होम
Particles and Nuclei: An Introduction to the Physical Concepts, 5th Edition
Particles and Nuclei: An Introduction to the Physical Concepts, 5th Edition
Bogdan Povh, Klaus Rith, Christoph Scholz, Frank Zetsche
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2006
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3540366830
ISBN 13:
9783540366836
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आप पुस्तक समीक्षा लिख सकते हैं और अपना अनुभव साझा कर सकते हैं. पढ़ूी हुई पुस्तकों के बारे में आपकी राय जानने में अन्य पाठकों को दिलचस्पी होगी. भले ही आपको किताब पसंद हो या न हो, अगर आप इसके बारे में ईमानदारी से और विस्तार से बताएँगे, तो लोग अपने लिए नई रुचिकर पुस्तकें खोज पाएँगे.
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Particles and Nuclei An Introduction to the Physical Concepts Bogdan Povh Klaus Rith Christoph Scholz Frank Zetsche Particles and Nuclei An Introduction to the Physical Concepts Translated by Martin Lavelle Fifth Edition With 148 Figures, 11 Tables, and 58 Problems and Solutions 123 Professor Dr. Bogdan Povh Professor Dr. Klaus Rith MaxPlanckInstitut für Kernphysik Postfach 10 39 80 69029 Heidelberg, Germany Physikalisches Institut der Universität ErlangenNürnberg ErwinRommelStrasse 1 91058 Erlangen, Germany Dr. Christoph Scholz Dr. Frank Zetsche SAP AG Postfach 1461 69185 Walldorf, Germany Universität Hamburg und Deutsches ElektronenSynchrotron Notkestrasse 85 22603 Hamburg, Germany Translator: Dr. Martin Lavelle Institut de Física d’Altes Energies Facultat de Ciències Universitat Autònoma de Barcelona 08193 Bellaterra (Barcelona), Spain Title of the original German Edition: B. Povh, K. Rith, C. Scholz, F. Zetsche: Teilchen und Kerne Eine Einführung in die physikalischen Konzepte. (7. Auﬂage) © Springer 1993, 1994, 1995, 1997, 1999, 2004 und 2006 Library of Congress Control Number: 2006932338 ISBN10 3540366830 5th Edition Springer Berlin Heidelberg New York ISBN13 9783540366836 5th Edition Springer Berlin Heidelberg New York ISBN 3540201688 4th Edition Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, speciﬁcally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microﬁlm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com © SpringerVerlag Berlin Heidelberg 1995, 199; 9, 2002, 2004, 2006 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a speciﬁc statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Jürgen Sawinski, Heidelberg Production: LETEX Jelonek, Schmidt & Vöckler GbR, Leipzig Cover: WMXDesign GmbH, Heidelberg SPIN 11801610 56/3100YL  5 4 3 2 1 0 Printed on acidfree paper Preface In the last two editions we included new results on the neutrino oscillations as evidence for a nonvanishing mass of the neutrinos. In the present edition we have rewritten the chapter on “Phenomenology of the Weak Interaction” (Chapter 10) in order to give a coherent presentation of the neutrino properties. Furthermore, we extended the chapter on “Nuclear Thermodynamics” (Chapter 19). Heidelberg, July 2006 Bogdan Povh Preface to the First Edition The aim of Particles and Nuclei is to give a uniﬁed description of nuclear and particle physics because the experiments which have uncovered the substructure of atomic nuclei and nucleons are conceptually similar. With the progress of experimental and theoretical methods, atoms, nuclei, nucleons, and ﬁnally quarks have been analysed during the course of this century. The intuitive assumption that our world is composed of a few constituents — an idea which seems attractive, but could not be taken for granted — appears to be conﬁrmed. Moreover, the interactions between these constituents of matter can be formulated elegantly, and are well understood conceptionally, within the socalled “standard model”. Once we have arrived at this underlying theory we are immediately faced with the question of how the complex structures around us are produced by it. On the way from elementary particles to nucleons and nuclei we learn that the “fundamental” laws of the interaction between elementary particles are less and less recognisable in composite systems because manybody interactions cause greater and greater complexity for larger systems. This book is therefore divided into two parts. In the ﬁrst part we deal with the reduction of matter in all its complication to a few elementary constituents and interactions, while the second part is devoted to the composition of hadrons and nuclei from their constituents. We put special emphasis on the description of the experimental concepts but we mostly refrain from explaining technical details. The appendix contains a short description of the principles of accelerators and detectors. The exercises predominantly aim at giving the students a feeling for the sizes of the phenomena of nuclear and particle physics. Wherever possible, we refer to the similarities between atoms, nuclei, and hadrons, because applying analogies has not only turned out to be a very eﬀective research tool but is also very helpful for understanding the character of the underlying physics. We have aimed at a concise description but have taken care that all the fundamental concepts are clearly described. Regarding our selection of topics, we were guided by pedagogical considerations. This is why we describe experiments which — from today’s point of view — can be interpreted in a straightforward way. Many historically signiﬁcant experiments, whose results can nowadays be much more simply obtained, were deliberately omitted. Particles and Nuclei (Teilchen und Kerne) is based on lectures on nuclear and particle physics given at the University of Heidelberg to students in their 6th semester and conveys the fundamental knowledge in this area, which is required of a student majoring in physics. On traditional grounds these lectures, and therefore this book, strongly emphasise the physical concepts. We are particularly grateful to J. Hüfner (Heidelberg) and M. Rosina (Ljubljana) for their valuable contributions to the nuclear physics part of the book. We would like to thank D. Dubbers (Heidelberg), A. Fäßler (Tübingen), G. Garvey (Los Alamos), H. Koch (Bochum), K. Königsmann (Freiburg), U. Lynen (GSI Darmstadt), G. Mairle (Mannheim), O. Nachtmann (Heidelberg), H. J. Pirner (Heidelberg), B. Stech (Heidelberg), and Th. Walcher (Mainz) for their critical reading and helpful comments on some sections. Many students who attended our lecture in the 1991 and 1992 summer semesters helped us through their criticism to correct mistakes and improve unclear passages. We owe special thanks to M. Beck, Ch. Büscher, S. Fabian, Th. Haller, A. Laser, A. Mücklich, W. Wander, and E. Wittmann. M. Lavelle (Barcelona) has translated the major part of the book and put it in the present linguistic form. We much appreciated his close collaboration with us. The English translation of this book was started by H. Hahn and M. Moinester (Tel Aviv) whom we greatly thank. Numerous ﬁgures from the German text have been adapted for the English edition by J. Bockholt, V. Träumer, and G. Vogt of the MaxPlanckInstitut für Kernphysik in Heidelberg. We would like to extend our thanks to SpringerVerlag, in particular W. Beiglböck for his support and advice during the preparation of the German and, later on, the English editions of this book. Heidelberg, May 1995 Bogdan Povh Klaus Rith Christoph Scholz Frank Zetsche Table of Contents 1 Hors d’œuvre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Fundamental Constituents of Matter . . . . . . . . . . . . . . . . . . . . . . 1.2 Fundamental Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Symmetries and Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . 1.4 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part I 1 1 2 4 5 6 Analysis: The Building Blocks of Matter 2 Global Properties of Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 The Atom and its Constituents . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Nuclides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Parametrisation of Binding Energies . . . . . . . . . . . . . . . . . . . . . . 2.4 Charge Independence of the Nuclear Force and Isospin . . . . . . Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 11 13 18 21 23 3 Nuclear Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 βDecay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 αDecay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Nuclear Fission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Decay of Excited Nuclear States . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 26 31 33 35 39 4 Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 General Observations About Scattering Processes . . . . . . . . . . . 4.2 Cross Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 The “Golden Rule” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Feynman Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 41 44 48 49 52 5 Geometric Shapes of Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Kinematics of Electron Scattering . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The Rutherford CrossSection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 The Mott CrossSection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 53 56 60 X Table of Contents 5.4 Nuclear Form Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 5.5 Inelastic Nuclear Excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 6 Elastic Scattering oﬀ Nucleons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Form Factors of the Nucleons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Quasielastic Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Charge Radii of Pions and Kaons . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 73 78 80 82 7 Deep Inelastic Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Excited States of the Nucleons . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Structure Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 The Parton Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Interpretation of Structure Functions in the Parton Model . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 83 85 88 91 94 8 Quarks, Gluons, and the Strong Interaction . . . . . . . . . . . . . . 8.1 The Quark Structure of Nucleons . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Quarks in Hadrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 The Quark–Gluon Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Scaling Violations of the Structure Functions . . . . . . . . . . . . . . Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 97 102 103 107 111 9 Particle Production in e+ e− Collisions . . . . . . . . . . . . . . . . . . . . 9.1 Lepton Pair Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Nonresonant Hadron Production . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Gluon Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 114 118 123 125 126 10 Phenomenology of the Weak Interaction . . . . . . . . . . . . . . . . . . 10.1 Properties of Leptons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 The Types of Weak Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Coupling Strength of the Weak Interaction . . . . . . . . . . . . . . . . 10.4 The Quark Families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 The Lepton families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6 Majorana Neutrino? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.7 Parity Violation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.8 Deep Inelastic Neutrino Scattering . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 128 132 134 139 142 144 144 147 150 Table of Contents 11 Exchange Bosons of the Weak Interaction . . . . . . . . . . . . . . . . 11.1 Real W and Z Bosons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Electroweak Uniﬁcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XI 151 151 156 163 12 The Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 Part II Synthesis: Composite Systems 13 Quarkonia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1 The Hydrogen Atom and Positronium Analogues . . . . . . . . . . . 13.2 Charmonium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3 Quark–Antiquark Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4 The Chromomagnetic Interaction . . . . . . . . . . . . . . . . . . . . . . . . . 13.5 Bottonium and Toponium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.6 The Decay Channels of Heavy Quarkonia . . . . . . . . . . . . . . . . . . 13.7 Decay Widths as a Test of QCD . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 171 174 177 180 181 183 185 187 14 Mesons Made from Light Quarks . . . . . . . . . . . . . . . . . . . . . . . . . 14.1 Mesonic Multiplets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 Meson Masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3 Decay Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4 Neutral Kaon Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 189 193 195 197 199 15 The Baryons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1 The Production and Detection of Baryons . . . . . . . . . . . . . . . . . 15.2 Baryon Multiplets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3 Baryon Masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4 Magnetic Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5 Semileptonic Baryon Decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.6 How Good is the Constituent Quark Concept? . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 201 207 210 213 217 225 226 16 The Nuclear Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.1 Nucleon–Nucleon Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2 The Deuteron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3 Nature of the Nuclear Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 230 234 237 243 XII Table of Contents 17 The Structure of Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.1 The Fermi Gas Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2 Hypernuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3 The Shell Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.4 Deformed Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.5 Spectroscopy Through Nuclear Reactions . . . . . . . . . . . . . . . . . . 17.6 βDecay of the Nucleus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.7 Double βdecay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 245 250 253 261 264 271 279 283 18 Collective Nuclear Excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.1 Electromagnetic Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.2 Dipole Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.3 Shape Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.4 Rotation States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 286 289 297 300 309 19 Nuclear Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.1 Thermodynamical Description of Nuclei . . . . . . . . . . . . . . . . . . . 19.2 Compound Nuclei and Quantum Chaos . . . . . . . . . . . . . . . . . . . 19.3 The Phases of Nuclear Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.4 Particle Physics and Thermodynamics in the Early Universe . 19.5 Stellar Evolution and Element Synthesis . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 312 314 317 322 330 336 20 ManyBody Systems in the Strong Interaction . . . . . . . . . . . . 337 A Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1 Accelerators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3 Combining Angular Momenta . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.4 Physical Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 341 348 358 360 Solutions to the Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385 1 Hors d’œuvre Nicht allein in Rechnungssachen Soll der Mensch sich Mühe machen; Sondern auch der Weisheit Lehren Muß man mit Vergnügen hören. Wilhelm Busch Max und Moritz (4. Streich) 1.1 Fundamental Constituents of Matter In their search for the fundamental building blocks of matter, physicists have found smaller and smaller constituents which in their turn have proven to themselves be composite systems. By the end of the 19th century, it was known that all matter is composed of atoms. However, the existence of close to 100 elements showing periodically recurring properties was a clear indication that atoms themselves have an internal structure, and are not indivisible. The modern concept of the atom emerged at the beginning of the 20th century, in particular as a result of Rutherford’s experiments. An atom is composed of a dense nucleus surrounded by an electron cloud. The nucleus itself can be decomposed into smaller particles. After the discovery of the neutron in 1932, there was no longer any doubt that the building blocks of nuclei are protons and neutrons (collectively called nucleons). The electron, neutron and proton were later joined by a fourth particle, the neutrino, which was postulated in 1930 in order to reconcile the description of βdecay with the fundamental laws of conservation of energy, momentum and angular momentum. Thus, by the midthirties, these four particles could describe all the then known phenomena of atomic and nuclear physics. Today, these particles are still considered to be the main constituents of matter. But this simple, closed picture turned out in fact to be incapable of describing other phenomena. Experiments at particle accelerators in the ﬁfties and sixties showed that protons and neutrons are merely representatives of a large family of particles now called hadrons. More than 100 hadrons, sometimes called the “hadronic zoo”, have thus far been detected. These hadrons, like atoms, can be classiﬁed in groups with similar properties. It was therefore assumed that they cannot be understood as fundamental constituents of matter. In the late sixties, the quark model established order in the hadronic zoo. All known hadrons could be described as combinations of two or three quarks. Figure 1.1 shows diﬀerent scales in the hierarchy of the structure of matter. As we probe the atom with increasing magniﬁcation, smaller and smaller structures become visible: the nucleus, the nucleons, and ﬁnally the quarks. 2 1 Hors d’œuvre Atom 3.0 Nucleus 0 10 10 m Fig. 1.1. Length scales and structural hierarchy in atomic structure. To the right, typical excitation energies and spectra are shown. Smaller bound systems possess larger excitation energies. [eV] Na Atom [MeV] Nucleus 3.0 Protons and Neutrons 10 14 m 0 208 Pb Nucleus [GeV] Proton 0.3 Quark 0 10 15 m Proton Leptons and quarks. The two fundamental types of building blocks are the leptons, which include the electron and the neutrino, and the quarks. In scattering experiments, these were found to be smaller than 10−18 m. They are possibly pointlike particles. For comparison, protons are as large as ≈ 10−15 m. Leptons and quarks have spin 1/2, i. e. they are fermions. In contrast to atoms, nuclei and hadrons, no excited states of quarks or leptons have so far been observed. Thus, they appear to be elementary particles. Today, however, we know of 6 leptons and 6 quarks as well as their antiparticles. These can be grouped into socalled “generations” or “families”, according to certain characteristics. Thus, the number of leptons and quarks is relatively large; furthermore, their properties recur in each generation. Some physicists believe these two facts are a hint that leptons and quarks are not elementary building blocks of matter. Only experiment will teach us the truth. 1.2 Fundamental Interactions Together with our changing conception of elementary particles, our understanding of the basic forces of nature and so of the fundamental interactions 1.2 Fundamental Interactions 3 between elementary particles has evolved. Around the year 1800, four forces were considered to be basic: gravitation, electricity, magnetism and the barely comprehended forces between atoms and molecules. By the end of the 19th century, electricity and magnetism were understood to be manifestations of the same force: electromagnetism. Later it was shown that atoms have a structure and are composed of a positively charged nucleus and an electron cloud; the whole held together by the electromagnetic interaction. Overall, atoms are electrically neutral. At short distances, however, the electric ﬁelds between atoms do not cancel out completely, and neighbouring atoms and molecules inﬂuence each other. The diﬀerent kinds of “chemical forces” (e. g., the VanderWaals force) are thus expressions of the electromagnetic force. When nuclear physics developed, two new shortranged forces joined the ranks. These are the nuclear force, which acts between nucleons, and the weak force, which manifests itself in nuclear βdecay. Today, we know that the nuclear force is not fundamental. In analogy to the forces acting between atoms being eﬀects of the electromagnetic interaction, the nuclear force is a result of the strong force binding quarks to form protons and neutrons. These strong and weak forces lead to the corresponding fundamental interactions between the elementary particles. Intermediate bosons. The four fundamental interactions on which all physical phenomena are based are gravitation, the electromagnetic interaction, the strong interaction and the weak interaction. Gravitation is important for the existence of stars, galaxies, and planetary systems (and for our daily life), it is of no signiﬁcance in subatomic physics, being far too weak to noticeably inﬂuence the interaction between elementary particles. We mention it only for completeness. According to today’s conceptions, interactions are mediated by the exchange of vector bosons, i.e. particles with spin 1. These are photons in electromagnetic interactions, gluons in strong interactions and the W+ , W− and Z0 bosons in weak interactions. The diagrams on the next page show examples of interactions between two particles by the exchange of vector bosons: In our diagrams we depict leptons and quarks by straight lines, photons by wavy lines, gluons by spirals, and W± and Z0 bosons by dashed lines. Each of these three interactions is associated with a charge: electric charge, weak charge and strong charge. The strong charge is also called colour charge or colour for short. A particle is subject to an interaction if and only if it carries the corresponding charge: – Leptons and quarks carry weak charge. – Quarks are electrically charged, so are some of the leptons (e. g., electrons). – Colour charge is only carried by quarks (not by leptons). The W and Z bosons, masses MW ≈ 80 GeV/c2 and MZ ≈ 91 GeV/c2 , are very heavy particles. According to the Heisenberg uncertainty principle, they can only be produced as virtual, intermediate particles in scattering 4 1 Hors d’œuvre J g Photon Mass=0 Gluon Mass=0 W Z0 WBoson 2 Mass 80 GeV/c ZBoson 2 Mass 91 GeV/c processes for extremely short times. Therefore, the weak interaction is of very short range. The rest mass of the photon is zero. Therefore, the range of the electromagnetic interaction is inﬁnite. The gluons, like the photons, have zero rest mass. Whereas photons, however, have no electrical charge, gluons carry colour charge. Hence they can interact with each other. As we will see, this causes the strong interaction to be also very short ranged. 1.3 Symmetries and Conservation Laws Symmetries are of great importance in physics. The conservation laws of classical physics (energy, momentum, angular momentum) are a consequence of the fact that the interactions are invariant with respect to their canonically conjugate quantities (time, space, angles). In other words, physical laws are independent of the time, the location and the orientation in space under which they take place. An additional important property in nonrelativistic quantum mechanics is reﬂection symmetry.1 Depending on whether the sign of the wave function changes under reﬂection or not, the system is said to have negative or positive parity (P ), respectively. For example, the spatial wave function of a bound system with angular momentum has parity P = (−1) . For those laws of nature with leftright symmetry, i.e., invariant under a reﬂection in space P, the parity quantum number P of the system is conserved. Conservation of parity leads, e. g., in atomic physics to selection rules for electromagnetic transitions. The concept of parity has been generalised in relativistic quantum mechanics. One has to ascribe an intrinsic parity P to particles and antiparticles. Bosons and antibosons have the same intrinsic parity, fermions and 1 As is well known, reﬂection around a point is equivalent to reﬂection in a plane with simultaneous rotation about an axis perpendicular to that plane. 1.4 Experiments 5 antifermions have opposite parities. An additional important symmetry relates particles and antiparticles. An operator C is introduced which changes particles into antiparticles and vice versa. Since the charge reverses its sign under this operation, it is called charge conjugation. Eigenstates of C have a quantum number Cparity which is conserved whenever the interaction is symmetric with respect to C. Another symmetry derives from the fact that certain groups (“multiplets”) of particles behave practically identically with respect to the strong or the weak interaction. Particles belonging to such a multiplet may be described as diﬀerent states of the same particle. These states are characterised by a quantum number referred to as strong or weak isospin. Conservation laws are also applicable to these quantities. 1.4 Experiments Experiments in nuclear and elementary particle physics have, with very few exceptions, to be carried out using particle accelerators. The development and construction of accelerators with ever greater energies and beam intensities has made it possible to discover more and more elementary particles. A short description of the most important types of accelerators can be found in the appendix. The experiments can be classiﬁed as scattering or spectroscopic experiments. Scattering. In scattering experiments, a beam of particles with known energy and momentum is directed toward the object to be studied (the target). The beam particles then interact with the object. From the changes in the kinematical quantities caused by this process, we may learn about the properties both of the target and of the interaction. Consider, as an example, elastic electron scattering which has proven to be a reliable method for measuring radii in nuclear physics. The structure of the target becomes visible via diﬀraction only when the de Broglie wavelength λ = h/p of the electron is comparable to the target’s size. The resulting diﬀraction pattern of the scattered particles yields the size of the nucleus rather precisely. Figure 1.1 shows the geometrical dimensions of various targets. To determine the size of an atom, Xrays with an energy of ≈ 104 eV suﬃce. Nuclear radii are measured with electron beams of about 108 eV, proton radii with electron beams of some 108 to 109 eV. Even with today’s energies, 9 · 1010 eV for electrons and 1012 eV for protons, there is no sign of a substructure in either quarks or leptons. Spectroscopy. The term “spectroscopy” is used to describe those experiments which determine the decay products of excited states. In this way, 6 1 Hors d’œuvre one can study the properties of the excited states as well as the interactions between the constituents. From Fig. 1.1 we see that the excitation energies of a system increase as its size decreases. To produce these excited states high energy particles are needed. Scattering experiments to determine the size of a system and to produce excited states require similar beam energies. Detectors. Charged particles interact with gases, liquids, amorphous solids, and crystals. These interactions produce electrical or optical signals in these materials which betray the passage of the particles. Neutral particles are detected indirectly through secondary particles: photons produce free electrons or electronpositron pairs, by the photoelectric or Compton eﬀects, and pair production, respectively. Neutrons and neutrinos produce charged particles through reactions with nuclei. Particle detectors can be divided into the following categories: – Scintillators provide fast time information, but have only moderate spatial resolution. – Gaseous counters covering large areas (wire chambers) provide good spatial resolution, and are used in combination with magnetic ﬁelds to measure momentum. – Semiconductor counters have a very good energy and spatial resolution. – Čherenkov counters and counters based on transition radiation are used for particle identiﬁcation. – Calorimeters measure the total energy at very high energies. The basic types of counters for the detection of charged particles are compiled in Appendix A.2. 1.5 Units The common units for length and energy in nuclear and elementary particle physics are the femtometre (fm, or Fermi) and the electron volt (eV). The Fermi is a standard SIunit, deﬁned as 10−15 m, and corresponds approximately to the size of a proton. An electron volt is the energy gained by a particle with charge 1e by traversing a potential diﬀerence of 1 V: 1 eV = 1.602 · 10−19 J . (1.1) For the decimal multiples of this unit, the usual preﬁxes are employed: keV, MeV, GeV, etc. Usually, one uses units of MeV/c2 or GeV/c2 for particle masses, according to the massenergy equivalence E = mc2 . Length and energy scales are connected in subatomic physics by the uncertainty principle. The Planck constant is especially easily remembered in 1.5 Units 7 the form · c ≈ 200 MeV · fm . (1.2) Another quantity which will be used frequently is the coupling constant for electromagnetic interactions. It is deﬁned by: α= 1 e2 ≈ . 4πε0 c 137 (1.3) For historical reasons, it is also called the ﬁne structure constant. A system of physical quantities which is frequently used in elementary particle physics has identical dimensions for mass, momentum, energy, inverse length and inverse time. In this system, the units may be chosen such that = c = 1. In atomic physics, it is common to deﬁne 4πε0 = 1 and therefore α = e2 (Gauss system). In particle physics, ε0 = 1 and α = e2 /4π is more commonly used (HeavysideLorentz system). However, we will utilise the SIsystem [SY78] used in all other ﬁelds of physics and so retain the constants everywhere. Part I Analysis: The Building Blocks of Matter Mens agitat molem. Vergil Aeneid 6, 727 2 Global Properties of Nuclei The discovery of the electron and of radioactivity marked the beginning of a new era in the investigation of matter. At that time, some signs of the atomic structure of matter were already clearly visible: e. g. the integer stoichiometric proportions of chemistry, the thermodynamics of gases, the periodic system of the elements or Brownian motion. But the existence of atoms was not yet generally accepted. The reason was simple: nobody was able to really picture these building blocks of matter, the atoms. The new discoveries showed for the ﬁrst time “particles” emerging from matter which had to be interpreted as its constituents. It now became possible to use the particles produced by radioactive decay to bombard other elements in order to study the constituents of the latter. This experimental ansatz is the basis of modern nuclear and particle physics. Systematic studies of nuclei became possible by the late thirties with the availability of modern particle accelerators. But the fundamental building blocks of atoms – the electron, proton and neutron – were detected beforehand. A precondition for these discoveries were important technical developments in vacuum techniques and in particle detection. Before we turn to the global properties of nuclei from a modern viewpoint, we will brieﬂy discuss these historical experiments. 2.1 The Atom and its Constituents The electron. The ﬁrst building block of the atom to be identiﬁed was the electron. In 1897 Thomson was able to produce electrons as beams of free particles in discharge tubes. By deﬂecting them in electric and magnetic ﬁelds, he could determine their velocity and the ratio of their mass and charge. The results turned out to be independent of the kind of cathode and gas used. He had in other words found a universal constituent of matter. He then measured the charge of the electron independently — using a method that was in 1910 signiﬁcantly reﬁned by Millikan (the drop method) — this of course also ﬁxed the electron mass. The atomic nucleus. Subsequently, diﬀerent models of the atom were discussed, one of them being the model of Thomson. In this model, the electrons, 12 2 Global Properties of Nuclei and an equivalent number of positively charged particles are uniformly distributed throughout the atom. The resulting atom is electrically neutral. Rutherford, Geiger and Marsden succeeded in disproving this picture. In their famous experiments, where they scattered αparticles oﬀ heavy atoms, they were able to show that the positively charged particles are closely packed together. They reached this conclusion from the angular distribution of the scattered αparticles. The angular distribution showed αparticle scattering at large scattering angles which was incompatible with a homogeneous charge distribution. The explanation of the scattering data was a central Coulomb ﬁeld caused by a massive, positively charged nucleus. The method of extracting the properties of the scattering potential from the angular distribution of the scattered projectiles is still of great importance in nuclear and particle physics, and we will encounter it repeatedly in the following chapters. These experiments established the existence of the atom as a positively charged, small, massive nucleus with negatively charged electrons orbiting it. The proton. Rutherford also bombarded light nuclei with αparticles which themselves were identiﬁed as ionised helium atoms. In these reactions, he was looking for a conversion of elements, i.e., for a sort of inverse reaction to radioactive αdecay, which itself is a conversion of elements. While bombarding nitrogen with αparticles, he observed positively charged particles with an unusually long range, which must have been ejected from the atom as well. From this he concluded that the nitrogen atom had been destroyed in these reactions, and a light constituent of the nucleus had been ejected. He had already discovered similar longranged particles when bombarding hydrogen. From this he concluded that these particles were hydrogen nuclei which, therefore, had to be constituents of nitrogen as well. He had indeed observed the reaction 14 N + 4 He → 17 O + p , in which the nitrogen nucleus is converted into an oxygen nucleus, by the loss of a proton. The hydrogen nucleus could therefore be regarded as an elementary constituent of atomic nuclei. Rutherford also assumed that it would be possible to disintegrate additional atomic nuclei by using αparticles with higher energies than those available to him. He so paved the way for modern nuclear physics. The neutron. The neutron was also detected by bombarding nuclei with αparticles. Rutherford’s method of visually detecting and counting particles by their scintillation on a zinc sulphide screen is not applicable to neutral particles. The development of ionisation and cloud chambers signiﬁcantly simpliﬁed the detection of charged particles, but did not help here. Neutral particles could only be detected indirectly. Chadwick in 1932 found an appropriate experimental approach. He used the irradiation of beryllium with αparticles from a polonium source, and thereby established the neutron as a fundamental constituent of nuclei. Previously, a “neutral radiation” had 2.2 Nuclides 13 been observed in similar experiments, but its origin and identity was not understood. Chadwick arranged for this neutral radiation to collide with hydrogen, helium and nitrogen, and measured the recoil energies of these nuclei in a ionisation chamber. He deduced from the laws of collision that the mass of the neutral radiation particle was similar to that of the proton. Chadwick named this particle the “neutron”. Nuclear force and binding. With these discoveries, the building blocks of the atom had been found. The development of ion sources and mass spectrographs now permitted the investigation of the forces binding the nuclear constituents, i.e., the proton and the neutron. These forces were evidently much stronger than the electromagnetic forces holding the atom together, since atomic nuclei could only be broken up by bombarding them with highly energetic αparticles. The binding energy of a system gives information about its binding and stability. This energy is the diﬀerence between the mass of a system and the sum of the masses of its constituents. It turns out that for nuclei this diﬀerence is close to 1 % of the nuclear mass. This phenomenon, historically called the mass defect, was one of the ﬁrst experimental proofs of the massenergy relation E = mc2 . The mass defect is of fundamental importance in the study of strongly interacting bound systems. We will therefore describe nuclear masses and their systematics in this chapter at some length. 2.2 Nuclides The atomic number. The atomic number Z gives the number of protons in the nucleus. The charge of the nucleus is, therefore, Q = Ze, the elementary charge being e = 1.6·10−19 C. In a neutral atom, there are Z electrons, which balance the charge of the nucleus, in the electron cloud. The atomic number of a given nucleus determines its chemical properties. The classical method of determining the charge of the nucleus is the measurement of the characteristic Xrays of the atom to be studied. For this purpose the atom is excited by electrons, protons or synchrotron radiation. Moseley’s law says that the energy of the Kα line is proportional to (Z − 1)2 . Nowadays, the detection of these characteristic Xrays is used to identify elements in material analysis. Atoms are electrically neutral, which shows the equality of the absolute values of the positive charge of the proton and the negative charge of the electron. Experiments measuring the deﬂection of molecular beams in electric ﬁelds yield an upper limit for the diﬀerence between the proton and electron charges [Dy73]: (2.1) ep + ee  ≤ 10−18 e . Today cosmological estimates give an even smaller upper limit for any diﬀerence between these charges. 14 2 Global Properties of Nuclei The mass number. In addition to the Z protons, N neutrons are found in the nucleus. The mass number A gives the number of nucleons in the nucleus, where A = Z +N . Diﬀerent combinations of Z and N (or Z and A) are called nuclides. – Nuclides with the same mass number A are called isobars. – Nuclides with the same atomic number Z are called isotopes. – Nuclides with the same neutron number N are called isotones. The binding energy B is usually determined from atomic masses [AM93], since they can be measured to a considerably higher precision than nuclear masses. We have: (2.2) B(Z, A) = ZM (1 H) + (A − Z)Mn − M (A, Z) · c2 . Here, M (1 H) = Mp + me is the mass of the hydrogen atom (the 13.6 eV binding energy of the Hatom is negligible), Mn is the mass of the neutron and M (A, Z) is the mass of an atom with Z electrons whose nucleus contains A nucleons. The rest masses of these particles are: Mp Mn me = 938.272 MeV/c2 = = 939.566 MeV/c2 = = 0.511 MeV/c2 . 1836.149 me 1838.679 me The conversion factor into SI units is 1.783 · 10−30 kg/(MeV/c2 ). In nuclear physics, nuclides are denoted by A X, X being the chemical symbol of the element. E.g., the stable carbon isotopes are labelled 12 C and 13 C; while the radioactive carbon isotope frequently used for isotopic dating A is labelled 14 C. Sometimes the notations A Z X or Z XN are used, whereby the atomic number Z and possibly the neutron number N are explicitly added. Determining masses from mass spectroscopy. The binding energy of an atomic nucleus can be calculated if the atomic mass is accurately known. At the start of the 20th century, the method of mass spectrometry was developed for precision determinations of atomic masses (and nucleon binding energies). The deﬂection of an ion with charge Q in an electric and magnetic ﬁeld allows the simultaneous measurement of its momentum p = M v and its kinetic energy Ekin = M v 2 /2. From these, its mass can be determined. This is how most mass spectrometers work. While the radius of curvature rE of the ionic path in an electrical sector ﬁeld is proportional to the energy: rE = M v2 · , Q E (2.3) in a magnetic ﬁeld B, the radius of curvature rM of the ion is proportional to its momentum: M v rM = · . (2.4) Q B 2.2 Nuclides 15 Ion source Detector Fig. 2.1. Doubly focusing mass spectrometer [Br64]. The spectrometer focuses ions of a certain speciﬁc charge to mass ratio Q/M . For clarity, only the trajectories of particles at the edges of the beam are drawn (1 and 2 ). The electric and magnetic sector ﬁelds draw the ions from the ion source into the collector. Ions with a diﬀerent Q/M ratio are separated from the beam in the magnetic ﬁeld and do not pass through the slit O. Figure 2.1 shows a common spectrometer design. After leaving the ion source, the ions are accelerated in an electric ﬁeld to about 40 keV. In an electric ﬁeld, they are then separated according to their energy and, in a magnetic ﬁeld, according to their momentum. By careful design of the magnetic ﬁelds, ions with identical Q/M ratios leaving the ion source at various angles are focused at a point at the end of the spectrometer where a detector can be placed. For technical reasons, it is very convenient to use the 12 C nuclide as the reference mass. Carbon and its many compounds are always present in a spectrometer and are well suited for mass calibration. An atomic mass unit u was therefore deﬁned as 1/12 of the atomic mass of the 12 C nuclide. We have: 1 M12 C = 931.494 MeV/c2 = 1.660 54 · 10−27 kg . 1u = 12 Mass spectrometers are still widely used both in research and industry. Nuclear abundance. A current application of mass spectroscopy in fundamental research is the determination of isotope abundances in the solar system. The relative abundance of the various nuclides as a function of their mass number A is shown in Fig. 2.2. The relative abundances of isotopes in 16 2 Global Properties of Nuclei Abundance [Si=106] Mass number A Fig. 2.2. Abundance of the elements in the solar system as a function of their mass number A, normalised to the abundance of silicon (= 106 ). terrestrial, lunar, and meteoritic probes are, with few exceptions, identical and coincide with the nuclide abundances in cosmic rays from outside the solar system. According to current thinking, the synthesis of the presently existing deuterium and helium from hydrogen fusion mainly took place at the beginning of the universe (minutes after the big bang [Ba80]). Nuclei up to 56 Fe, the most stable nucleus, were produced by nuclear fusion in stars. Nuclei heavier than this last were created in the explosion of very heavy stars (supernovae) [Bu57]. Deviations from the universal abundance of isotopes occur locally when nuclides are formed in radioactive decays. Figure 2.3 shows the abundances of various xenon isotopes in a drill core which was found at a depth of 10 km. The isotope distribution strongly deviates from that which is found in the earth’s atmosphere. This deviation is a result of the atmospheric xenon being, for the most part, already present when the earth came into existence, while the xenon isotopes from the core come from radioactive decays (spontaneous ﬁssion of uranium isotopes). 2.2 Nuclides 17 Events Fig. 2.3. Mass spectrum of xenon isotopes, found in a roughly 2.7·109 year old gneiss sample from a drill core produced in the Kola peninsula (top) and, for comparison, the spectrum of Xeisotopes as they occur in the atmosphere (bottom). The Xeisotopes in the gneiss were produced by spontaneous ﬁssion of uranium. (Picture courtesy of Klaus Schäfer, MaxPlanckInstitut für Kernphysik.) Mass number A Determining masses from nuclear reactions. Binding energies may also be determined from systematic studies of nuclear reactions. Consider, as an example, the capture of thermal neutrons (Ekin ≈ 1/40 eV) by hydrogen, n + 1H → 2H + γ . (2.5) The energy of the emitted photon is directly related to the binding energy B of the deuterium nucleus 2 H: B = (Mn + M1 H − M2 H ) · c2 = Eγ + Eγ2 = 2.225 MeV, 2M2 H c2 (2.6) where the last term takes into account the recoil energy of the deuteron. As a further example, we consider the reaction 1 H + 6 Li → 3 He + 4 He . The energy balance of this reaction is given by E1 H + E6Li = E3He + E4He , (2.7) where the energies EX each represent the total energy of the nuclide X, i.e., the sum of its rest mass and kinetic energy. If three of these nuclide masses are known, and if all of the kinetic energies have been measured, then the binding energy of the fourth nuclide can be determined. The measurement of binding energies from nuclear reactions was mainly accomplished using lowenergy (van de Graaﬀ, cyclotron, betatron) accelerators. Following two decades of measurements in the ﬁfties and sixties, the 18 2 Global Properties of Nuclei Z B/A [MeV] B/A [MeV] N Mass number A Mass number A Fig. 2.4. Binding energy per nucleon of nuclei with even mass number A. The solid line corresponds to the Weizsäcker mass formula (2.8). Nuclei with a small number of nucleons display relatively large deviations from the general trend, and should be considered on an individual basis. For heavy nuclei deviations in the form of a somewhat stronger binding per nucleon are also observed for certain proton and neutron numbers. These socalled “magic numbers” will be discussed in Sect. 17.3. systematic errors of both methods, mass spectrometry and the energy balance of nuclear reactions, have been considerably reduced and both now provide high precision results which are consistent with each other. Figure 2.4 shows schematically the results of the binding energies per nucleon measured for stable nuclei. Nuclear reactions even provide mass determinations for nuclei which are so shortlived that that they cannot be studied by mass spectroscopy. 2.3 Parametrisation of Binding Energies Apart from the lightest elements, the binding energy per nucleon for most nuclei is about 89 MeV. Depending only weakly on the mass number, it can 2.3 Parametrisation of Binding Energies 19 be described with the help of just a few parameters. The parametrisation of nuclear masses as a function of A and Z, which is known as the Weizsäcker formula or the semiempirical mass formula, was ﬁrst introduced in 1935 [We35, Be36]. It allows the calculation of the binding energy according to (2.2). The mass of an atom with Z protons and N neutrons is given by the following phenomenological formula: M (A, Z) = N Mn + ZMp + Zme − av A + as A2/3 + ac with Z2 (N − Z)2 δ + 1/2 + a a 4A A1/3 A (2.8) N =A−Z. The exact values of the parameters av , as , ac , aa and δ depend on the range of masses for which they are optimised. One possible set of parameters is given below: 15.67 MeV/c2 17.23 MeV/c2 0.714 MeV/c2 93.15 MeV/c2 ⎧ ⎨ −11.2 MeV/c2 for even Z and N (eveneven nuclei) 0 MeV/c2 for odd A (oddeven nuclei) δ= ⎩ +11.2 MeV/c2 for odd Z and N (oddodd nuclei). av as ac aa = = = = To a great extent the mass of an atom is given by the sum of the masses of its constituents (protons, neutrons and electrons). The nuclear binding responsible for the deviation from this sum is reﬂected in ﬁve additional terms. The physical meaning of these ﬁve terms can be understood by recalling that the nuclear radius R and mass number A are connected by the relation R ∝ A1/3 . (2.9) The experimental proof of this relation and a quantitative determination of the coeﬃcient of proportionality will be discussed in Sect. 5.4. The individual terms can be interpreted as follows: Volume term. This term, which dominates the binding energy, is proportional to the number of nucleons. Each nucleon in the interior of a (large) nucleus contributes an energy of about 16 MeV. From this we deduce that the nuclear force has a short range, corresponding approximately to the distance between two nucleons. This phenomenon is called saturation. If each nucleon would interact with each of the other nucleons in the nucleus, the total binding energy would be proportional to A(A − 1) or approximately to A2 . Due to saturation, the central density of nucleons is the same for all nuclei, with few exceptions. The central density is 20 2 Global Properties of Nuclei 3 0 ≈ 0.17 nucleons/fm3 = 3 · 1017 kg/m . (2.10) The average nuclear density, which can be deduced from the mass and radius (see 5.56), is smaller (0.13 nucleons/fm3 ). The average internucleon distance in the nucleus is about 1.8 fm. Surface term. For nucleons at the surface of the nucleus, which are surrounded by fewer nucleons, the above binding energy is reduced. This contribution is proportional to the surface area of the nucleus (R2 or A2/3 ). Coulomb term. The electrical repulsive force acting between the protons in the nucleus further reduces the binding energy. This term is calculated to be 3 Z(Z − 1) α c . (2.11) ECoulomb = 5 R This is approximately proportional to Z 2 /A1/3 . Asymmetry term. As long as mass numbers are small, nuclei tend to have the same number of protons and neutrons. Heavier nuclei accumulate more and more neutrons, to partly compensate for the increasing Coulomb repulsion by increasing the nuclear force. This creates an asymmetry in the number of neutrons and protons. For, e.g., 208 Pb it amounts to N –Z = 44. The dependence of the nuclear force on the surplus of neutrons is described by the asymmetry term (N −Z)2 /(4A). This shows that the symmetry decreases as the nuclear mass increases. We will further discuss this point in Sect. 17.1. The dependence of the above terms on A is shown in Fig. 2.5. Pairing term. A systematic study of nuclear masses shows that nuclei are more stable when they have an even number of protons and/or neutrons. This observation is interpreted as a coupling of protons and neutrons in pairs. The pairing energy depends on the mass number, as the overlap of the wave functions of these nucleons is smaller, in larger nuclei. Empirically this is described by the term δ · A−1/2 in (2.8). All in all, the global properties of the nuclear force are rather well described by the mass formula (2.8). However, the details of nuclear structure which we will discuss later (mainly in Chap. 17) are not accounted for by this formula. The Weizsäcker formula is often mentioned in connection with the liquid drop model . In fact, the formula is based on some properties known from liquid drops: constant density, shortrange forces, saturation, deformability and surface tension. An essential diﬀerence, however, is found in the mean free path of the particles. For molecules in liquid drops, this is far smaller than the size of the drop; but for nucleons in the nucleus, it is large. Therefore, the nucleus has to be treated as a quantum liquid, and not as a classical one. At low excitation energies, the nucleus may be even more simply described as a Fermi gas; i. e., as a system of free particles only weakly interacting with each other. This model will be discussed in more detail in Sect. 17.1. B/A [MeV] 2.4 Charge Independence of the Nuclear Force and Isospin 21 Fig. 2.5. The diﬀerent contributions to the binding energy per nucleon versus mass number A. The horizontal line at ≈ 16 MeV represents the contribution of the volume energy. This is reduced by the surface energy, the asymmetry energy and the Coulomb energy to the effective binding energy of ≈ 8 MeV (lower line). The contributions of the asymmetry and Coulomb terms increase rapidly with A, while the contribution of the surface term decreases. Volume energy Surface energy Coulomb energy Asymmetry energy Total binding energy A 2.4 Charge Independence of the Nuclear Force and Isospin Protons and neutrons not only have nearly equal masses, they also have similar nuclear interactions. This is particularly visible in the study of mirror nuclei. Mirror nuclei are pairs of isobars, in which the proton number of one of the nuclides equals the neutron number of the other and vice versa. Figure 2.6 shows the lowest energy levels of the mirror nuclei 146 C8 and 14 14 14 8 O6 , together with those of 7 N7 . The energylevel diagrams of 6 C8 and 14 P of the levels 8 O6 are very similar with respect to the quantum numbers J as well as with respect to the distances between them. The small diﬀerences and the global shift of the levels as a whole in 146 C8 , as compared to 148 O6 can be explained by diﬀerences in the Coulomb energy. Further examples of mirror nuclei will be discussed in Sect. 17.3 (Fig. 17.7). The energy levels of 146 C8 and 148 O6 are also found in the isobaric nucleus 147 N7 . Other states in 147 N7 have no analogy in the two neighbouring nuclei. We therefore can distinguish between triplet and singlet states. These multiplets of states are reminiscent of the multiplets known from the coupling of angular momenta (spins). The symmetry between protons and neutrons may therefore be described by a similar formalism, called isospin I. The proton and neutron are treated as two states of the nucleon which form a doublet (I = 1/2). proton: I3 = +1/2 (2.12) Nucleon: I = 1/2 neutron: I3 = −1/2 Formally, isospin is treated as a quantum mechanical angular momentum. For example, a protonneutron pair can be in a state of total isospin 1 or 0. The third (z) component of isospin is additive: I3nucleus = I3nucleon = Z −N . 2 (2.13) 2 Global Properties of Nuclei E [MeV] 22 Fig. 2.6. Lowlying energy levels of the three most stable A = 14 isobars. Angular momentum J and parity P are shown for the most important levels. The analogous states of the three nuclei are joined by dashed lines. The zero of the energy scale is set to the ground state of 147 N7 . This enables us to describe the appearance of similar states in Fig. 2.6: 146 C8 and 148 O6 , have respectively I3 = −1 and I3 = +1. Therefore, their isospin cannot be less than I = 1. The states in these nuclei thus necessarily belong to a triplet of similar states in 146 C8 , 147 N7 and 148 O6 . The I3 component of the nuclide 147 N7 , however, is 0. This nuclide can, therefore, have additional states with isospin I = 0. Since 147 N7 is the most stable A = 14 isobar, its ground state is necessarily an isospin singlet since otherwise 146 C8 would possess an analogous state, which, with less Coulomb repulsion, would be lower in energy and so more stable. I = 2 states are not shown in Fig. 2.6. Such states would have analogous states in 145 B9 and in 149 F5 . These nuclides, however, are very unstable (i. e., highly energetic), and lie above the energy range of the diagram. The A = 14 isobars are rather light nuclei in which the Coulomb energy is not strongly felt. In heavier nuclei, the inﬂuence of the Coulomb energy grows, which increasingly disturbs the isospin symmetry. The concept of isospin is of great importance not only in nuclear physics, but also in particle physics. As we will see quarks, and particles composed of quarks, can be classiﬁed by isospin into isospin multiplets. In dynamical processes of the stronginteraction type, the isospin of the system is conserved. Problem 23 Problem 1. Isospin symmetry One could naively imagine the three nucleons in the 3 H and 3 He nuclei as being rigid spheres. If one solely attributes the diﬀerence in the binding energies of these two nuclei to the electrostatic repulsion of the protons in 3 He, how large must the separation of the protons be? (The maximal energy of the electron in the β −decay of 3 H is 18.6 keV.) 3 Nuclear Stability Stable nuclei only occur in a very narrow band in the Z − N plane (Fig. 3.1). All other nuclei are unstable and decay spontaneously in various ways. Isobars with a large surplus of neutrons gain energy by converting a neutron into a proton. In the case of a surplus of protons, the inverse reaction may occur: i.e., the conversion of a proton into a neutron. These transformations are called βdecays and they are manifestations of the weak interaction. After dealing with the weak interaction in Chap. 10, we will discuss these decays in more detail in Sects. 15.5 and 17.6. In the present chapter, we will merely survey certain general properties, paying particular attention to the energy balance of βdecays. sion s fis Estable nuclides ou ane nt spo punstable nunstable Fig. 3.1. βstable nuclei in the Z − N plane (from [Bo69]). Fe and Niisotopes possess the maximum binding energy per nucleon and they are therefore the most stable nuclides. In heavier nuclei the binding energy is smaller because of the larger Coulomb repulsion. For still heavier 26 3 Nuclear Stability masses nuclei become unstable to ﬁssion and decay spontaneously into two or more lighter nuclei should the mass of the original atom be larger than the sum of the masses of the daughter atoms. For a twobody decay, this condition has the form: M (A, Z) > M (A − A , Z − Z ) + M (A , Z ) . (3.1) This relation takes into account the conservation of the number of protons and neutrons. However, it does not give any information about the probability of such a decay. An isotope is said to be stable if its lifetime is considerably longer than the age of the solar system. We will not consider manybody decays any further since they are much rarer than twobody decays. It is very often the case that one of the daughter nuclei is a 4 He nucleus, i. e., A = 4, Z = 2. This decay mode is called αdecay, and the Helium nucleus is called an αparticle. If a heavy nucleus decays into two similarly massive daughter nuclei we speak of spontaneous ﬁssion. The probability of spontaneous ﬁssion exceeds that of αdecay only for nuclei with Z > ∼ 110 and is a fairly unimportant process for the naturally occurring heavy elements. Decay constants. The probability per unit time for a radioactive nucleus to decay is known as the decay constant λ. It is related to the lifetime τ and the half life t1/2 by: τ= 1 λ and t1/2 = ln 2 . λ (3.2) The measurement of the decay constants of radioactive nuclei is based upon ﬁnding the activity (the number of decays per unit time): A=− dN = λN dt (3.3) where N is the number of radioactive nuclei in the sample. The unit of activity is deﬁned to be 1 Bq [Becquerel] = 1 decay /s. (3.4) For shortlived nuclides, the falloﬀ over time of the activity: A(t) = λN (t) = λN0 e−λt where N0 = N (t = 0) (3.5) may be measured using fast electronic counters. This method of measuring is not suitable for lifetimes larger than about a year. For longerlived nuclei both the number of nuclei in the sample and the activity must be measured in order to obtain the decay constant from (3.3). 3.1 βDecay Let us consider nuclei with equal mass number A (isobars). Equation 2.8 can be transformed into: 3.1 M (A, Z) = α · A − β · Z + γ · Z 2 + where δ A1/2 , βDecay 27 (3.6) aa , 4 β = aa + (Mn − Mp − me ) , ac aa + 1/3 , γ= A A α = Mn − av + as A−1/3 + δ = as in (2.8) . The nuclear mass is now a quadratic function of Z. A plot of such nuclear masses, for constant mass number A, as a function of Z, the charge number, yields a parabola for odd A. For even A, the masses of the eveneven and the oddodd nuclei are found to lie on two vertically shifted parabolas. The odd√ odd parabola lies at twice the pairing energy (2δ/ A) above the eveneven one. The minimum of the parabolas is found at Z = β/2γ. The nucleus with the smallest mass in an isobaric spectrum is stable with respect to βdecay. βdecay in odd mass nuclei. In what follows we wish to discuss the diﬀerent kinds of βdecay, using the example of the A = 101 isobars. For this mass number, the parabola minimum is at the isobar 101 Ru which has 101 Z = 44. Isobars with more neutrons, such as 101 42 Mo and 43 Tc, decay through the conversion: (3.7) n → p + e− + ν e . The charge number of the daughter nucleus is one unit larger than that of the the parent nucleus (Fig. 3.2). An electron and an eantineutrino are also produced: 101 42 Mo 101 43 Tc → → 101 − 43 Tc + e + ν e , 101 − 44 Ru + e + ν e . Historically such decays where a negative electron is emitted are called β − decays. Energetically, β − decay is possible whenever the mass of the daughter atom M (A, Z + 1) is smaller than the mass of its isobaric neighbour: M (A, Z) > M (A, Z + 1) . (3.8) We consider here the mass of the whole atom and not just that of the nucleus alone and so the rest mass of the electron created in the decay is automatically taken into account. The tiny mass of the (anti)neutrino (< 15 eV/c2 ) [PD98] is negligible in the mass balance. Isobars with a proton excess, compared to 101 44 Ru, decay through proton conversion: (3.9) p → n + e+ + νe . The stable isobar 101 44 Ru is eventually produced via M [MeV/c 2] 28 3 Nuclear Stability Fig. 3.2. Mass parabola of the A = 101 isobars (from [Se77]). Possible βdecays are shown by arrows. The abscissa coordinate is the atomic number, Z. The zero point of the mass scale was chosen arbitrarily. Eunstable stable 5 4 A=101 3 2 1 42 Mo 43 Tc 44 Ru 45 Rh 101 46 Pd 101 45 Rh 46 Pd → → 47 Ag 101 45 Rh 101 44 Ru + e+ + νe , + e+ + νe . and Such decays are called β + decays. Since the mass of a free neutron is larger than the proton mass, the process (3.9) is only possible inside a nucleus. By contrast, neutrons outside nuclei can and do decay (3.7). Energetically, β + decay is possible whenever the following relationship between the masses M (A, Z) and M (A, Z − 1) (of the parent and daughter atoms respectively) is satisﬁed: (3.10) M (A, Z) > M (A, Z − 1) + 2me . This relationship takes into account the creation of a positron and the existence of an excess electron in the parent atom. βdecay in even nuclei. Even mass number isobars form, as we described above, two separate (one for eveneven and one for oddodd nuclei) parabolas which are split by an amount equal to twice the pairing energy. Often there is more than one βstable isobar, especially in the range A > 70. Let us consider the example of the nuclides with A = 106 (Fig. 3.3). The 106 106 eveneven 106 46 Pd and 48 Cd isobars are on the lower parabola, and 46 Pd is the 106 stablest. 48 Cd is βstable, since its two oddodd neighbours both lie above it. The conversion of 106 48 Cd is thus only possible through a double βdecay into 106 46 Pd: 106 106 + 48 Cd → 46 Pd + 2e + 2νe . The probability for such a process is so small that 106 48 Cd may be considered to be a stable nuclide. Oddodd nuclei always have at least one more strongly bound, eveneven neighbour nucleus in the isobaric spectrum. They are therefore unstable. The M [MeV/c 2] 3.1 E+ 4 3 E– 29 Fig. 3.3. Mass parabolas of the A = 106isobars (from [Se77]). Possible βdecays are indicated by arrows. The abscissa coordinate is the charge number Z. The zero point of the mass scale was chosen arbitrarily. oddodd 5 βDecay eveneven 2 1 E unstable A = 106 stable 43 Tc 44 Ru 45 Rh 46 Pd 47 Ag 48 Cd 49 In only exceptions to this rule are the very light nuclei 21 H, 63 Li, 105 B and 147 N, which are stable to βdecay, since the increase in the asymmetry energy would exceed the decrease in pairing energy. Some oddodd nuclei can undergo both β − decay and β + decay. Wellknown examples of this are 40 19 K (Fig. 3.4) and 64 29 Cu. Electron capture. Another possible decay process is the capture of an electron from the cloud surrounding the atom. There is a ﬁnite probability of ﬁnding such an electron inside the nucleus. In such circumstances it can combine with a proton to form a neutron and a neutrino in the following way: p + e− → n + νe . (3.11) This reaction occurs mainly in heavy nuclei where the nuclear radii are larger and the electronic orbits are more compact. Usually the electrons that are captured are from the innermost (the “K”) shell since such Kelectrons are closest to the nucleus and their radial wave function has a maximum at the centre of the nucleus. Since an electron is missing from the Kshell after such a Kcapture, electrons from higher energy levels will successively cascade downwards and in so doing they emit characteristic Xrays. Electron capture reactions compete with β + decay. The following condition is a consequence of energy conservation M (A, Z) > M (A, Z − 1) + ε , (3.12) where ε is the excitation energy of the atomic shell of the daughter nucleus (electron capture always leads to a hole in the electron shell). This process has, compared to β + decay, more kinetic energy (2me c2 − ε more) available to it and so there are some cases where the mass diﬀerence between the initial and ﬁnal atoms is too small for conversion to proceed via β + decay and yet Kcapture can take place. 30 3 Nuclear Stability Energy 2 MeV 2+ EC t1/2 = 1.27.109 a (11 %) 4– 40 19K E– (89 %) 1 MeV 0+ 40 18Ar E+ (0.001 %) 0+ 40 20Ca Fig. 3.4. The βdecay of 40 K. In this nuclear conversion, β −  and β + decay as well as electron capture (EC) compete with each other. The relative frequency of these decays is given in parentheses. The bent arrow in β + decay indicates that the production of an e+ and the presence of the surplus electron in the 40Ar atom requires 1.022 MeV, and the remainder is carried oﬀ as kinetic energy by the positron and the neutrino. The excited state of 40Ar produced in the electron capture reaction decays by photon emission into its ground state. Lifetimes. The lifetimes τ of βunstable nuclei vary between a few ms and 1016 years. They strongly depend upon both the energy E which is released (1/τ ∝ E 5 ) and upon the nuclear properties of the mother and daughter nuclei. The decay of a free neutron into a proton, an electron and an antineutrino releases 0.78 MeV and this particle has a lifetime of τ = 886.7 ± 1.9 s [PD98]. No two neighbouring isobars are known to be βstable.1 A wellknown example of a longlived βemitter is the nuclide 40 K. It transforms into other isobars by both β −  and β + decay. Electron capture in 40 K also competes here with β + decay. The stable daughter nuclei are 40Ar and 40 Ca respectively, which is a case of two stable nuclei having the same mass number A (Fig. 3.4). The 40 K nuclide was chosen here because it contributes considerably to the radiation exposure of human beings and other biological systems. Potassium is an essential element: for example, signal transmission in the nervous system functions by an exchange of potassium ions. The fraction of radioactive 40 K in natural potassium is 0.01 %, and the decay of 40 K in the human body contributes about 16 % of the total natural radiation which we are exposed to. 1 In some cases, however, one of two neighbouring isobars is stable and the other is extremely longlived. The most common isotopes of indium (115 In, 96 %) and rhenium (187 Re, 63 %) β − decay into stable nuclei (115 Sn and 187 Os), but they are so longlived (τ = 3 · 1014 yrs and τ = 3 · 1011 yrs respectively) that they may also be considered stable. 3.2 αDecay 31 3.2 αDecay Protons and neutrons have binding energies, even in heavy nuclei, of about 8 MeV (Fig. 2.4) and cannot generally escape from the nucleus. In many cases, however, it is energetically possible for a bound system of a group of nucleons to be emitted, since the binding energy of this system increases the total energy available to the process. The probability for such a system to be formed in a nucleus decreases rapidly with the number of nucleons required. In practice the most signiﬁcant decay process is the emission of a 4 He nucleus; i. e., a system of 2 protons and 2 neutrons. Contrary to systems of 2 or 3 nucleons, this socalled αparticle is extraordinarily strongly bound — 7 MeV/nucleon (cf. Fig. 2.4). Such decays are called αdecays. Figure 3.5 shows the potential energy of an αparticle as a function of its separation from the centre of the nucleus. Beyond the nuclear force range, the αparticle feels only the Coulomb potential VC (r) = 2(Z − 2)αc/r, which increases closer to the nucleus. Within the nuclear force range a strongly attractive nuclear potential prevails. Its strength is characterised by the depth of the potential well. Since we are considering αparticles which are energetically allowed to escape from the nuclear potential, the total energy of this αparticle is positive. This energy is released in the decay. The range of lifetimes for the αdecay of heavy nuclei is extremely large. Experimentally, lifetimes have been measured between 10 ns and 1017 years. These lifetimes can be calculated in quantum mechanics by treating the αparticle as a wave packet. The probability for the αparticle to escape from the nucleus is given by the probability for its penetrating the Coulomb barrier (the tunnel eﬀect). If we divide the Coulomb barrier into thin potential walls and look at the probability of the αparticle tunnelling through one of these (Fig. 3.6), then the transmission T is given by: where κ = 2mE − V / , (3.13) T ≈ e−2κ∆r and ∆r is the thickness of the barrier and V is its height. E is the energy of the αparticle. A Coulomb barrier can be thought of as a barrier composed of V(r) Vc = 2(Z2) Dhc r E 0 R 'r r1 r Fig. 3.5. Potential energy of an αparticle as a function of its separation from the centre of the nucleus. The probability that it tunnels through the Coulomb barrier can be calculated as the superposition of tunnelling processes through thin potential walls of thickness ∆r (cf. Fig. 3.6). 32 3 Nuclear Stability Fig. 3.6. Illustration of the tunnelling probability of a wave packet with energy E and velocity v faced with a potential barrier of height V and thickness ∆r. v r 'r v v r 'r a large number of thin potential walls of diﬀerent heights. The transmission can be described accordingly by: T = e−2G . (3.14) The Gamow factor G can be approximated by the integral [Se77]: G= 1 r1 R π · 2 · (Z − 2) · α , 2mE − V  dr ≈ β (3.15) where β = v/c is the velocity of the outgoing αparticle and R is the nuclear radius. The probability per unit time λ for an αparticle to escape from the nucleus is therefore proportional to: the probability w(α) of ﬁnding such an αparticle in the nucleus, the number of collisions (∝ v0 /2R) of the αparticle with the barrier and the transmission probability: v0 −2G e λ = w(α) , (3.16) 2R where v0 is the velocity of the αparticle in the nucleus (v0 ≈ 0.1 c). The large variation in the lifetimes √ is explained by the Gamow factor in the exponent: since G ∝ Z/β ∝ Z/ E, small diﬀerences in the energy of the αparticle have a strong eﬀect on the lifetime. Most αemitting nuclei are heavier than lead. For lighter nuclei with A < ∼ 140, αdecay is energetically possible, but the energy released is extremely small. Therefore, their nuclear lifetimes are so long that decays are usually not observable. An example of a αunstable nuclide with a long lifetime, 238 U, is shown in Fig. 3.7. Since uranium compounds are common in granite, uranium and its radioactive daughters are a part of the stone walls of buildings. They therefore contribute to the environmental radiation background. This is particularly true of the inert gas 222 Rn, which escapes from the walls and is inhaled into the lungs. The αdecay of 222 Rn is responsible for about 40 % of the average natural human radiation exposure. 3.3 N 146 234Pa E– 142 6.66 h 234U 2.5.105 a D 230Th 8.104 a D 140 226Ra 138 1620 a D 222Rn 3.8 d D 136 218Po 134 3.05 min D E– 214Pb 26.8 min 130 126 4.5.109 a E– 24.1d 144 128 33 238U D 234Th 132 Nuclear Fission D 210Pb E– 19.4 a 214Bi 19.7 min E– 214Po 164 Ps 210Bi 3.0.106 a D 206Tl 4.2 min E– 124 206Pb stable 80 82 84 86 88 90 92 Z 238 Fig. 3.7. Illustration of the U decay chain in the N –Z plane. The half life of each of the nuclides is given together with its decay mode. 3.3 Nuclear Fission Spontaneous ﬁssion. The largest binding energy per nucleon is found in those nuclei in the region of 56 Fe. For heavier nuclei, it decreases as the nuclear mass increases (Fig. 2.4). A nucleus with Z > 40 can thus, in principle, split into two lighter nuclei. The potential barrier which must be tunnelled through is, however, so large that such spontaneous ﬁssion reactions are generally speaking extremely unlikely. The lightest nuclides where the probability of spontaneous ﬁssion is comparable to that of αdecay are certain uranium isotopes. The shape of the ﬁssion barrier is shown in Fig. 3.8. It is interesting to ﬁnd the charge number Z above which nuclei become ﬁssion unstable, i.e., the point from which the mutual Coulombic repulsion of the protons outweighs the attractive nature of the nuclear force. An estimate can be obtained by considering the surface and the Coulomb energies 34 3 Nuclear Stability V(r) ( ) Z 2 Dhc Vc = r 2 R r Fig. 3.8. Potential energy during diﬀerent stages of a ﬁssion reaction. A nucleus with charge Z decays spontaneously into two daughter nuclei. The solid line corresponds to the shape of the potential in the parent nucleus. The height of the barrier for ﬁssion determines the probability of spontaneous ﬁssion. The ﬁssion barrier disappears for nuclei with Z 2 /A > ∼ 48 and the shape of the potential then corresponds to the dashed line. during the ﬁssion deformation. As the nucleus is deformed the surface energy increases, while the Coulomb energy decreases. If the deformation leads to an energetically more favourable conﬁguration, the nucleus is unstable. Quantitatively, this can be calculated as follows: keeping the volume of the nucleus constant, we deform its spherical shape into an ellipsoid with axes a = R(1 + ε) and b = R(1 − ε/2) (Fig. 3.9). The surface energy then has the form: 2 Es = as A2/3 1 + ε2 + · · · 5 , (3.17) while the Coulomb energy is given by: 1 Ec = ac Z 2 A−1/3 1 − ε2 + · · · 5 . (3.18) ε2 2as A2/3 − ac Z 2 A−1/3 . 5 (3.19) Hence a deformation ε changes the total energy by: ∆E = If ∆E is negative, a deformation is energetically favoured. The ﬁssion barrier disappears for: Z2 2as ≥ ≈ 48 . (3.20) A ac This is the case for nuclei with Z > 114 and A > 270. 3.4 Decay of Excited Nuclear States 35 b R b a Fig. 3.9. Deformation of a heavy nucleus. For a constant volume V (V = 4πR3 /3 = 4πab2 /3), the surface energy of the nucleus increases and its Coulomb energy decreases. Induced ﬁssion. For very heavy nuclei (Z ≈ 92) the ﬁssion barrier is only about 6 MeV. This energy may be supplied if one uses a ﬂow of low energy neutrons to induce neutron capture reactions. These push the nucleus into an excited state above the ﬁssion barrier and it splits up. This process is known as induced nuclear ﬁssion. Neutron capture by nuclei with an odd neutron number releases not just some binding energy but also a pairing energy. This small extra contribution to the energy balance makes a decisive diﬀerence to nuclide ﬁssion properties: in neutron capture by 238 U, for example, 4.9 MeV binding energy is released, which is below the threshold energy of 5.5 MeV for nuclear ﬁssion of 239 U. Neutron capture by 238 U can therefore only lead to immediate nuclear ﬁssion if the neutron possesses a kinetic energy at least as large as this diﬀerence (“fast neutrons”). On top of this the reaction probability is proportional to v −1 , where v is the velocity of the neutron (4.21), and so it is very small. By contrast neutron capture in 235 U releases 6.4 MeV and the ﬁssion barrier of 236 U is just 5.5MeV. Thus ﬁssion may be induced in 235 U with the help of lowenergy (thermal) neutrons. This is exploited in nuclear reactors and nuclear weapons. Similarly both 233 Th and 239 Pu are suitable ﬁssion materials. 3.4 Decay of Excited Nuclear States Nuclei usually have many excited states. Most of the lowestlying states are understood theoretically, at least in a qualitative way as will be discussed in more detail in Chaps. 17 and 18. Figure 3.10 schematically shows the energy levels of an eveneven nucleus with A ≈ 100. Above the ground state, individual discrete levels with speciﬁc J P quantum numbers can be seen. The excitation of eveneven nuclei generally corresponds to the break up of nucleon pairs, which requires about 1–2 MeV. Eveneven nuclei with A > ∼ 40, therefore, rarely possess excitations 36 3 Nuclear Stability E [MeV] 20 Giant resonance Continuum 10 V(J,n) VTOT (n) AX(J,n)A–1X Z Z Discrete States A–1X + n Z E1 E2 E2,M1 5–– 3+ 4+ 0+ 2 2+ E2 0 AX Z 0+ Fig. 3.10. Sketch of typical nuclear energy levels. The example shows an eveneven nucleus whose ground state has the quantum numbers 0+ . To the left the total crosssection for the reaction of the nucleus A−1 Z X with neutrons (elastic scattering, inelastic scattering, capture) is shown; to the right the total crosssection for γA−1 induced neutron emission A ZX + γ → Z X + n. below 2 MeV.2 In oddeven and oddodd nuclei, the number of lowenergy states (with excitation energies of a few 100 keV) is considerably larger. Electromagnetic decays. Low lying excited nuclear states usually decay by emitting electromagnetic radiation. This can be described in a series expansion as a superposition of diﬀerent multipolarities each with its characteristic angular distribution. Electric dipole, quadrupole, octupole radiation etc. are denoted by E1, E2, E3, etc. Similarly, the corresponding magnetic multipoles are denoted by M1, M2, M3 etc. Conservation of angular momentum and parity determine which multipolarities are possible in a transition. A photon of multipolarity E has angular momentum and parity (−1) , an M photon has angular momentum and parity (−1)(+1) . In a transi2 Collective states in deformed nuclei are an exception to this: they cannot be understood as single particle excitations (Chap. 18). 3.4 Decay of Excited Nuclear States 37 Table 3.1. Selection rules for some electromagnetic transitions. Multipolarity E Dipole Quadrupole Octupole E1 E2 E3 Electric ∆J  ∆P M − + − M1 M2 M3 1 2 3 Magnetic ∆J  ∆P 1 2 3 + − + tion Ji → Jf , conservation of angular momentum means that the triangle inequality Ji − Jf  ≤ ≤ Ji + Jf must be satisﬁed. The lifetime of a state strongly depends upon the multipolarity of the γtransitions by which it can decay. The lower the multipolarity, the larger the transition probability. A magnetic transition M has approximately the same probability as an electric E( + 1) transition. A transition 3+ → 1+ , for example, is in principle a mixture of E2, M3, and E4, but will be easily dominated by the E2 contribution. A 3+ → 2+ transition will usually consist of an M1/E2 mixture, even though M3, E4, and M5 transitions are also possible. In a series of excited states 0+ , 2+ , 4+ , the most probable decay is by a cascade of E2transitions 4+ → 2+ → 0+ , and not by a single 4+ → 0+ E4transition. The lifetime of a state and the angular distribution of the electromagnetic radiation which it emits are signatures for the multipolarity of the transitions, which in turn betray the spin and parity of the nuclear levels. The decay probability also strongly depends upon the energy. For radiation of multipolarity it is proportional to Eγ2+1 (cf. Sect. 18.1). The excitation energy of a nucleus may also be transferred to an electron in the atomic shell. This process is called internal conversion. It is most important in transitions for which γemission is suppressed (high multipolarity, low energy) and the nucleus is heavy (high probability of the electron being inside the nucleus). 0+ → 0+ transitions cannot proceed through photon emission. If a nucleus is in an excited 0+ state, and all its lower lying levels also have 0+ quantum numbers (e. g. in 16 O or 40 Ca – cf. Fig. 18.6), then this state can only decay in a diﬀerent way: by internal conversion, by emission of 2 photons or by the emission of an e+ e− pair, if this last is energetically possible. Parity conservation does not permit internal conversion transitions between two levels with J = 0 and opposite parity. The lifetime of excited nuclear states typically varies between 10−9 s and −15 s, which corresponds to a state width of less than 1 eV. States which 10 can only decay by low energy and high multipolarity transitions have considerably longer lifetimes. They are called isomers and are designated by an “m” superscript on the symbol of the element. An extreme example is the second excited state of 110Ag, whose quantum numbers are J P = 6+ and excitation energy is 117.7 keV. It relaxes via an M4transition into the ﬁrst excited state 38 3 Nuclear Stability (1.3 keV; 2− ) since a decay directly into the ground state (1+ ) is even more improbable. The half life of 110Agm is extremely long (t1/2 = 235 d) [Le78]. Continuum states. Most nuclei have a binding energy per nucleon of about 8 MeV (Fig. 2.4). This is approximately the energy required to separate a single nucleon from the nucleus (separation energy). States with excitation energies above this value can therefore emit single nucleons. The emitted nucleons are primarily neutrons since they are not hindered by the Coulomb threshold. Such a strong interaction process is clearly preferred to γemission. The excitation spectrum above the threshold for particle emission is called the continuum, just as in atomic physics. Within this continuum there are also discrete, quasibound states. States below this threshold decay only by (relatively slow) γemission and are, therefore, very narrow. But for excitation energies above the particle threshold, the lifetimes of the states decrease dramatically, and their widths increase. The density of states increases approximately exponentially with the excitation energy. At higher excitation energies, the states therefore start to overlap, and states with the same quantum numbers can begin to mix. The continuum can be especially eﬀectively investigated by measuring the crosssections of neutron capture and neutron scattering. Even at high excitation energies, some narrow states can be identiﬁed. These are states with exotic quantum numbers (high spin) which therefore cannot mix with neighbouring states. Figure 3.10 shows schematically the crosssections for neutron capture and γinduced neutron emission (nuclear photoelectric eﬀect). A broad resonance is observed, the giant dipole resonance, which will be interpreted in Sect. 18.2. Problems 39 Problems 1. αdecay The αdecay of a 238 Pu (τ =127 yrs) nuclide into a long lived 234 U (τ = 3.5 · 105 yrs) daughter nucleus releases 5.49 MeV kinetic energy. The heat so produced can be converted into useful electricity by radiothermal generators (RTG’s). The Voyager 2 space probe, which was launched on the 20.8.1977, ﬂew past four planets, including Saturn which it reached on the 26.8.1981. Saturn’s separation from the sun is 9.5 AU; 1 AU = separation of the earth from the sun. a) How much plutonium would an RTG on Voyager 2 with 5.5 % eﬃciency have to carry so as to deliver at least 395 W electric power when the probe ﬂies past Saturn? b) How much electric power would then be available at Neptune (24.8.1989; 30.1 AU separation)? c) To compare: the largest ever “solar paddles” used in space were those of the space laboratory Skylab which would have produced 10.5 kW from an area of 730 m2 if they had not been damaged at launch. What area of solar cells would Voyager 2 have needed? 2. Radioactivity Naturally occuring uranium is a mixture of the 238 U (99.28 %) and 235 U (0.72 %) isotopes. a) How old must the material of the solar system be if one assumes that at its creation both isotopes were present in equal quantities? How do you interpret this result? The lifetime of 235 U is τ = 1.015 · 109 yrs. For the lifetime of 238 U use the data in Fig. 3.7. b) How much of the 238 U has decayed since the formation of the earth’s crust 2.5·109 years ago? c) How much energy per uranium nucleus is set free in the decay chain 238 U → 206 Pb? A small proportion of 238 U spontaneously splits into, e. g., 142 54 Xe und 96 38 Sr. 3. Radon activity After a lecture theatre whose walls, ﬂoor and ceiling are made of concrete (10×10×4 m3 ) has not been aired for several days, a speciﬁc activity A from 222 Rn of 100 Bq/m3 is measured. a) Calculate the activity of 222 Rn as a function of the lifetimes of the parent and daughter nuclei. b) How high is the concentration of 238 U in the concrete if the eﬀective thickness from which the 222 Rn decay product can diﬀuse is 1.5 cm? 4. Mass formula Isaac Asimov in his novel The Gods Themselves describes a universe where the 186 stablest nuclide with A = 186 is not 186 74 W but rather 94 Pu. This is claimed to be a consequence of the ratio of the strengths of the strong and electromagnetic interactions being diﬀerent to that in our universe. Assume that only the electromagnetic coupling constant α diﬀers and that both the strong interaction and the nucleon masses are unchanged. How large must α be in order that 186 82 Pb, 186 186 88 Ra and 94 Pu are stable? 40 3 Nuclear Stability 5. αdecay The binding energy of an α particle is 28.3 MeV. Estimate, using the mass formula (2.8), from which mass number A onwards αdecay is energetically allowed for all nuclei. 6. Quantum numbers An eveneven nucleus in the ground state decays by α emission. Which J P states are available to the daughter nucleus? 4 Scattering 4.1 General Observations About Scattering Processes Scattering experiments are an important tool of nuclear and particle physics. They are used both to study details of the interactions between diﬀerent particles and to obtain information about the internal structure of atomic nuclei and their constituents. These experiments will therefore be discussed at length in the following. In a typical scattering experiment, the object to be studied (the target) is bombarded with a beam of particles with (mostly) welldeﬁned energy. Occasionally, a reaction of the form a+b→c+d between the projectile and the target occurs. Here, a and b denote the beamand target particles, and c and d denote the products of the reaction. In inelastic reactions, the number of the reaction products may be larger than two. The rate, the energies and masses of the reaction products and their angles relative to the beam direction may be determined with suitable systems of detectors. It is nowadays possible to produce beams of a broad variety of particles (electrons, protons, neutrons, heavy ions, . . . ). The beam energies available vary between 10−3 eV for “cold” neutrons up to 1012 eV for protons. It is even possible to produce beams of secondary particles which themselves have been produced in high energy reactions. Some such beams are very shortlived, such as muons, π– or Kmesons, or hyperons (Σ± , Ξ− , Ω− ). Solid, liquid or gaseous targets may be used as scattering material or, in storage ring experiments, another beam of particles may serve as the target. Examples of this last are the electronpositron storage ring LEP (Large Electron Positron collider) at CERN1 in Geneva (maximum beam energy at present: Ee+ ,e− = 86 GeV), the “Tevatron” protonantiproton storage ring at the Fermi National Accelerator Laboratory (FNAL) in the USA (Ep,p = 900 GeV) and HERA (HadronElektronRinganlage), the electronproton storage ring at DESY2 in Hamburg (Ee = 30 GeV, Ep = 920 GeV), which last was brought online in 1992. 1 2 Conseil Européen pour la Recherche Nucléaire Deutsches ElektronenSynchrotron 42 4 Scattering Figure 4.1 shows some scattering processes. We distinguish between elastic and inelastic scattering reactions. Elastic scattering. In an elastic process (Fig. 4.1a): a + b → a + b , the same particles are presented both before and after the scattering. The target b remains in its ground state, absorbing merely the recoil momentum and hence changing its kinetic energy. The apostrophe indicates that the particles in the initial and in the ﬁnal state are identical up to momenta and energy. The scattering angle and the energy of the a particle and the production angle and energy of b are unambiguously correlated. As in optics, conclusions about the spatial shape of the scattering object can be drawn from the dependence of the scattering rate upon the beam energy and scattering angle. It is easily seen that in order to resolve small target structures, larger beam energies are required. The reduced deBroglie wavelength λ– = λ/2π of a particle with momentum p is given by λ– = c = ≈ 2 2 p 2mc Ekin + Ekin j p for Ekin mc2 / 2mEkin c/Ekin ≈ c/E for Ekin mc2 . (4.1) The largest wavelength that can resolve structures of linear extension ∆x, is of the same order: λ– < ∼ ∆x . a) b) a' a' a b a b b* b' d c c) d) c a b d e c a b d Fig. 4.1. Scattering processes: (a) elastic scattering; (b) inelastic scattering – production of an excited state which then decays into two particles; (c) inelastic production of new particles; (d) reaction of colliding beams. 4.1 General Observations About Scattering Processes p O 1 TeV/c 1am 1 GeV/c p 1 MeV/c 1fm D P Fig. 4.2. The connection between kinetic energy, momentum and reduced wavelength of photons (γ), electrons (e), muons (µ), protons (p), and 4 He nuclei (α). Atomic diameters are typically a few Å (10−10 m), nuclear diameters a few fm (10−15 m). 1pm e J q 1A 1 keV/c 1keV 43 1MeV 1GeV 1TeV E kin From Heisenberg’s uncertainty principle the corresponding particle momentum is: c 200 MeV fm pc > . (4.2) p > ∼ ∆x ≈ ∼ ∆x , ∆x Thus to study nuclei, whose radii are of a few fm, beam momenta of the order of 10 − 100 MeV/c are necessary. Individual nucleons have radii of about 0.8 fm; and may be resolved if the momenta are above ≈ 100 MeV/c. To resolve the constituents of a nucleon, the quarks, one has to penetrate deeply into the interior of the nucleon. For this purpose, beam momenta of many GeV/c are necessary (see Fig. 1.1). Inelastic scattering. In inelastic reactions (Fig. 4.1b): a + b → a + b∗ → c + d , part of the kinetic energy transferred from a to the target b excites it into a higher energy state b∗ . The excited state will afterwards return to the ground state by emitting a light particle (e. g. a photon or a πmeson) or it may decay into two or more diﬀerent particles. A measurement of a reaction in which only the scattered particle a is observed (and the other reaction products are not), is called an inclusive measurement. If all reaction products are detected, we speak of an exclusive measurement. When allowed by the laws of conservation of lepton and baryon number (see Sect. 8.2 and 10.1), the beam particle may completely disappear in the reaction (Fig. 4.1c,d). Its total energy then goes into the excitation of the 44 4 Scattering target or into the production of new particles. Such inelastic reactions represent the basis of nuclear and particle spectroscopy, which will be discussed in more detail in the second part of this book. 4.2 Cross Sections The reaction rates measured in scattering experiments, and the energy spectra and angular distributions of the reaction products yield, as we have already mentioned, information about the dynamics of the interaction between the projectile and the target, i. e., about the shape of the interaction potential and the coupling strength. The most important quantity for the description and interpretation of these reactions is the socalled crosssection σ, which is a yardstick of the probability of a reaction between the two colliding particles. Geometric reaction crosssection. We consider an idealised experiment, in order to elucidate this concept. Imagine a thin scattering target of thickness d with Nb scattering centres b and with a particle density nb . Each target particle has a crosssectional area σb , to be determined by experiment. We bombard the target with a monoenergetic beam of pointlike particles a. A reaction occurs whenever a beam particle hits a target particle, and we assume that the beam particle is then removed from the beam. We do not distinguish between the ﬁnal target states, i. e., whether the reaction is elastic or inelastic. The total reaction rate Ṅ , i. e. the total number of reactions per unit time, is given by the diﬀerence in the beam particle rate Ṅa upstream and downstream of the target. This is a direct measure for the crosssectional area σb (Fig. 4.3). We further assume that the beam has crosssectional area A and particle density na . The number of projectiles hitting the target per unit area and per unit time is called the ﬂux Φa . This is just the product of the particle density and the particle velocity va : Φa = Ṅa = na · va , A (4.3) and has dimensions [(area×time)−1 ]. The total number of target particles within the beam area is Nb = nb ·A·d. Hence the reaction rate Ṅ is given by the product of the incoming ﬂux and the total crosssectional area seen by the particles: Ṅ = Φa · Nb · σb . (4.4) This formula is valid as long as the scattering centres do not overlap and particles are only scattered oﬀ individual scattering centres. The area presented by a single scattering centre to the incoming projectile a, will be called the geometric reaction crosssection: in what follows: 4.2 Cross Sections 45 d A Va )a =nava Nb =nbAd Fig. 4.3. Measurement of the geometric reaction crosssection. The particle beam, a, coming from the left with velocity va and density na , corresponds to a particle ﬂux Φa = na va . It hits a (macroscopic) target of thickness d and crosssectional area A. Some beam particles are scattered by the scattering centres of the target, i. e., they are deﬂected from their original trajectory. The frequency of this process is a measure of the crosssectional area of the scattering particles. σb = = Ṅ Φa · Nb (4.5) number of reactions per unit time beam particles per unit time per unit area × scattering centres . This deﬁnition assumes a homogeneous, constant beam (e. g., neutrons from a reactor). In experiments with particle accelerators, the formula used is: σb = number of reactions per unit time beam particles per unit time × scattering centres per unit area , since the beam is then generally not homogeneous but the area density of the scattering centres is. Cross sections. This naive description of the geometric reaction crosssection as the eﬀective crosssectional area of the target particles, (if necessary convoluted with the crosssectional area of the beam particles) is in 46 4 Scattering many cases a good approximation to the true reaction crosssection. An example is highenergy protonproton scattering where the geometric extent of the particles is comparable to their interaction range. The reaction probability for two particles is, however, generally very different to what these geometric considerations would imply. Furthermore a strong energy dependence is also observed. The reaction rate for the capture of thermal neutrons by uranium, for example, varies by several orders of magnitude within a small energy range. The reaction rate for scattering of (pointlike) neutrinos, which only feel the weak interaction, is much smaller than that for the scattering of (also pointlike) electrons which feel the electromagnetic interaction. The shape, strength and range of the interaction potential, and not the geometric forms involved in the scattering process, primarily determine the eﬀective crosssectional area. The interaction can be determined from the reaction rate if the ﬂux of the incoming beam particles, and the area density of the scattering centres are known, just as in the model above. The total crosssection is deﬁned analogously to the geometric one: σtot = number of reactions per unit time beam particles per unit time × scattering centres per unit area . In analogy to the total crosssection, crosssections for elastic reactions σel and for inelastic reactions σinel may also be deﬁned. The inelastic part can be further divided into diﬀerent reaction channels. The total crosssection is the sum of these parts: (4.6) σtot = σel + σinel . The crosssection is a physical quantity with dimensions of [area], and is independent of the speciﬁc experimental design. A commonly used unit is the barn, which is deﬁned as: 1 barn = 1 b = 10−28 m2 1 millibarn = 1 mb = 10−31 m2 etc. Typical total crosssections at a beam energy of 10 GeV, for example, are σpp (10 GeV) ≈ 40 mb (4.7) for protonproton scattering; and σνp (10 GeV) ≈ 7 · 10−14 b = 70 fb for neutrinoproton scattering. (4.8) 4.2 Cross Sections 47 Luminosity. The quantity L = Φa · Nb (4.9) is called the luminosity. Like the ﬂux, it has dimensions of [(area×time)−1 ]. From (4.3) and Nb = nb · d · A we have L = Φa · Nb = Ṅa · nb · d = na · va · Nb . (4.10) Hence the luminosity is the product of the number of incoming beam particles per unit time Ṅa , the target particle density in the scattering material nb , and the target’s thickness d; or the beam particle density na , their velocity va and the number of target particles Nb ex