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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

Max-Planck-Institut für Kernphysik
Postfach 10 39 80
69029 Heidelberg, Germany

Physikalisches Institut
der Universität Erlangen-Nürnberg
Erwin-Rommel-Strasse 1
91058 Erlangen, Germany

Dr. Christoph Scholz

Dr. Frank Zetsche

SAP AG
Postfach 1461
69185 Walldorf, Germany

Universität Hamburg und
Deutsches Elektronen-Synchrotron
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

ISBN-10 3-540-36683-0 5th Edition Springer Berlin Heidelberg New York
ISBN-13 978-3-540-36683-6 5th Edition Springer Berlin Heidelberg New York
ISBN 3-540-20168-8 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
© Springer-Verlag 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: LE-TEX Jelonek, Schmidt & Vöckler GbR, Leipzig
Cover: WMXDesign GmbH, Heidelberg
SPIN 11801610

56/3100YL - 5 4 3 2 1 0

Printed on acid-free paper

Preface

In the last two editions we included new results on the neutrino oscillations
as evidence for a non-vanishing 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 so-called “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 many-body 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 Max-Planck-Institut
für Kernphysik in Heidelberg.
We would like to extend our thanks to Springer-Verlag, 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 β-Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 α-Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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 Cross-Section . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 The Mott Cross-Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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 Quasi-elastic 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 Non-resonant 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 β-Decay of the Nucleus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.7 Double β-decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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 Many-Body 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 β-decay
with the fundamental laws of conservation of energy, momentum and angular
momentum.
Thus, by the mid-thirties, 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 point-like 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 so-called “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 Van-der-Waals force) are thus expressions of the electromagnetic force.
When nuclear physics developed, two new short-ranged forces joined the
ranks. These are the nuclear force, which acts between nucleons, and the
weak force, which manifests itself in nuclear β-decay. 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

W-Boson
2
Mass |80 GeV/c

Z-Boson
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 non-relativistic 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 left-right 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 C-parity 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, X-rays 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 electron-positron 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 SI-unit, 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 mass-energy 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 (Heavyside-Lorentz 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 pre-condition 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 α-particles 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 α-particles. The angular distribution showed α-particle 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 α-particles 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 α-decay, which itself is a conversion of elements. While bombarding
nitrogen with α-particles, 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 long-ranged 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 α-particles
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
α-particles. 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
α-particles 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 α-particles.
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 X-rays 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 X-rays 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 H-atom 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 Xe-isotopes as
they occur in the atmosphere
(bottom). The Xe-isotopes in
the gneiss were produced by
spontaneous fission of uranium. (Picture courtesy of
Klaus Schäfer, Max-PlanckInstitut 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 low-energy (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 so-called “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 short-lived 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 8-9 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 semi-empirical 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 (even-even nuclei)
0 MeV/c2 for odd A (odd-even nuclei)
δ=
⎩
+11.2 MeV/c2 for odd Z and N (odd-odd 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 inter-nucleon 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, short-range 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 energy-level 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 proton-neutron 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. Low-lying 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 strong-interaction 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 β-decays 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 β-decays.

sion

s fis

E-stable nuclides

ou
ane

nt

spo

p-unstable

n-unstable

Fig. 3.1. β-stable nuclei in the Z − N plane (from [Bo69]).

Fe- and Ni-isotopes 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 two-body 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 many-body
decays any further since they are much rarer than two-body 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 α-decay, and the Helium nucleus
is called an α-particle. If a heavy nucleus decays into two similarly massive
daughter nuclei we speak of spontaneous fission. The probability of spontaneous fission exceeds that of α-decay 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 short-lived nuclides, the fall-off 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 longer-lived 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 β-Decay
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

,

β-Decay

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 even-even and the
odd-odd nuclei are found to lie on two vertically shifted
parabolas. The odd√
odd parabola lies at twice the pairing energy (2δ/ A) above the even-even
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 β-decay.
β-decay in odd mass nuclei. In what follows we wish to discuss the
different kinds of β-decay, 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 e-antineutrino 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 β-decays are
shown by arrows. The abscissa
co-ordinate is the atomic number, Z. The zero point of the
mass scale was chosen arbitrarily.

E-unstable
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.
β-decay in even nuclei. Even mass number isobars form, as we described
above, two separate (one for even-even and one for odd-odd nuclei) parabolas
which are split by an amount equal to twice the pairing energy.
Often there is more than one β-stable isobar, especially in the range A >
70. Let us consider the example of the nuclides with A = 106 (Fig. 3.3). The
106
106
even-even 106
46 Pd and 48 Cd isobars are on the lower parabola, and 46 Pd is the
106
stablest. 48 Cd is β-stable, since its two odd-odd neighbours both lie above
it. The conversion of 106
48 Cd is thus only possible through a double β-decay
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.
Odd-odd nuclei always have at least one more strongly bound, even-even
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 = 106-isobars (from
[Se77]). Possible β-decays are
indicated by arrows. The abscissa coordinate is the charge
number Z. The zero point of
the mass scale was chosen arbitrarily.

odd-odd

5

β-Decay

even-even

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 β-decay, since the increase in the asymmetry energy would
exceed the decrease in pairing energy. Some odd-odd nuclei can undergo both
β − -decay and β + -decay. Well-known 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 K-electrons 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 K-shell after
such a K-capture, electrons from higher energy levels will successively cascade
downwards and in so doing they emit characteristic X-rays.
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
K-capture 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 β-decay 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 β-unstable 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 β-stable.1
A well-known example of a long-lived β-emitter 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 long-lived. 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 long-lived (τ = 3 · 1014 yrs and τ = 3 · 1011 yrs respectively) that they may
also be considered stable.

3.2

α-Decay

31

3.2 α-Decay
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 so-called α-particle is extraordinarily strongly bound
— 7 MeV/nucleon (cf. Fig. 2.4). Such decays are called α-decays.
Figure 3.5 shows the potential energy of an α-particle as a function of its
separation from the centre of the nucleus. Beyond the nuclear force range, the
α-particle 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 α-particles which are energetically allowed to escape from the nuclear potential, the total energy of this
α-particle is positive. This energy is released in the decay.
The range of lifetimes for the α-decay 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 α-particle 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 α-particle tunnelling through one of these
(Fig. 3.6), then the transmission T is given by:

where κ = 2m|E − 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 α-particle. A Coulomb barrier can be thought of as a barrier composed of
V(r)
Vc = 2(Z--2)

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
,
2m|E − V | dr ≈
β

(3.15)

where β = v/c is the velocity of the outgoing α-particle and R is the nuclear
radius.
The probability per unit time λ for an α-particle to escape from the
nucleus is therefore proportional to: the probability w(α) of finding such an
α-particle in the nucleus, the number of collisions (∝ v0 /2R) of the α-particle
with the barrier and the transmission probability:
v0 −2G
e
λ = w(α)
,
(3.16)
2R
where v0 is the velocity of the α-particle 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 α-particle
have a strong effect on the lifetime.
Most α-emitting nuclei are heavier than lead. For lighter nuclei with A <
∼
140, α-decay 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 α-unstable 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 α-decay 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 α-decay 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 lowest-lying 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 even-even nucleus
with A ≈ 100. Above the ground state, individual discrete levels with specific J P quantum numbers can be seen. The excitation of even-even nuclei
generally corresponds to the break up of nucleon pairs, which requires about
1–2 MeV. Even-even 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 even-even
nucleus whose ground state has the quantum numbers 0+ . To the left the total
cross-section for the reaction of the nucleus A−1
Z X with neutrons (elastic scattering,
inelastic scattering, capture) is shown; to the right the total cross-section for γA−1
induced neutron emission A
ZX + γ →
Z X + n.

below 2 MeV.2 In odd-even and odd-odd nuclei, the number of low-energy
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
γ-transitions 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 E2-transitions 4+ → 2+ → 0+ , and not by a single 4+ → 0+
E4-transition. 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 γ-emission 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 M4-transition 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 γ-emission.
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, quasi-bound states. States below this threshold decay only by
(relatively slow) γ-emission 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 cross-sections 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 cross-sections for neutron capture and
γ-induced 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. α-decay
The α-decay 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 radio-thermal 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. α-decay
The binding energy of an α particle is 28.3 MeV. Estimate, using the mass formula (2.8), from which mass number A onwards α-decay is energetically allowed
for all nuclei.
6. Quantum numbers
An even-even 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) well-defined 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 K-mesons, 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 electron-positron storage ring LEP
(Large Electron Positron collider) at CERN1 in Geneva (maximum beam
energy at present: Ee+ ,e− = 86 GeV), the “Tevatron” proton-antiproton storage ring at the Fermi National Accelerator Laboratory (FNAL) in the USA
(Ep,p = 900 GeV) and HERA (Hadron-Elektron-Ringanlage), the electronproton storage ring at DESY2 in Hamburg (Ee = 30 GeV, Ep = 920 GeV),
which last was brought on-line in 1992.
1
2

Conseil Européen pour la Recherche Nucléaire
Deutsches Elektronen-Synchrotron

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 de-Broglie wave-length 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 wave-length
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 π-meson) 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 so-called cross-section σ, which is
a yardstick of the probability of a reaction between the two colliding particles.
Geometric reaction cross-section. 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 cross-sectional area σb , to be determined by experiment. We
bombard the target with a monoenergetic beam of point-like 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 cross-sectional
area σb (Fig. 4.3).
We further assume that the beam has cross-sectional 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 cross-sectional 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 cross-section: in what follows:

4.2

Cross Sections

45

d
A
Va

)a =nava

Nb =nbAd

Fig. 4.3. Measurement of the geometric reaction cross-section. 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 cross-sectional
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 cross-sectional 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 cross-sectional area of the target particles, (if necessary convoluted with the cross-sectional area of the beam particles) is in

46

4

Scattering

many cases a good approximation to the true reaction cross-section. An example is high-energy proton-proton 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
(point-like) neutrinos, which only feel the weak interaction, is much smaller
than that for the scattering of (also point-like) 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 cross-sectional 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
cross-section 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 cross-section, cross-sections 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 cross-section is
the sum of these parts:
(4.6)
σtot = σel + σinel .
The cross-section 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 cross-sections at a beam energy of 10 GeV, for example, are
σpp (10 GeV) ≈ 40 mb

(4.7)

for proton-proton scattering; and
σνp (10 GeV) ≈ 7 · 10−14 b = 70 fb
for neutrino-proton 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