Theoretical Nuclear Physics John M. Blatt Victor F. Weisskopf Theoretical Nuclear Physics I Springer-Verlag New York Heidelberg Berlin John M. Blatt Department of Applied Mathematic. University of New South Wales Kensington. N.S.W. Australia Victor F. Weisskopf Department of Physics Massachusetts Institute of Technology Cambridge. Massachusetts 02139 U.S.A. Library of Con,rellll Catalo,in, in Publication Data Blatt. lohn Markus. Theoretical nuclear physics. Bibliography: p. Includes index. 1. Nuclear physics. I. Weisskopf, Victor Frederick, 1908- joint author. II. Title. QC776.B53 1979 539.7 794268 ISBN -13: 978-1-4612-9961-5 This book was first published by lohn Wiley & Sons, Inc., 1952. All rights reserved. No part of this book may be translated or reproduced in any form without written permi8sion of Springer·Veriag. © 1979 by Springer·Verlag New York Inc. Softcover reprint of the hardcover 1st edition 1979 9 8 7 6 5 432 1 ISBN-13: 978-1-4612-9961-5 e-ISBN-13: 978-1-4612-9959-2 DOl: 10.1007/978-1-4612-9959-2 PREFACE The last twenty years have witnessed an enormous development of nuclear physics. A large number of data have accumulated and many experimental facts are known. As the experimental techniques have achieved greater and greater perfection, the theoretical analysis and interpretation of these data have become correspondingly more accurate and detailed. The development of nuclear physics has depended on the development of physics as a whole. While there were interesting speculations about nuclear constitution as early as 1922, it was impossible to make any quantitative theory of even the simplest nucleus until the discovery of quantum mechanics on the one hand, and the development of experimental methods sufficiently sensitive to detect the presence of a neutral particle (the neutron) on the other hand. The further development of our understanding of the nucleus has depended, and still depends, on the development of ever more powerful experimental techniques for measuring nuclear properties and more powerful theoretical techniques for correlating these properties. Practically every "simple," "reasonable," and "plausible" assumption made in theoretical nuclear physics has turned out to be in need of refinement; and the numerous attempts to derive nuclear forces and the properties of nuclei from a more" fundamental" approach than the analysis of the data have proved unsuccessful so far. Nuclear physics is by no means a finished edifice. It is very much to be hoped that simple fundamental laws can be discovered which will account for all the known properties of nuclei, and will allow us to predict new, unknown properties successfully. At present we must restrict ourselves to the investigation and correlation of all known nuclear properties on a semi-empirical basis. This book is devoted in its entirety to this task. Its subject matter is theoretical nuclear physics, by which we mean the theoretical con cepts, methods, and considerations which have been devised in order to interpret the experimental material and to advance our ability to predict and control nuclear phenomena. Obviously, this book does not pretend to cover all aspects of theo retical nuclear physics. We are forced to omit many details and special developments. The omissions are due partly to the lack of space and partly to the authors' lack of special knowledge. We hope v VI Preface that the study of this book will mak~ it somewhat easier for the reader to understand the original literature containing the material which is not covered in this book. We have restricted ourselves to phenomena involving energies below about 50 Mev, a region which is sometimes called classical nuclear physics. Thus we exclude the nuclear phenomena in cosmic rays as well as the phenomena associated with the production and absorption of mesons. The only exception to this rule is Chapter IV, which deals with nucleon-nucleon scattering experiments at energies up to 350 Mev and their interpretation in terms of nuclear forces. In general, we have omitted theoretical considerations which are not concerned directly with the properties of the nucleus itself. Thus we exclude, for example, the theory of the stopping of charged particles in matter, the theory of the diffraction and slowing down of neutrons, the theory underlying molecular beam and magnetic resonance experiments, and the theory of atomic hyperfine structure. Although all these subjects are important from an experimental point of view, their inclusion would have lengthened the book too much. We have also excluded subjects generally referred to as nuclear engineering, such as the theory of nuclear reactors. In so far as the relevant mate rial has been declassified, adequate textbooks are already available. Unfortunately, the theory of nuclear fission (which properly belongs in this book) could be treated only in a very cursory manner, since too many relevant facts are still unavailable. We have completely omitted the discussion of the theories of nuclear forces based on the various meson field theories. The numerous attempts to predict nuclear forces on the basis of meson fields have led to brilliant insights and predictions regarding mesons, but they have failed so far to reproduce quantitatively the observed forces between nuclear particles. This subject seemed to us not yet suffi ciently developed to warrant a systematic treatment in this book. It was our constant aim throughout the book to keep it on a level which is understandable to the experimental physicist who works in the field of nuclear physics or to a graduate student who knows the essential concepts and problems of nuclear physics. A one-term course in quantum mechanics, based on a book such as Schiff (49) should suffice as a prerequisite. Some parts of the book may be hard reading for students unaccustomed to theoretical work. Sections which are difficult and can be omitted without loss of understanding of subsequent material are indicated by the symbol ~. We have concentrated our efforts on a better understanding and a critical analysis of the different subjects. This has led in some cases Preface vii to new developments which are not yet published elsewhere. Since we are dealing with a growing and changing part of science, we must expect that many ideas which today are considered valid will turn out to be incorrect before long. Hence, many assumptions and state ments found in this book should be regarded as preliminary. It contains a great deal of information of which we are far from sure, and which is included only because nothing better is available at present. A characteristic example is the information about nuclear level densities given in Chapter VIII. In some instances recent developments have changed the emphasis from one way of description to another more successful one. For example, nuclear spectroscopy today is being based to an increasing extent upon the shell theory of nuclear structure, as indicated in Chapter XIV. At the time this book was conceived the concepts of Wigner's supermultiplet theory were the main tools for the understanding of nuclear spectra; they are the basis of the discussions in Chapter VI. We make no pretense of having a complete list of references. We have tried to include the basic theoretical papers in each field, such other theoretical papers as we happened to come across, and experi mental papers only in so far as they illustrate some points made in the discussion or substantiate values of nuclear constants used in the text. We have not made a systematic search of the whole litera ture. This applies especially to papers which appeared in journals other than The Physical Review. If a relevant reference is missing, it is very probable that we did not know of its existence. The manu script was revised for the last time in the spring of 1951; it contains only occasional references to later work. We are quite aware of the possibility that this book contains errors, not all of which may be trivial or typographical in nature. We have tried to keep a reasonable balance between the effort to eliminate errors and the effort to understand the subject matter and to clarify its presentation. At the end of every chapter is a list of symbols with a short explana tion of the meaning of each symbol, and the number of the equation in which the symbol is introduced and defined. If the symbol in question is defined in the body of the text, the section number is given in the symbol list. We have made an effort to have these lists of symbols comprehensive, but some minor symbols, which occur only a few times in the chapter, have been omitted. It would be impossible to acknowledge in detail the enormous amount of generous help and constructive criticism which we have received from friends and colleagues. We would like, however, to viii Preface express our special gratitude to H. Bethe, M. Deutsch, H. Feshbach, and E. Wigner for their extensive advice. We are greatly indebted to Miss lnge Reethof for her constant help and patience in the typing and retyping of the manuscript. J. M. BLATT V. F. WEISSKOPF June, 1952 CONTENTS I. GENERAL PROPERTIES OF THE NUCLEUS 1. INTRODUCTION 1 2. QUANTUM STATES, BINDING ENERGY, BINDING FRACTION 3 3. STABLE AND UNSTABLE NUCLEI, FISSION, ALPHA-DECAY, BETA-DECAY A. "Dynamical" Instability 7 B. Beta-Radioactivity 9 4. SIZE OF THE NUCLEI 13 A. Scattering of High-Energy Neutrons by Nuclei 14 B. The Yield of Nuclear Reactions Initiated by Protons or Alpha- Particles 14 C. Alpha-Decay Lifetimes 14 D. Maximum Energy of Some Beta-Rays 14 5. THE COULOMB BARRIER 16 6. ANGULAR MOMENTUM, SPIN 21 7. ELECTRIC AND MAGNETIC MOMENTS A. Electric Moments 23 B. Magnetic Moments 30 8. STATISTICS 39 SYMBOLS 44 II. TWO-BODY PROBLEMS AT LOW ENERGIES 1. INTRODUCTION 48 2. THE GROUND STATE OF THE DEUTERON; SIMPLIFIED DISCUSSION (CEN- TRAL FORCES ASSUMED) 49 3. NEUTRON-PROTON SCATTERING A. Simple Theory 56 B. Comparison with Experiment: The Spin Dependence of Nuclear Forces 65 C. The Effect of Chemical Binding 71 D. Coherent Scattering of Neutrons by Protons 80 4. PROTON-PROTON SCATTERING 86 5. THE TENSOR FORCE A. Experimental Discovery of the Existence of Non-central Forces 94 B. General Form of the Non-central Force 96 C. Properties of the Tensor Force 97 D. The Ground State of the Deuteron: Dynamics 99 E. The Ground State of the Deuteron: Quadrupole Moment 105 F. The Ground State of the Deuteron: Magnetic Moment 108 G. Neutron-Proton Scattering below 10 Mev 110 SYMBOLS 113 III. NUCLEAR FORCES 1. INTRODUCTION 119 2. STABILITY OF A NUCLEUS AGAINST COLLAPSE. THE IMPOSSIBILITY OF ATTRACTIVE FORCES' BETWEEN ALL PAIRS 121 IX x Contents 3. ExCHANGE FORCES A. Qualitative Considerations 127 B. Formal Definition of Exchange Forces 135 4. THE SATURATION CONDITIONS A. The Comparison Theorem 140 B. Saturation Conditions for Mixed Wigner and Majorana Forces 146 C. The Complete Saturation Conditions for Central Forces 149 D. Saturation Conditions for Tensor Forces 150 5. THE ISOTOPIC SPIN FORMALISM 153 SYMBOLS 162 IV. TWO-BODY PROBLEMS AT HIGH ENERGIES 1. INTRODUCTION 168 2. NEUTRON-PROTON SCATTERING AT ENERGIES BETWEEN 10 AND 30 MEV 171 3. NEUTRON-PROTON SCATTERING AT ENERGIES LARGER THAN 30 MEV 182 4. PROTON-PROTON SCATTERING 186 SYMBOLS 188 V. THREE- AND FOUR-BODY PROBLEMS 1. INTRODUCTION 191 2. THE GROUND STATE OF THE TRITON; CENTRAL FORCES 193 3. THE GROUND STATE OF THE ALPHA-PARTICLE; CENTRAL FORCES 202 4. HJ AND HeJ: THE EQUALITY OF NEUTRON-NEUTRON AND PROTON-PROTON FORCES 203 5. THE GROUND STATE OF THE TRITON; TENSOR FORCES 206 SYMBOLS 208 VI. NUCLEAR SPECTROSCOPY: I. GENERAL THEORY 1. THE SYSTEM~TICS OF STABLE NUCLEI A. Stability Conditions 211 B. Discussion of Stable Nuclei 217 2. THE SEMI-EMPIRICAL MASS FORMULA OF WEIZSACKER 225 3. DETAILED STUDY OF THE SYMMETRY EFFECT 233 4. THE SYMMETRY ENERGY AND THE SYSTEMATICS OF STABLE NUCLEI 241 5. NUCLEAR MAGNETIC MOMENTS IN LIGHT ELEMENTS 248 6. THE SPECTROSCOPIC CLASSIFICATION OF NUCLEAR ENERGY LEVELS 254 SYMBOLS 262 VII. NUCLEAR SPECTROSCOPY: II. SPECIAL MODELS 1. INTRODUCTION 266 2. THE UNIFORM MODEL OF WIGNER A. Theory 266 B. Comparison with Experiment 274 3. THE INDEPENDENT-PARTICLE MODEL A. Introduction 278 B. The P Shell Configurations 281 C. The Energy of the Ground State 285 D. Nuclear Magnetic Moments on the Independent-Particle Model 289 E. Criticism of the Independent-Particle Model 291 4. THE ALPHA-PARTICLE MODEL OF THE NUCLEUS A. Outline of the Theory 292 B. Criticism of the Alpha-Particle Model 298 Contents Xl 5. THE LIQUID DROP MODEL 300 SYMBOLS 305 VIII. NUCLEAR REACTIONS: GENERAL THEORY 1. INTRODUCTION A. Description of a Nuclear Reaction 311 B. Channels 313 C. Energy Relations 315 2. CROSS SECTIONS A. Geometrical Limitations on Reaction and Scattering Cross Sections 317 B. The Determination of Cross Sections from the Conditions at the Nuclear S\U"face, for Neutrons with l =0 325 C. The Determination of Cross Sections from the Conditions at the Nuclear Surface. General Case 329 D. The Angular Distribution of Elastically Scattered Particles 335 E. The Reciprocity Theorem for Nuclear Reactions 336 3. THE COMPOUND NUCLEUS, CONTINUUM THEORY A. The Bohr Assumption 340 B. Nuclear Reactions, Cross Sections, and Emission Rates 342 4. DETERMINATION OF CROSS SECTIONS, CONTINUUM THEORY 345 5. TRANSMISSION OF POTENTIAL BARRIERS 358 6. THE DECAY OF THE COMPOUND NUCLEUS A. Competition; Evaporation Model 365 B. Secondary Nuclear Reactions 374 7. RESONANCE THEORY; QUALITATIVE TREATMENT A. The Occurrence of Resonances 379 B. The Compound Nucleus, Level Widths, Qualitative Description 383 C. Interpretation of D and r 386 D. Cross Sections for Nuclear Reactions 391 E. Behavior of Nuclear Cross Sections near Threshold 394 8. RESONANCE THEORY; DETERMINATION OF CROSS SECTIONS A. Pure Resonance Scattering 398 B. Resonance Scattering and Resonance Reactions 406 9. RESONANCE THEORY; DECAYING STATES OF THE COMPOUND NUCLEUS A. The Potential Well Model 412 B. The Actual Nucleus 417 10. SPIN AND ORBITAL ANGULAR MOMENTUM A. l =0 Neutrons 422 B. Particles with arbitrary l 426 SYMBOLS 441 IX. NUCLEAR REACTIONS; APPLICATION OF THE THEORY TO EXPERIMENTS 1. INTRODUCTION 458 2. NEUTRON-INDUCED REACTIONS 462 A. Low and Intermediate Energy, Intermediate Nuclei 463 B. Low Energy, Heavy Nuclei 471 C. Intermediate Energy, Heavy Nuclei 476 D. High Energy, Intermediate and Heavy Nuclei 480 E. Very High Energy, Intermediate and Heavy Nuclei 481