei and the angular distribution of emitted radiation can then be measured with respect to the direction of the applied field. • Selectively picking out only those nuclei whose spins lie in a preferred direction. This can be realized for a nucleus that decays by successive emission of two radiations, R1 and R2. By observing R1 in a fixed direction, an ensemble of

3.3 Angular Distributions and DCO-Ratios

49

nuclei having an anisotropic distribution of spin orientations is selected. The succeeding radiation R2 then shows a definite distribution with respect to R1. This method is called angular correlation of successive emission of radiation. • Utilizing nuclear reactions like coulomb excitation or fusion evaporation to populate nuclear states of interest. By default, the ensemble of nuclei produced post these reactions have spins that are aligned perpendicular to the direction of the incoming beam. This alignment can then be employed to measure the angular distribution of the emitted radiation. The present work utilizes this technique to measure angular distributions for γ rays and its corresponding method is discussed hereafter. The theory of angular distribution is given in Sect. 2.7, and combining the information obtained from Eqs. (2.41) to (2.51), the fit function is obtained as: W (θ ) = A0 (1 + A2 P2 (cos θ ) + A4 P4 (cos θ )) ,

(3.6)

where A0 , A2 and A4 are the angular distribution parameters. A0 is the normalized intensity obtained from the area under the peaks observed in the γ -ray spectra. Reference [7] gives the typical values that A2 and A4 take for pure dipole and pure quadrupole transitions. Obtaining the peak areas as a function of θ and fitting the distribution with Eq. (3.6), the experimental A2 and A4 coefficients are obtained. Using the obtained values of the A2 and A4 coefficients, Eq. (2.46) can be solved to extract the mixing ratio, δ, and the corresponding percentage of mixing between the two possible angular momentum transfers associated with the γ -ray transition. Like angular distributions, Directional Correlation of Oriented states in nuclei (DCO) analysis, can also be used to determine the multipolarities and mixing ratios of the γ rays involved. For the DCO analysis, however, two successive γ -ray transitions (γ1 and γ2 ) are selected. γ1 and γ2 are, respectively, observed in two different detectors placed at θ1 and θ2 with respect to the beam. The ratio of the intensity of γ2 in the detector at θ1 with a gate on γ1 in the detector at θ2 and the intensity of γ2 in the detector at θ2 with a gate on γ1 in the detector at θ1 relates to the DCO function [8]. RDCO =

γ

γ

γ

γ

Iθ12 (Gateθ21 ) Iθ22 (Gateθ11 )

(3.7)

Another quantity of interest is the DCO-like ratio which is analogous to the DCO ratio defined in Eq. (3.7). DCO-like ratio is the ratio of the intensity of γ2 in the detector at θ1 from a gate on γ1 in all detectors and the intensity of γ2 in the detector at θ2 from a gate on γ1 in all detectors [9]. RDCO−like =

γ

γ

γ

γ

Iθ12 (Gateall1 detectors ) Iθ22 (Gateall1 detectors )

(3.8)

50 Table 3.2 Typical values of DCO-like ratios (r) for γ rays with Dipole (D) and Quadrupole (Q) multipolarity

3 Experimental Details I 2 1 0 1 0

L Q D D D+Q D+Q

DCO-like ratio (r) 1.2 0.8 1.3 0.4