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BOOK 

THE ATOMIC NUCLEUS 



Robley D. Evans, Ph.D. 

PROFESSOR OF PHYSICS 
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 



TATA McGRAW-HILL PUBLISHINC COMPANY LTD. 
Bombay New D*lhi 



THE ATOMIC NUCLEUS 



1955 by McGraw-Hill, Inc. 



All Rights Reserved 

This book, or parts thereof, may not be reproduced in any form 
without permission of the publishers. 



T M H Edition 



Reprinted in India by arrangement with the McGraw-Hill, Inc. 

New York. 



This edition can be exported from India only by the Publishers, 
Tata McGraw-Hill Publishing Company Ltd. 



Published by Tata McGraw-Hill Publishing Company Limited and 
Printed by Mohan Makhijani at Rekha Printers Pvt. Ltd., New Delhi-15. 



Preface 



This book represents the present content of a two-semester course 
in nuclear physics which the author has taught at the Massachusetts 
Institute of Technology for the past twenty years. During this time 
nuclear physics has expanded greatly in depth and breadth. Nuclear 
physics was originally a subject which represented the research interests 
of a small number of academic scientists, and whose modest size permitted 
easy coverage in a one-year graduate course. Now pure and applied 
nuclear physics is a gigantic area of research and engineering. Numerous 
subtopics have grown rapidly into large and separate fields of professional 
competence, but each of these derives its strength and nourishment 
from fundamental experimental and theoretical principles. It is this 
fundamental core material which is discussed -here. Even this central 
body of empirical knowledge and of theoretical interpretation has grown 
to be very large. This book embraces more material than my students 
and I are now able to cover, with adequate regard for depth of under- 
standing, in a one-year course of ninety class hours. Those topics 
which seem most lively and timely are selected from it by each year's 
group of students. Material which has to be excluded from the course 
is thus fully available for reference purposes. 

This text is an experimentalist's approach to the understanding of 
nuclear phenomena. It deals primarily with the area in which theory 
and experiment meet and ib intermediate between the limiting cases of a 
theoretical treatise and of a detailed handbook of experimental tech- 
niques. It undertakes to strike that compromise in viewpoint which 
has been adopted by the majority of working physicists. 

Detailed attention is given in the early chapters to several funda- 
mental concepts, so that the student may learn to think in center-of- 
mass coordinates and may visualize clearly the phenomena of barrier 
transmission, particle interactions during collisions, and collision cross 
sections. The physical aspects receive emphasis in the main text, while 
the corresponding mathematical details are treated more fully in appen- 
dixes. This reiteration, with varied emphasis and viewpoint, has been 
preserved because of the experiences of students and colleagues. 

As to prerequisites, it is expected that the reader has had at least 
an introductory course in atomic physics and that his mathematical 



vi Preface 

equipment is in working order through the calculus and differential 
equations. Prior experience in wave mechanics is not assumed, and 
the necessary mathematical and conceptual portions of this subject are 
developed from first principles as the need and application arise. 

I have been repeatedly impressed by the varied preparation and by 
the nonuniform backgrounds of seniors and first-year graduate students 
as they enter this course. Each student is well prepared in some areas 
but is blank in others. In an average class of fifty students there is a 
nearly random distribution of areas of competence and areas of no 
previous experience. These observations have dictated the level of 
approach. It. must be assumed that each subfield is a new area to the 
majority. With this experience in mind, the discussion of each topic 
usually begins at an introductory level. Within each subfield, the dis- 
cussion extends through the intermediate level and into the area of the 
most recent advances in current research. The aim is to bring the 
student to a level of competence from which he can understand the 
current research literature, ran profitably read advanced treatises and 
the many excellent monographs which are now appearing, and can under- 
take creative personal research. To help encourage early familiarity 
with the original papers, numerous references to the pertinent periodical 
literature appear throughout. 

Nuclear physics today embraces many topics which are strongly 
interdependent, such as nucloar moments and ft decay, and some topics 
which are nearly independent fields, such as some aspects of mass spec- 
troscopy. An optimum sequential arrangement of these topics is a 
difficult, if not insolvable, problem. The collection of indisputably 
nuclear topics definitely does not form a linear array, in which one may 
start at A and proceed to B, C, D, . . . , without having to know about 
Q in the meantime. The order of topics which is used here is that which 
has developed in the classroom as an empirical solution involving "mini- 
mum regret." 

I begin as Bethe and Bacher have done, with the fundamental prop- 
erties of nuclei. These are the characteristics which are measurable for 
any particular nuclide and which comprise the entries in any complete 
table of the ground-level nuclear properties: charge, size, mass, angular 
momentum, magnetic dipole moment, electric quadrupole moment, 
isobaric spin, parity, and statistics. In order to evaluate even these 
" static " properties of nuclei, it is necessary to invoke many types of 
experimental and theoretical studies of the "dynamic" behavior of 
nuclei, including a decay, ft decay, and nuclear reactions. The result 
is that those aspects of nuclear dynamics which enlighten the static 
properties are referred to early. This might have been done by saying, 
"It can be shown ..." or "We shall see later that . . . ," but it 
has proved more satisfactory to give, a reasonable, account of the per- 
tinent dynamic aspect at the place where it is first needed. This has 
been found to lead to better understanding, although it does give rise to 
occasional duplication, or "varied reiteration," and, in some instances, 
to division of dynamic topics, such as a decay and ft decay, into two parts. 



Preface vii 

Cross references appear throughout these topics, in order to reinforce 
the integration of the dynamic subjects. 

The middle of the book deals with the systematics of nuclei, with 
binding energy and separation energy, with intrrnucleon forces and 
illustrative nuclear models, and with the dynamics of nuclear reactions, 
a-ray spectra, ft decay, and radioactive-series transformations. Chapters 
18 through 25 treat the behavior of charged particles and of photons 
while passing through matter, concluding with a chapter containing 
baric material on a group of "practical" scientific, military, and indus- 
trial problems on the physical evaluation of penetrating radiation fields. 
The final three chapters drill with the statistical theory of fluctuations 
and uncertainties due to the randomicity of nuclear events, which is so 
often n governing factor in the design of imclr.ar experiment. Practical 
topics given detailed treatment, include the effects of resolving time, 
random coincidences, sealer and count ing-rate-meter fluHualiaiiH, and 
the statistics of rapidly decaying sources. 

Keferencr tables of many of the reasonably well-established nuclear 
properties accompany the corresponding text. For more comprehensive 
tables, explicit references arc made to the voluminous and valuable 
standard compilations. For the latest data, thcsn compilations must 
be augmented by the 1 Miminarics of new nuclear data published quarterly 
in NurJcar iS'ci'rwr Abstracts. 

Kvory worker in nuclear physics faces ihe opportunity of making a 
signilicant ,n-\v discovery. It is useful in know how discoveries have 
lii-eu marie by thuM* who have preceded us. Most of ihe history of 
nuclear physics ic. very recoiil and has occurred within the memory of 
people still working in the field. In order to illuminate the "anatomy 
of discovery" and at the stunt; time to focus on fundamental physical 
principles, some chapters, such as Chap. 13, Nuclear Reactions, Illus- 
trated by H IU (arj;) and Jts Associates, have been arranged with due 
regard to the history of nuclear physics and to the pitfalls and accidental 
triumphs of research. This was done to encourage the student to develop 
a feeling for the stapes t hrough which nuclear science has progressed and a 
sense of the conditions under which new discoveries are made. 

Problems are offered for solution at the end of many sections. These 
have been selected from homework and quizzes and are the type which 
one likes to work through in order to see that the principles 'of the subject 
are understood. Many problems supplement the text by containing 
their own answers, in the well-known "show that." style of Miles H. 
Sherrill and the late Arthur A. Noyes. 

Much help, both explicit, and general, has been received from pro- 
fessional colleagues, especially Profs. V. F. Weisakopf, H. Feshbach, 
and W. A. Fowler, and from the hundreds of students who have taken 
the course over the many years during which this book has been in 
preparation. The students' experiences have determined the content, 
the order of presentation, the amount of detail needed on particular 
topics, the nature and number of problems, and the topics which should 
be transferred to other new courses in specialized aspects of pure or 



viii Preface 

applied nuclear physics. Some former students may find that their 
favorite topic has been deleted altogether, in order to make space for 
the remainder in an already vast field. 

Each year one or more graduate students have collaborated closely 
in developing and presenting certain sections of the course, and to these 
men I welcome this opportunity of recalling our joint experiences of 
the past two decades and of recording my thanks, especially to Alfredo 
Banos, Keith Boyer, Sanborn Brown, Gordon Brownell, Randall Caswell, 
Eric Clarke, Franklin Cooper, Martin Deutsch, Robert Dudley, Lloyd 
Elliott, Wilfred Good, Clark Goodman, Arthur Kip, Alexander Langsdorf, 
Melvin Lax, John Marshall, Otto Morriiiigstar, Robert Osborne, Wendell 
Peacock, Norman Rasmusseii, Norman Rudnick, Leonard Schiff, and 
Marvin Van Dilla. Special thanks go to Norman Rasmussen for exten- 
sive work on semifinal revisions of the chapters dealing with the inter- 
action of radiation and matter. 

Miss Mary Margaret Shanahan has been tireless, accurate, and patient 
in editing and typing a series of hcotographed partial editions for student 
use and in preparing the entire final manuscript. The assistance of 
Miss Betsy Short, Mrs. Elizabeth Backofen, Mrs. Grace Rowe, Joel 
Bulkley, and Harry Watters has been invaluable. Transcending all 
this, the unbounded patience, insight, and encouragement of my wife, 
Gwendolyn Aldrich Evans, have made it possible to put this volume 
together. 

ROBLEY D. EVANS 



Contents 



Preface V 

INTRODUCTION 

Historical Sketch of the Development of the Concept of the Atomic Nucleus I 

CHAPTER 1 
CHARGE or ATOMIC NUCLEI 

Introduction 6 

1. Chemical Origin of Atomic Number 6 

2. Number of Electrons per Atom. X-ray Scattering 7 

3. Charge on the Atomic Nucleus. a-Ray Scattering . 11 

4. Frequency of K- and L-series X Rays 21 

5. The Displacement Law 25 

CHAPTER 2 
RADIUS or NUCLEI 

Introduction 28 

1. The Growth of Concepts Concerning the Size of Nuclei 28 

2. Coulomb-energy Difference between Isobars . 31 

3. Coulomb Potential inside a Nucleus 38 

4. The Nuclear Potential Barrier .... 45 

5. Wave Mechanics and the Penetration of Potential Barriers 49 

6. Lifetime of a-Ray Emitters .... 74 

7. Anomalous Scattering of a Particles ... 81 

8. Cross Sections for Nuclear Reactions Produced by Charged Particles . . 89 

9. Nuclear Cross Sections for the Attenuation of Fast Neutrons .... 94 

CHAPTER 3 
MASS OF NUCLEI AND OF NEUTRAL ATOMS 

Introduction 96 

1. The Discovery of Isotopes and Isobars . 96 

2. Nomenclature of Nuclei 98 

3. Mass Spectroscopy 101 

4. Atomic Mass from Nuclear Disintegration Energies 117 

5. Tables of Atomic Mass 135 

CHAPTER 4 
NUCLEAR MOMENTS, PARITY, AND STATISTICS 

Introduction 140 

1. Nuclear Angular Momentum 141 



x Contents 

2. Nuclear Magnetic Dipole Moment 148 

3. Anomalous Magnetic Moments of Free Nucleons 151 

4. Relationships between / and M 155 

5. Electric Quadrupole Moment 163 

6. Parity . . 174 

7. The Statistics of Nuclear Particles 177 

CHAPTER 5 

ATOMIC AND MOLECULAR EFFECTS OF NUCLEAR MOMENTS, 
PARITY, AND STATISTICS 

Introduction. ... 1S1 

1. Extraimclear Effects of Nuclear Angular Momentum iiml StiitisLicb. 181 

2. Extranuclcar Effects of Nuclear Magnetic Dipole Moment . 1D1 

3. Extranuclear Effects of Nuclear Electric Quadrupole Moment . 11)7 

CHAPTER 6 
EFFECTS or NUCLEAR MOMENTS AVD PARITY ON 

N U C LEA H T R A N S I TI O N S 

Introduction . .... 202 

1. Conservation of Parity and Angular Moiufutuiii . . . 2(M 

2. Penetration of Nuc.lrur Harrier . . . 201 

3. Lifetime in tf Decay . - 20f> 

4. Radiative Transitions in Nuclei . 211 

5. Internal Conversion 218 

6. Nutlfur Ipomers 22*. * 

7. Determination oi Angulnr Momentum and Purity of Exeited Level,? frorrj p- 

and 7- Transit inn Probabilities , 2H2 

8. Angular Corn-hit ion of Successive liadintiruis , 234 

9. Angular Distribution I'L Nuclear Reactions. . 214 

CHAPTER 7 
ISOTOPIC ABUNDANCE RATIOS 

Introduction. ... ... ... 250 

1. Ratios from Mass Spectrosoopy . . 250 

2. Isotope Shift in Line Spectra . .... 2f)(> 

3. Isotope Stiift in the Hand Spectra of Diatomic Molecules .... 2. r >8 

4. Isotope Ratios from Radioactive Decay Constants . ... 2(2 

5. Chemical and Physical Scales of Atomic Weight . 202 

6. Mass-spectrograph ic Identification of Nuclides in Nuclear Reactions . 264 

7. The Separation of Isotopes by Direct Selection Methods . . 2<<i 

8. Thr Separation of Isotopes by Enrichment Methods . 2h'9 

9. Szilard-Chalmera Reaction for the Enrichment of Radioactive Isotopes . 273 
10. Separation of Radioactive Isomers 275 

CHAPTER 8 
SYSTEMATIC^ OF STABLE NUCLEI 

Introduction 276 

1. Constituents of Atomic Nuclei 276 

2. Relative Abundance of the Chemical Elements 279 

3. Empirical Rules of Nuclear Stability .... 284 



Contents xi 

CHAPTER 9 
BINDING ENERGY OF NUCLEI 

Introduction. ... 294 

1. Packing Fraction . . 294 

2. Total Binding Energy .... 295 

3. Average Binding Energy . 297 

4. Separation Energy for One Nucleon 302 

CHAPTER 10 
FORCES BETWEEN NUCLEONS 

Introduction. ... 309 

1. General Characteristics of Specifically Nuclear Force*! .... 309 

2. Ground Level of the Deuteron . . ... 313 

3. Neutron-Proton Scattering at to 10 Mev . 317 

4. Electromagnetic Transitions in the n-p System . ... 330 

5. The Proton-Proton Force at to 10 Mev . . . 338 
G. Equivalence of (/in) and (pp) Forces . . . 344 

7. Summary of Central Forces . . 345 

8. Effects of Tensor Forces . . ... 348 

9 High-energy n-p and p-p Scattering 350 

CHAPTER 11 
MODELS OF NUCLEI 

Introduction 357 

1. Summary of Experimental Evidence Which Should Be Represented by the 
Model . . ... 357 

2. The Nuclear Shell Model 358 

3. The Liquid-drop Model . . 365 

4. Statistical Model of Excited Levels 397 

CHAPTER 12 
CONSERVATION LAWS FOR NUCLEAR REACTIONS 

Introduction . . . ... ... . 408 

1. Physical Quantities Which Are Conserved in Nuclear Ktwtions . . 408 

2. Determination of the Q Value for Nuclear Reactions . .... 410 

CHAPTER 13 

NUCLEAR REACTIONS, ILLUSTRATED BY B l (,p) 
AND TTS ASSOCIATES 

Introduction .... . 422 

1. Energy Distribution of Protons from B H \rt,p)C 113 423 

2. Discovery of the Neutron from B -+- r* . . 420 
\\. Discovery of Artificial Kiidumctivity from H + tr . 430 

4. Resonances in the Formation of the Compound Nucleus . . 434 

5. Energy Loss in Inelastic Scattering . . . 438 

6. Summary of the Del 'Tini nation of Nuclear Hni'rgy Levels from Inaction 
Energetics ..... .440 

CHAPTER 14 
ENERGY 1 fiorEMnwE UK NurLE/ui-itrAr'iio\ (Yv.s SUCTIONS 

Introduction. . . 441 

1. Resonance Theory r>f Nuclear Cross Sections . . . 444 

2. Continuum Theory of Nuclear Cross Sections . 45'- 



xii Contents 

CHAPTER 15 
RADIOACTIVE-SERIES DECAY 

Introduction. .... . ......... 470 

1. Decay of a. Single Radioactive Nuclide . .... . 470 

2. Radioactive-Henes D^cay. Growth of a Daughter Product . 477 

3. Accumulation of Daughter Atoms . .... . 478 

4. Time of Maximum Activity of Daughter Product. Ideal Equilibrium . 479 

5. Ratio of Activity of Parent and Daughter. Transient Equilibrium 480 

6. Yield of a Radioactive Nuriide Produced by Nuclear Bombardment 484 

7. Growih of a Granddaughter Product . . 486 

8. General Equations of Radioactive-series Growth and Decay ..... 490 

9. Accumulation of Stable End Products ...... 494 

10. Summation Rules ...... .... . .496 

11. Approximate Method? for Short Accumulation Times. . 500 

12. Graphical Methods for Series Growth and Decay ..... , 502 

CHAPTER 16 
SPECTRA 



Introduction . . ... ......... 511 

1. Fine Structure of a-Ray Spectra . ......... 511 

2. Genealogy of Nuclides Which Emit a Ra} r s ...... 517 

3. The Nuclear Energy Surface, for Heavy Nuclides ...... 523 

4. System a tics of a Decay Energies . ......... 527 

CHAPTER 17 
0-RAY SPECTRA 

Introduction. . . ... ....... 536 

1. Experimental Characteristics of the /3-Ray Continuum ...... 536 . 

2. The Neutrino . . ....... 541 

3. Fermi Theory of Decay . . ............ 548 

CHAPTER 18 
lONIZATION OF MATTER BY ClIARQED PARTICLES 

Introduction. . . ...... . .... 567 

1. Classical Theory of Inelastic Collisions with Atomic Electrons .... 570 

2. Quantum-mechanical Theories of Inelastic Collisions with Atomic Electrons 574 

3. Comparison of Classical and Quantum-mechanical Theories . . 584 

4. Energy Loss per Ion Pair by Primary and Secondary lonization. . . 586 

5. Dependence of Collision Losses on the Physical and Chemical State of the 
Absorber . .......... ...... 587 

6. Certiiikov Radiation ................. 589 

CHAPTER 19 
ELASTIC SCATTERING OF ELECTRONS AND POSITRONS 

1. Scattering of Electrons by Nuclei ...... ..... 592 

2. Scattering of Swift Electrons by Electrons . ....... 597 

CHAPTER 20 

RADIATIVE COLLISIONS OF ELECTRONS WITH ATOMIC NUCLEI 

Introduction. . ............ ... 600 

1. Theory of Bremsstrahlung ............... 600 

2. Comparison of Various Interactions between Swift Electrons and Atoms . . 606 



Contents ziii 

CHAFTEB 21 
STOPPING OF ELECTRONS BY THICK ABSORBERS 

Introduction 611 

1. Path Length and Range of Electrons 611 

2. Thick-target Bromsstrahlung . 614 

3. Range-Energy Relations for Electrons 621 

4. Annihilation Radiation 629 

CHAPTER 22 
PASSAGE OF HEAVY CHARGED PARTICLES THROUGH MATTER 

Introduction. . 632 

1. Capture and Loss of Electrons . 633 

2. Energy Loss per Unit Path Length 637 

3. Range- Energy Relationships 647 

4. lonization of Gases. 654 

5. Straggling . . 660 

6. Range of Fission Fragments 668 

CHAPTER 23 
THE INTERACTION OF ELECTROMAGNETIC 

RADIATIONS WITH MATTER. 
COMPTON SCATTERING AND ABSORPTION 

Introduction. . . . 672 

1. Compton Collision and the Conservation Laws 674 

2. Klein-Nishina Cross Sections for Polarized and Unpolarized Radiation . . 677 

3. Compton Attenuation Coefficients . . . ... 684 

4. Angular Distribution of Compton Scattered Photons and Recoil Electrons 690 

5. Energy Distribution of Compton Electrons and Photons 692 

CHAPTER 24 
PHOTOELECTRIC EFFECT AND PAIR PRODUCTION 

1. Photoelectric Effect. . . 695 

2. Pair Production by Photons 701 

CHAPTER 25 
ATTENUATION AND ABSORPTION OF ELECTROMAGNETIC RADIATION 

Introduction. . 711 

1. Attenuation Coefficients 711 

2. Energy Absorption 719 

3. Multiple Scattering of Photons 728 

4. Distributed y-Ray Sources 736 

CHAPTER 26 
STATISTICAL FLUCTUATIONS IN NUCLEAR PROCESSES 

Introduction 746 

1. Frequency Distributions 747 

2. Statistical Characterization of Data 757 

3. Composite Distributions . 766 

CHAPTER 27 
STATISTICAL TESTS FOR GOODNESS OF FIT 

Introduction. .... . .... 774 

1. Lexis' Divergence Coefficient 774 



xiv Contents 

2. Pearson's Chi-square Test ............... 775 

3. An Extension of the Chi-squnre Test ............ 777 

4. Examples of Random Fluctuations ............ 777 

CHAPTER 28 

APPLICATIONS OF POISSON STATISTICS TO SOME INSTRUMENTS USED 
IN NUCLEAR PHYSICS 

Introduction. ................. 7B5 

1. Effects of the Finite Resolving Time of Counting Instruments . . . 7B5 

2. Scaling Circuits. . . ... 794 

3. Counting-rate Meters . ... . ..... 803 

4. lonizatinn Chambers . .......... . 810 

5. Rapid Decay of a Single Radionurlide ........... 812 

6. Radioactive-series Disintegrations ............ 818 

APPENDIX A 

THOMSON SCATTERING AS AN ILLUSTRATION OF THE WAVE 
AND CORPUSCULAR CONCEPTS OF CROSS SECTION 

Introduction. . . . 819 

1. Thomson Scattering .... . ....... 819 

2. Comparison of Wave and Corpuscular Concepts of Cross Section . . . 821 

APPENDIX B 

CENTER-OF-MASS COORDINATES. AND THE NONRELATIVISTIC 
ELASTIC COLLISION IN CLASSICAL MECHANICS 

Introduction . ........ ...... 828 

1. Relations between L (Laboratory! and C (Center-of-mass) Coordinates. . 828 

2. Equation of the Hyperbola in Polar Coordinates . . . 836 

3. Klastiu Collision between Charged Particles. . .... . 838 

4. Cross Sections for Elastic Scattering by Coulomb Forces. . . . 847 

5. Summary of Principal Symbols and Results ........ 851 

APPENDIX C 
THE WAVE MECHANICS OF NUCLEAR POTENTIAL BARRIERS 

Introduction ...... . . . . 852 

1. Exact Solution of Schrodinger's Equations for a One-dimensional Rectangu- 

lar Barrier .... . 852 

2. iSchrodmKiT\s Equation for a Central Field . 860 
II. KfjireKentatjrui of the Plane Wvi- in Spherical Polar Coordinates . 8GG 
1. Physical Correspondence between Partial Waves and OlassiVal Impact 



3. Transmission through a Nurlrur Pnleriti.'il Barrier . 874 

ti. Elastic- ScutU'riiiK of Particles Incident cm a Nuclear Potential Barrier. . 878 

AlTJfiNIlIX D 

Rf ifttivistie Krltniunsliips between Muss, Mum on turn, Energy, and Mag- 
nHi< Iligiility .... . ........... 890 

APPENDIX E 
Pom* tt;^n Minnie and Nuclear Constants .......... 898 



Contents xv 

APPENDIX F 

Table of the Elements 900 

APPENDIX G 

Somo Useful Inofficiont Statistics 902 

Bibliography 905 

Glossary of Principal tfymhols 930 

Index 953 



INTRODUCTION 



Historical Sketch of the Development of the Concept 
of the Atomic Nucleus 



The earliest speculations on the atomic hypothesis of the ultimate 
structure of mattci are ascribed to the Ionian philosophers of the fifth 
century B.C. Anaxagoras, Leucippus, and Democritus postulated that 
all matter is made up of a set of particles which were called atoms to 
denote their presumed indivisibility. Their concept of a world made 
up of invisible, incompressible, eternal atoms in motion is best known 
now through the writings of the Latin poet Lucretius (98 to 55 B.C.), 
especially through his six-book scientific poem " Concerning the Nature 
of Things 1 ' (De Rerum Natura) (Dl).f 

Bodies of things are safe 'till they receive 

A force which may their proper thread unweave, 

Nought then returns to nought, but parted falls 

To Bodies. of their prime Originals. 

. . . Then nothing sure its being quite forsakes, 

Since Nature one thing, from another makes; 

. . . LUCRETIUS 

Through the subsequent centuries many philosophers speculated on the 
ultimate structure of matter. Because nearly every possible guess was 
made by one person or another, it is no surprise that some of them 
were close to the truth, but all these theories lacked any experimental 
foundation. 

At the beginning of the nineteenth century the researches on chem- 
ical combining weights by John Dalton and his contemporaries (C54) 
led to his enunciation, on experimental grounds, of the atomic theory ol 
matter in his great book : 'A New System of Chemical Philosophy" 
(1808). Three years later, Avogadro, professor of physics at Turin, 
distinguished clearly between atoms and molecules and filled the only 
gap in Dal ton's logic when he pointed out that equal volumes of differ- 
ent gases contain equal numbers of molecules when the temperatures and 
pressures are equal. Then followed the first hypothesis concerning the 
structure of the atoms themselves. Prout, an Englishman, as was 

t For references in parentheses, see the Bibliography at the end of the book, which 
is arranged alphabetically and by number. 

1 



2 The Atomic Nucleus 

Dalton, suggested in 1815 that the atoms of all elements were made up 
of atoms of hydrogen. Prout's hypothesis was soon discredited by the 
more accurate atomic-weight measurements of the later nineteenth cen- 
tury, only to be reestablished, in modified form, after the discovery of 
isotopes during the early part of the present century. This discovery 
required the introduction of the concept of mass number. 

Modern atomic physics had its inception in the discovery of X rays 
by Rontgen (R26) in 1895, of radioactivity by Becquerel (H25) in 1800, 
and of the electron by J. ,1. Thomson (T22) in 1897. J. J. Thomson's 
measurement of c/m for the electron and H. A.. Wilson's determination 
(W04, M46) of the electronic charge c by the cloud method showed the 
mass of the electron to be about 10~ 2T g. The value of r, combined with 
Faraday's electrolysis laws, showed that the hydrogen atom wan of the 
order of 1,800 times as heavy as the electron. Thomson's studies had 
shown that all atoms contained electrons, and Barkln's (H12) experi- 
ments on X-ray scattering showed that the number of electrons in each 
atom (except hydrogen) is approximately equal to half the atomic weight. 

It was then evident that the mass of the atom is principally associ- 
ated with the positire charge which it contains. Xagaoka's (Nl ) nuclear 
atom model, with rings of rotating electrons, had attracted few endorse- 
ments because, from considerations of classical electromagnetic theory, 
the revolving electrons should continually radiate, because of their 
centripetal acceleration, and should eventually fall into the central 
nucleus. J. J. Thomson circumvented this difficulty with his "chargcd- 
cloud 1f atom model, consisting of "a case in which the positive electricity 
is distributed in the way most amenable to mathematical calculation; 
i.e., when it occurs as a .sphere of uniform density, throughout which the 
corpuscles (electrons) are distributed" (T23). 

By this time a rays from radioactive substances were under intensive 
study. Following Rutherford's (R42) semiquantitativc observation of 
the scattering of a rays by air or by a thin foil of mica, ( JeigiT (() 10) found 
the most probable angular deflection suffered by a rays in pas.sing through 
0.0005-mm gold foils to be of the order of 1. (loiger and Marsden 
(G13) had shown that 1 a ray in 8,000 is deflected more than 00 by a thin 
platinum film. The Thomson model had predicted only small deflec- 
tions for single scattering and an extremely minute probability for large 
deflections resulting from multiple scattering. The predict ioius of the 
Thomson model fell short of these experimental results by at least a 
factor of 10 in . Accordingly, Rutherford proposed (It 43) that the charge 
of the atom (aside from the electrons) was concentrated into a very ,-mall 
central body, and he showed that such a model could explain the la .-go 
deflections of a rays observed by f li'iger and Marsden. Whereas Thoir, - 
son's positive cloud has atomic dimensions (<~10~ s cm), ItuthorfordV: 
atomic nucleus has a diameter of less than 10~ 12 cm. Rutherford 1 ^ 
theory did not predict the sign of the nuclear chtirRi*, but the electronic 
mass and the X-ray and spectral data indicated thai it must, be positive, 
with the negative electrons distributed about it to form the neutral atom. 

The quantitative dependence of the 1 intensity of -ray scattering on 



Introduction 3 

the angular deflection, foil thickness, nuclear charge, and a-ray energy 
was predicted by Rutherford's theory a prediction completely confirmed 
by Geiger and Marsden's (G14) later experiments. In agreement with 
Barkla's experiments on X-ray scattering, and with Moseley's (M60) 
brilliant pioneer work on X-ray spectra, Geiger and Marsden's experi- 
ments showed that "the number of elementary charges composing the 
center of the atom is (approximately) equal to half the atomic weight. 19 
Thus the concept of atomic number Z became recognized as the charge 
on the nucleus; with its aid the few irregularities in Mendeleev's periodic 
table (M42) were resolved. 

Once the existence of a small, massive, positive nucleus and an array 
of external electrons had been established, it became obligatory to 
abandon classical electromagnetic theory and to postulate nonradiating 
electronic orbits. Bohr (B92) took the step and, by combining Planck's 
quantum postulate with Nicholson's (B27) suggestion of the constancy 
of angular momentum, succeeded in describing the then observed hydro- 
gen spectra in detail, as well as in deriving the numerical value of Ryd- 
berg's constant entirely theoretically. These striking successes estab- 
lished the Rutherford-Bohr atom model and the existence of the small, 
massive, positively charged atomic nucleus. 

Soddy, Fajans, and others established the so-called displacement law 
(S58), according to which the emission of an a ray is accompanied by a 
change in the chemical properties of an atom by an amount corresponding 
to a leftward displacement of two columns in the Mendeleev periodic 
table of the elements (Appendix F). Similarly, a 0-ray transformation 
corresponds to a displacement of one column in the opposite direction. 
Since the emitted a ray carries a double positive charge, whereas the 
ray carries a single negative charge, it was evident that radioactive 
emission was a spontaneous nuclear disintegration process. Moreover, 
two elements differing from each other by one a-ray and two 0-ray 
emissions would have the same nuclear charge, hence the same chemical 
properties, but would exhibit a mass difference due to the loss of the 
heavy a particle. Thus the existence of isotopes was postulated by 
Soddy as early as 1910 from chemical and physical studies (A36) of the 
heavy radioactive elements. J. J. Thomson (T24) had succeeded in 
obtaining positive-ion beams of several of the light elements, and their 
deflection in magnetic and electrostatic fields proved that all atoms of a 
given type have the same mass. In 1912 Thomson, by his "parabola 
method," discovered the existence of two isotopes of neon, later shown 
by Aston (A36) to have masses of 20 and 22. 

Chadwick's (C12) proof of the existence of neutrons now permits us 
to contemplate the a particle as a close combination of two protons and 
two neutrons, and the nuclei of all elements as composed basically of 
protons and neutrons. Spectroscopy has dealt with the structure of the 
extranuclear swarm of electrons and, in so doing, has found it necessary 
to make at least two refinements in the Rutherford-Bohr atom model. 
The wave-mechanical treatment of the electrons has removed the definite- 
ness of planetlike electronic orbits, substituting a cloudlike distribution 



The Atomic Nucleus 

CHRONOLOGICAL REVIEW OF SOME MAJOR STEPS IN THE ACCRETION OF 
EXPERIMENTAL KNOWLEDGE CONCERNING THE ATOMIC NUCLEUS 



Advance 


Date 


By whom 


Where 


First experimental basis for the atomic 
hypothesis. Chemical combining weights 
Atoms and molecules distinguished. Gas 
laws unified 


1808 
1811 


Dalton 
Avogadro 


England 
Italy 


Precursor of mass number. Hydrogen as a 
basic unit in structure of heavy atoms. 
Periodic chemical classification of the elements 
Discovery of continuous X rays 
Discovery of radioactivity of urn-ilium. 


1815 
1868 
1895 
1896 


Prout 
Mendeleev 
Rontgen 
Becquerel 


England 
Russia 
Germany 
France 


Discovery of electron as constituent of all 
atoms . . ... ... .... 


1897 


J. J. Thomson 


England 


Charge of electron measured by cloud method. 
Avopadro's number estimated . . 
Identification of <* particle as a helium nucleus 
Equivalence of mass and energy 
Number of electrons per atom estimated from 
X-ray scattering 
Isotopes, isobars identified 
Discovery of stable isotopes of Ne 20 - 22 . . . 
Atomic nucleus discovered by interpretation 
of a-ray -scattering results 

Nuclear atom model "completed" by expla- 
nation of origin of spectra. Quantization 
of atomic states . 
Assignment of atomic numbers, from X-ray 
spectra 
Nuclear transmutation induced; proton iden- 
tified 


1903 
1909 
1905 
1904- 
1911 
1911 
1912 
1911- 
1913 

1913 
1913 
1919 


H. A. Wilson 
Rutherford 
Einstein 
Barkla 

Soddy 
Thomson 
Rutherford, 
Geiger, and 
Marsden 

Bohr 
Moseley 
Rutherford 


England 
England 
Switzerland 
England 

England 
England 
England 

Denmark 
England 
England 


Compton effect 


1923 


A. H. Compton 


U.S.A. 


Wavelength proposed for corpuscles 


1924 


dc Broglie 


France 


The wave equation . . 


1926 


Schrodingcr 


Germany 


Uncertainty principle 


1927 


Hoi sen berg 


Germany 


De Broglie wavelength observed when elec- 
trons diffracted by crystals 
a-ray decay explained as wave penetration of 
a nuclear barrier 

Discovery of deuterium 


1927 
1928 

1932 


Davisdon and 
Germer 
Gamow, 
Condon, and 
Gurney 
Urey 


U.S.A. 

Germany, 
U.S.A. 

U.S.A. 


Discovery of the neutron . . . 


1932 


Chad wick 


England 


Nuclear transmutation by artificially acceler- 
ated particles 
Positron discovered 


1932 
1932 


Cockcroft and 
Walton 
Anderson 


England 
U.S.A. 


Anomalous magnetic dipole moment of proton 
discovered 
/Neutrino hypothesis 


1933 
1933 


R. Frisch and 
0. Stern 
Pauli 


Germany 
Switzerland 


1 Theory of /9 decay. ... 


1934 


Fermi 


Italv 











Introduction 

CHBONOLOGICAL REVIEW OF SOME MAJOR STEPS IN THE ACCRETION or 

EXPERIMENTAL KNOWLEDGE CONCERNING THE ATOMIC NUCLEUS 

(Continued) 



Advance 


Date 


By whom 


Where 


/Radioactive light nu elides discovered 


1934 


I. Curie and F. 


France 


1 




Joliot 




| Radioactive nuclides produced by acceler- 


1934 


Lawrence et al., 


U.S.A. 


V ated particles 




Lauritsen et al. 




Transformation of nuclei by neutron capture 


1934 


Fermi 


Italy 


Anomalous proton-proton scattering 


1936 


White, Tuve, 


U.S.A. 






Hafstad, Herb, 








Breit, etc. 




p meson discovered ... 


1936 


Anderson and 


U.S.A. 






Ncddermeyer 




Precise measurements of nuclear moments 








by molecular-beam magnetic-resonance 








methods 


1938 


Rabi 


U.S.A. 


Nuclear fission discovered 


1939 


O. Hahn and F. 


Germany 






Strassmann 




Measurement of magnetic moment of the 


1940 


Alvarez and 


U.S.A. 


neutron 




Bloch 




+ v meson discovered . . 


1947 


Powell et al. 


England 


Artificial production of v mesons ... 


1948 


Gardner and 


U.S.A. 






Lattes 





of position probabilities for the extranuclear electrons. Secondly, 
detailed examination (hyperfine structure) of line spectra has shown that 
at least three more properties must be assigned to nuclei. These are the 
mechanical moment of momentum, the magnetic dipole moment, and 
an electric quadrupole moment. 

The nuclear transmutation experiments of Rutherford (R46), of Cock- 
croft and Walton (C27), of I. Curie and Joliot (C62), and of Fermi (F33) 
opened up a vast field of investigation and suggested new experimental 
attacks on the basic problems of nuclear structure the identification 
of the component particles within nuclei and of the forces which bind 
these particles together, the determination of the energy states of nuclei 
and their transition probabilities, and the investigation of the nature and 
uses of the radiations associated with these transitions. These are the 
problems with which we shall deal in the following chapters. 



CHAPTER 1 

Charge of Atomic Nuclei 



The number 7. of positive elementary charges (r = 4.8 X 10~ in esu, 
or 1.0 X 10~ lu coulomb) carried by the nuclei of all i.solopcw of an ele- 
ment Is called the atomic number of Miai element. At least five* different 
experimental approaches have been needed for the ultimate 1 assignment 
of atomic numbcra to all the chemical elements. 

Originally, the atomic number wan simply a serial ;mmber which was 
assigned to the known elrmenLs when arranged in :i sequence of inciviiR- 
ing iitomic weight. Tht* connection between thewe serial numbers and 
the quantitative rstr'vural properties of the atoms remained unrli,ycov- 
ered for half a century. At present, Z is probably the only nuclear 
quantify which is kmr.vu 'without error 1 ' for all nuclei. Of course, the 
actual charyr Zc contains (he experimental uncertainty of the best 
determinations of the elementary charge c. Thus the absolute nuclear 
charge, like everything else in physics, is known only within expen mental 
accuracy. 

1. Chemical Origin of Atomic Number 

About the time of the American Civil War the Russian chemist 
D. I. Mendeleev proposed his now well-known periodic, tablet of the 
elements. Mendeleev's successful classification of all elements into 
columns exhibiting similar chemical properties, and into rows with pro- 
gressively increasing atomic weights, dictated several revisions in the 
previously accepted atomic weights. The chemical atomic, weight of 
multivalent elements is determined by multiplying the observed chemical 
combining weight by the smallest integer which is compatible with other 
known evidence. For example, indium has a chemical combining weight 
of 38.3 and had been incorrectly assigned ail atomic weight of twice this 
figure; the progressions of chemical properties in the periodic system 
showed that the atomic weight of indium must be three times the com- 
bining weight, or 114.8. After minor readjustments of this type, and the 

t The American physician James Blake, by observing the effects of all available 
chemical elements on the circulation, respiration, and central nervous system of dogs, 
arranged the elements in chemical groups [Am. J. Mcd. JSci., 15: 03 (1848)] but the 
periodicities were first shown two decades later by Mendeleev (B64). 

6 



2] 



Charge of Atomic Nuclei 



subsequent discovery of helium and argon, which required the addition 
of the eighth and final column to the original table, the periodic table 
became a systematic pattern of the elements in which successive whole 
numbers, known as the atomic number, could be assigned confidently to all 
the light elements, on a basis of increasing atomic weight. Because the 
total number of rare-earth elements was unestablished, it was impossible 
to be certain of the atomic numbers of elements heavier than these, 
though tentative assignments could be made. Outside the rare-earth 
group, the periodic system successfully predicted the existence and prop- 
erties of several undiscovered elements and properly reserved atomic 
numbers for these. 

Three inversions were noted in the uniform increase of atomic num- 
ber with atomic weight. Because of their chemical properties, it was 
necessary to assume that the three pairs K (39.1) and A (39.9), Co (58.9) 
and Ni (58.7), Te (127.6) and I (126.9) were exceptions in which the 
element with lower atomic weight has the higher atomic number. These 
inversions arc now fully explained by the relative abundance of the iso- 
topes of these particular elements. For example, Table 1.1 shows that 
while argon contains some atoms which are lighter than any of those of 
potassium, the heaviest argon isotope is the most abundant. Also, while 
potassium contains some atoms which are heavier than any of argon, the 
lightest potassium isotope happens to be most abundant. 

TABLE 1.1. THE RELATIVE ABUNDANCE OF THE ISOTOPES OF ARGON AND 

OF POTASSIUM 



Element 


Atomic 
number 


Miiws numbers arid their relative abundance 


Average 
atomic 
weight 


:3(i 


37 


38 


39 


40 


41 


A 


18 


0.3 




0.06 




99.6 




39.9 


K 


19 






. L . 


93.4 


0.01 


6.6 


39.1 



There seems little room to doubt the completeness of the chemical 
evidence of the light elements, and on this basis the first 13 atomic num- 
bers were assigned to the elements from hydrogen to aluminum. From 
aluminum upward, the atomic numbers have been assigned on a basis of a 
variety of mutually consistent physical methods. Final confirmation of 
even the lowest atomic numbers has been obtained from observations of 
the scattering of X rays and of a rays and from spectroscopic evidence. 
The atomic numbers for the 103 elements which are now well established 
will be found in the periodic table of Appendix F. 

2. Number of Electrons per Atom. X-ray Scattering 

a. Scattering of X Rays by Atomic Electrons. One of the earliest 
experiments undertaken with X rays was the unsuccessful effort to reflect 
them from the surface of a mirror. It was found instead that the X rays 



8 The Atomic Nucleus [CH. 1 

were diffusely scattered, more or less in all directions, by the mirror or, 
indeed, by a slab of paraffin or any other object on which the X rays 
impinged. J. J. Thomson interpreted this simple observation as prob- 
ably due to the interaction of the X rays with the electrons which he had 
only recently shown to be present in all atoms. Treating the X ray as a 
classical electromagnetic wave, Thomson derived an expression for the 
scattering which should be produced by each electron. In this classical 
theory, each atomic electron is regarded as free to respond to the force 
produced on it by the electric vector of the electromagnetic wave. Then 
each electron oscillates with a frequency which is the same as that of the 
incident X ray. This oscillating charge radiates as an oscillating dipole, 
and its radiation is the scattered X radiation. 

From classical electromagnetic theory, Thomson showed that each 
electron should radiate, or "scatter, 11 a definite fraction of the energy 
flux which is incident on the electron. In Thomson's theory, the fraction 
of the incident radiation scattered by each electron is independent rf the 
wavelength of the X ray. It is now known that this is true only for 
electromagnetic radiation whose quantum energy hv is large compared 
with the binding energy of the atomic electrons, yet small compared 
with the rest energy, m c 2 = 0.51 Mev,t of an electron. 

b. Cross Section for Thomson Scattering. A derivation which fol- 
lows in principle that performed by J. J. Thomson is given in Appendix A. 
It is found that each electron scatters an energy t Q ergs when it is trav- 
ersed for a time t sec by a plane wave of X rays whose intensity is I 
ergs/ (cm 2 ) (sec). The scattered radiation has the same frequency as the 
incident radiation. The rate at which energy is scattered by each elec- 
tron, i.e., the scattered power ,Q/t ergs/sec, is found to be 



where e = electronic charge 
m = rest mass of electron 

c = velocity of light 

e z /m c 2 = "classical radius" of electron = 2.818 X 10~ 13 cm 
The proportionality constant between the incident intensity (or power 
per unit area) and the power scattered by each electron appears in the 
square bracket of Eq. (2.1) and is represented by the symbol c <r. It will 
be noted that & has the dimensions of an area, i.e., 

ergs/sec =cm 2 
ergs/ (cm 2 ) (sec) 

It is the area on which enough energy falls from the plane wave to equal 
the energy scattered by one electron. Each electron in the absorber 
scatters independently of the other electrons. Therefore e a is called the 
Thomson electronic cross section. When the most probable values of the 
fundamental physical constants (see Appendix E) e } ra , and c are sub- 

fFor definitions of abbreviations and mathematical symbols, see Glossary of 
Principal Symbols at the end of the book. 



2] Charge of Atomic Nuclei 9 

stituted, the Thomson cross section has the numerical value 

= 0.6652 X 10- 24 cmVelectron (2.2) 



The popular and now officially recognized international unit of cross sec- 
tion is the barn,} which is defined as 

1 barn = 10~ 24 cm 2 

Then the Thomson cross section is very close to 

f <r = barn /electron 

c. Linear and Mass Attenuation Coefficients. In a thin absorbing 
foil of thickness Ax, containing N atoms/cm 3 , there arc (NZ) elec- 
trons/cm 3 and (NZ AT) electrons/cm 2 of absorber area as seen by an 
incident beam of X rays. If each electron has an effective cross sectior 
of t <T cm a /clectron, then the total effective scattering area in 1 cm 2 of area 
of absorbing foil is (NZ AJT) rcr "cm 2 of electrons "/cm 2 of foil. Thus 
(NZ AT) ,<r is ike fraction of the superficial area of the foil which appears 
to be "opaque" to the incident X rays. 

Then if an X-ray intensity / is incident normally on the foil, Af is 
the fraction of this intensity which will not be present in the transmitted 
beam, the corresponding energy having been scattered more or less in all 
directions b}^ the electrons in the foil. This decrease in the intensity 
of the collimated beam is therefore 

A/ = - JNZ a Ax 

The quantity (NZ ,c) has dimensions of cm" 1 and is often called the 
linear attenuation coefficient a. Then we may write 

~=-adx (2.3) 

Integrating this equation, we find that, if an intensity /" is incident on a 
scattering foil of thickness jc cm, the transmitted unscattered intensity / 
is given by the usual exponential expression 

= er a * = e-( ff "M"> (2.4) 

/o 

In practice, the thickness of absorbing foils is often expressed in terms 
of mass per unit area. Then if p g/cm 3 is the density of the foil material, 

t The origin of the barn unit is said to lie in the American colloquialism "big as a 
barn," which WD.S first applied to the cross sections for the interaction of slow neutrons 
with certain atomic nuclei during the Manhattan District project of World War II 
The international Joint Commission on Standards, Units, and Constants of Radio- 
activity recommended in 1950 the international acceptance of the term "barn" for 
1C-' 1 ' cm' because of its common usage in the United States [F. A. Paneth, Nature, 
166: 931 (1950); Nucleonics, 8 (5): 38 (1951)]. 




10 The Atomic Nucleus [en. 1 

the "thickness" is (xp) g/cm 2 , and the mass attenuation coefficient is 
(cr/p) cm 2 /g. 

d. Number of Electrons per Atom. Barkla first carried out quanti- 
tative experiments on the attenuation suffered by a beam of X rays in 
passing through absorbing layers of various light materials, especially 
carbon, The number of atoms of carbon per gram N/p is simply Avo- 
gadro's number divided by the atomic weight of carbon. Hence the 
number of electrons Z per atom can be computed from the measured 
X-ray transmission ///u, assuming only the validity of Thomson's theory 
of X-rajr scattering. 

Actually, at least two other phenomena contribute significantly to 
The atom of oxygen the attenuation of low-energy X rays 

m 9 in carbon. These are the excitation 

of fluorescence radiation following 
photoelectric absorption of the X 

O rays by K and L electrons, and the 

* coherent and diffuse scattering from 

m m the crystal planes in graphite. The 

crystal effects were unknown at the 
Thomson Rutherford-Bohr t j me rf Barkla > g WQrkj but thcy ap _ 

Fig. 2.1 The atom model of J. J. *> to h fve been fortuitously aver- 
Thomson (T23) distributed the doc- a 6 ed out bv the combined effects of 
trons, shown as black dots, inside a wavelength inhomogeneity in the in- 
large sphere of uniform positive clectri- cident X rays and wide-angle geome- 
fication. The Rutherford-Bohr model try in the detection system. Barkla 
compressed all the positive charge, and recognized the influence of photoc- 
its associated large mass, into a small lectric absorption, which is strongly 
central nucleus, with thr electrons per- dependent on wavelength, and un- 
forming Copernicanlike orbits at dis- dertook to extrapolate this effect out 
tanres of the order of 10- to W time. fc compari ng <r/ p for carbon at 
the nurlcar radius. J i Vi* , i ,\ ^- 

several different wavelengths. Fi- 
nally, the theoretical value of the Thomson cross section depends on 
e 2 /m n c 2 and iuiice on measured values of both e and e/m$. The numer- 
ical values of e and e/m Q were known only approximately in Barkla's 
time. They were sufficiently accurate to show unambiguously that the 
X-ray scattering would be produced by the atomic electrons, because 
of their small mass, and not by the positively charged parts of the atom. 

In fact, the X-ray scattering does noi depend on the disposition of the 
positive charges in the atom, as long as these are associated with the mas- 
sive parts of the atom, as can be seen from the 1/mjj factor in Eq. (2.2). 
Barkla's experiments were done while Thomson's atom model, Fig. 2. 1 , 
was in vogue, but the results are equally valid on the Rutherford-Bohr 
nuclear model. In the nuclear model, it is obvious that an electrically 
neutral atom must contain the same number of electrons as there are 
elementary charges Z in the nucleus. 

It is interesting to note that Barkla's first values, obtained in 1904, 
ran to 100 to 200 electrons per molecule of air; by 1907 (T23) his results 
were down to 16 electrons per molecule of air. Improvements in tech- 



3] Charge of Atomic Nuclei 11 

niquc, and better values of e and r/m , led Barkla in 1911 (B12) to con- 
clude that, the mass attenuation coefficient cr/p for Thomson scattering 
by carbon is about 0.2 cm 2 /g, which corresponds to the currently accept- 
able value of six electrons per atom of carbon. For other light elements, 
Barkla concluded correctly that "the number of scattering electrons per 
atom is about half the atomic weight of the element." 

It should be remarked that Barkla's results would have been incorrect 
if he had applied the Thomson theory to atoms of such large, Z that the 
(then unknown) electron binding energies were comparable with the 
relatively s'nall quantum energy [~40 kev (kiloelectron volts)] of his 
X rays. Secondly, if Barkla't* X-ray quantum energy had been suffi- 
ciently large HO that it was comparable with w n r 2 = 0.51 Mev, the 
Thomson formula would also have been invalid because it omits consider- 
ation of the recoil of the electron, which was discovered much later by 
Oompton. At the time of Barkla's work, many phenomena now regarded 
as fundamental in atomic physics were unknown. The atomic nucleus 
had not yet been discovered, and Thomson's model of the atom was still 
fashionable. Bohr's explanation of atomic spectra and of the binding 
energy of atomic electrons and Compton's explanation of the interaction 
of higher-energy photons with electrons were yet unborn. The prin- 
ciples of Bragg diffraction of X rays by crystal planes were unknown. 

With all these factors in view, Hewlett, in 1922 (H49), found a/p = 0.2 
cm 2 /g for 17.5-kev X rays in carbon, and consequently six electrons per 
atom of carbon, and this result is acceptable from all viewpoints. 

In 1928 Klein and NLshina applied the Dirac rclativiwtic electron 
theory to the problem of the scattering of high-energy photons by atomic 
electrons. The details of this work arc discussed later, in (/hap. 23. 
Here we may note that the excellent agreement, between this theory 
and the experimental observations on the scattering of photons up to as 
much as 100 Mev constitutes a fairly direct modern measurement of the 
number of electrons per atom for a wide variety of elements. In all 
cases, of course, Z is found to agree with the atomic numbers which have 
been assigned in the meantime cm a basis of other types of evidence. 

Problems 

1. Assuming only Thomson scattering, calculate the fractional transmission 
of low-energy X rays through 5 mm of graphite, assuming that carbon has six 
electrons per atom. 

2. What transmission would be expected if carbon had 12 electrons per atom? 

3. About what photon energy should be use?! in this measurement of Z, if 
competition with Thomson scattering, due to both photoelectric absorption and 
Oompton losses, are to be minimized ? 

4. What is the fractional transmission if the graphite slab in Prob. 1 is tilted 
so that the X-ray beam strikes the slab at 30 with the normal? 

3. Charge on the Atomic Nucleus. a-Ray Scattering 

a. Qualitative Character of the Rutherford -Bohr Atom Model. 
Rutherford, in 1906, first noticed that the deflections experienced by a 



12 The Atomic Nucleus [en. 1 

rays while passing through air, mica, arid gold were occasionally much 
greater than could be accounted for by the Thomson model of the atom. 
Rutherford's first mathematical paper on the a-roy scattering appeared 
in 1911. This is a classic (1143, Bf>3) which should be read in its original 
form by every serious student. 

He assumed that each atom contains a small central nucleus, whose 
radius is less than 10~ 12 cm, whereas the radius of the entire atom was 
known to be of the order of 10~ 8 cm. Although it is now evident that the 
nucleus is positively charged, Rutherford left the sign of the charge on 
the nucleus as an open question in 1911 and pointed out that the angular 
distribution of scattered a rays is independent of the sign of the nuclear 
charge. If the nucleus be regarded as having a positive charge Ze and 
if an equal amount of negative charge be distributed throughout the 
volume of the entire a, torn, all a-ray deflections greater than about 1 
were shown to be attributable to nuclear scattering and to have an 
intensity proportional to Z z . 

The mass of the atom is now known to be found primarily in the 
nucleus, but this fact was not needed in order to explain the early a-ray- 
scattering results and was not used in Rutherford's original theory. It 
was only necessary to make the tacit assumption that tho atom was not 
disrupted by the collision; thus the nucleus was simply the center of mass 
of the atom. The essentially new feature in Rutherford's model of the 
atom was the concentration of all the positive charge Ze into a nucleus, 
or central region, smaller than 10~ 12 cm in radius, with an equal amount 
of charge of the opposite sign distributed throughout the entire atom in a 
sphere whose radius is much greater than that of the nucleus. He simply 
deprived Thomson's atom model of its uniform sphere of positive elec- 
tricity and concentrated all this charge at the center of his new atom 
model. 

Two years later Bohr (B92) put the atom's mass into the nucleus, 
gave quantized energy states to the atomic electrons, produced his suc- 
cessful theory of the origin of spectra, and essentially completed the 
Rutherford-Bohr nuclear model for the atom. Darwin (D5) later gen- 
eralized Rutherford's theory of scattering by giving tho solutions on 
classical theory for collisions in which the mass of the struck atom is 
comparable with that of the incident ray, and for collisions in which the 
force varies as the inverse nth power of the separation. In 1920 Chad- 
wick (Cll) showed experimentally that n = 2.00 0.03 for the scatter- 
ing of a. rays by heavy nuclei and therefore that Coulomb's law is valid 
for these collisions. 

b. Scattering in Center -of-mass Coordinates and in Laboratory 
Coordinates. All collision problems between free particles are greatly 
simplified through the use of a coordinate system whose origin is at the 
center of mass of the colliding particles. This coordinate system is known 
as the " particle coordinates," the "zero-momentum coordinates," the 
" ecu ter-of -mass coordinates," or most simply the "C coordinates." 
Physically, it is usually more realistic to visualize the collision in the C 
coordinates than in the laboratory, or "L coordinates." The words 



3] 



Charge of Atomic Nuclei 



13 



"projectile'' and "target" particle have a meaning only in the L coordi- 
nates. In the C coordinates neither particle is the aggressor; both 
particles approach their common center of mass with equal and opposite 
momenta, interact with each other, and depart from the scene of the 
"collision." The total linear momentum of the colliding particles is always 
zero in the C coordinates. We shall discuss here only nonrelativistic 
collisions. The corresponding transformation equations for the rela- 
tivistic case have been given by Bergmann (B35), Blaton (B65), Morrison 
(M57), and others. 

The use of C coordinates has a 
profound mathematical advantage. 
In both classical mechanics and wave 
mechanics, the use of C coordinates 
reduces any two-body collision prob- 
lem to a one-body problem, namely, 
the interaction of one particle having 
the reduced mass M and velocity V 
with a potential field which can 
always be considered as centered at 
the origin of the C coordinates. The 
reduced mass M of a system of two 
particles having masses MI and M 2 is 




J_=_L + J_ 

M ~ Mi M t 



or 



+ M, 



(3.1) 



Fig. 3.1 Coulomb elastic scattering of 
an a ray (Mi) by an oxygen nucleus 
(Af 2 ), seen in the laboratory coordi- 
nates. The center of mass, marked C, 
moves through the laboratory with a 
constant velocity V c which is one-fifth 
the initial velocity of the a ray. The 
impact parameter x is the separation 
at which the particles would pass if 
Lhore were no interaction between them. 
Note that the initial direction of Af 2 
is away from MI. The trajectories 
are no I simple hyperbolas in the L 
coordinates. 



Therefore the reduced mass always 
lies between 0.5 and 1.0 times the 
mass of the lighter particle. 

The analytical relationships be- 
tween various collision parameters in 
the C and L coordinates are derived 
in Appendix B. Here we quote only 
some of the principal results. In the L coordinates, a typical colli- 
sion is the encounter of a projectile particle having mass MI and 
initial velocity V with a target particle having mass M z and being 
initially at rest. This pair of particles must always share the initial 
momentum MiV; therefore their center of mass moves through the 
laboratory at a constant velocity V c = M\V /(M\ + M 2 ) which is always 
parallel to the initial direction of V. This state of affairs is illustrated 
in Fig. 3.1, where for dcfiniteness we have shown an elastic collision 
between an a ray (Mi = 4) and an oxygen nucleus (Mz = 16). As a 
result of the collision, the a ray is deflected through an angle d, while the 
oxygen nucleus is projected at an angle y> with the original direction of 
the incident a ray. In the L coordinates, thn analytical relationships 
which connect the scattering angles t9 and p with Mio impact parameter x 



14 



The Atomic Nucleus 



[CH. 1 



and with the charges, masses, and velocities of MI and M 2 are unduly 
complicated and are too cumbersome for use in the general case. Indeed, 
the relationships are derived by solving the problem first in the C coordi- 
nates and then transforming the motion to the L coordinates. 

In the C coordinates, all parameters are measured with respect to an 
origin at the center of mass. The motion of the particles in the C 
coordinates can always be transformed to motion in the L coordinates 
by noting that the (/-coordinate system moves through the laboratory 
with the same uniform velocity T c which the center of mass possesses in 

the L coordinates. 

In the C coordinates, both par- 
ticles initially approach each other, 
as shown in Fig. 3.2. They move 
in such a way that their total linear 
momentum is alwaj^s zero. Their 
total angular momentum about the 
origin at their center of mass is 
always-- J/oV-r, where x is the impact 
parameter. The initial velocit}' of 
Mi in the C coordinates is 




r 



Fig. 3.2 The same collision as Fip. 3.1 
but now seen as the particles actually 
experience it, in the center-of-mass coor- 
dinates. The center of mas.s, marked C, 
is now at rest. The total linear momen- 
tum is zero. Each particle traverses a 
true hj r perbolic orbit about (' as its exter- 
nal focus. The deflection angle () is the 
same for both particles. Note that I lie 
initial direction of Mz is toward M\, or 
opposite to the motion of M-> in the L 
coordinates of Fig. 3.1. 



v _ 
1 



(3.2) 



to the right, while the initial ve- 
locity of M z is 



V. = V 



(3.3) 



to the loft in Fig. 3.2. The mutual 
velocity with which M } and A/ 2 ap- 
proach each other initially is there- 
fore V, which is the same as in the 
L coordinates. 

In the C coordinates, both particles are scattered through the same 
angle 0, and their final velocities are equal to their initial velocities. 
Xeither of these simple relationships holds in the L coordinates. The 
angular deflection tf of Mi in the L coordinates turns out to be given by 



cot',T= ^ 



cot0 



(3.4) 



Then in general & < 0, The relationship between tf and is simple 
only in two special c&& y which are 

forM, Afi, tf~ 



for M, = 



3] Charge of Atomic Nuclei 15 

The laboratory angle <? through which M 2 is projected in theL coordinates 
is given, for elastic collisions only, by 

Tf \J ff\ r\ 



Finally, the angle between the final directions of Mi and M t is always 
180 in the C coordinates but in the L coordinates has the values 

for .17, = A/,, <P + a = * 

for .l/i < Jl/2, v + >'> < " + | 

for J/ t Jl/j, *> + * ^ ][ + ? 

^ ^ 

All these angular relationships are consequences of the conservation of 
momentum arid energy and art* independent of the force laws which may 
govern the scattering for the particular type of collision involved. The 
nature of the interaction between the particles determines only the cross 
section for the collision. 

c. Elastic Scattering by Coulomb Forces. Tt can be shown quite 
generally (see Appendix B) that, when any incident nonrclativistic par- 
ticle interacts with a target particle according, to an inverse-square law 
of force (cither attractive or repulsive), both particles must, in order to 
conserve angular momentum, traverse hyperbolic orbits in a coordinate 
system whose origin is at the center of muss of the colliding particles. 
(Note that the incident particle's path in the laboratory-coordinate sys- 
tem is not necessarily hyperbolic.) AA'hon the restriction is added that 
the sum of the kinetic energy and potential energy of the two particles i.^ 
constant, it is found that the angle of deflection (?) in the center-of-mass 
coordinates is given by 

b . B /0 px 

-r = - cot -- (3.0) 

where the "impact parameter" 1 a- is the distance at Avhich the two particles 
would pass each other if there were no interaction between them, and 
where b is the collision diameter denned by 



where ze = charge on incident particle 

Zc = charge on target particle 
V mutual velocity of approach 

Mo = reduced mass of colliding particles, i^jq. (3.T) 

The absolute value of Zz is to be taken, without regard to sign. Tin- 
collision radius b/2 is the value of the impact parameter for which the 
scattering angle is just 90 in the center-of-mass coordinates, both for 



16 The Atomic Nucleus [CH. 1 

attractive and for repulsive forces. For the special case of repulsive 
forces, as in the nuclear scattering of a rays, the collision diameter b is 
also equal to the closest possible distance of approach, i.e., to the minimum 
separation between the particles during a head-on collision. At this 
minimum separation the particles are stationary with respect to one 
another, and therefore their initial kinetic energy ^M Q V 2 is just equal 
to their mutual electrostatic potential energy Zze z /b. 

d. Cross Section for Rutherford Scattering. In all collisions for 
which the minimum distance of approach b is significantly greater than 
the radius of the nucleus, the only force acting will be the inverse-square 
coulomb force, and Eq. (3.6) will be valid. All collisions for which the 
impact parameter lies between and x will result in scattering of the 
incident particle through an angle between 180 and 0. Then the cross 
section, r(> 0), for scattering through an angle equal to or greater than 
is the area of a disk of radius x, or 

w 

*(> 0) =7rz 2 = ~6 2 cot 2 | (3.8) 

Thus the cross section for backscattering (0 > 90) is simply ir6 2 /4, 
which is the area of a disk whose radius equals the collision radius 6/2. 

For = 0, <r and x are both infinite; thus every a ray appears to 
suffer some slight deflection. Physically, this situation does not occur, 
because for very large impact parameters the nuclear coulomb field is 
neutralized, or "screened," by the field of the atomic electrons. 

The differential cross section da for nuclear scattering between angles 
and + d0 is the area of a ring of radius x and width dx, or 

da = \2 v x dx\ = *- 6 2 cot f esc 2 f d (3.9) 

4 22 

The solid angle d!2, into which particles scattered between and + d0 
are deflected, is 

dfi = 2ir sin d 

A ' *M 

= \if sin cos d 
J 4 

Therefore the differential cross section for scattering into the solid angle 
dft at mean angle is 

, __&_'[ !_' 



Equations (3.8) to (3.10) are various equivalent forms which all represent 
Rutherford (i.e., classical) scattering. Each is best suited to particular 
types of experiments. Each exhibits the marked predominance of for- 
ward scattering which is generally characteristic of long-range forces, 
such as the inverse-square interaction. 

e. Single Scattering by a Foil. A scattering foil of thickness As cm, 
containing N atoms/cm*, will present N A* scattering centers per square 



3] Charge of Atomic Nuclei 17 

centimeter to normally incident a rays. If the cross section of each 
scattering center is a cm 2 /atom, then the scattering centers comprise the 
fraction crN As of the total area of the foil. Then if no a rays are incident 
normally on the foil, n a. rays will be scattered in the directions repre- 
sented by the particular value of the cross section a being used. The 
fraction so scattered is simply 

- = <r(N As) or = AT (N Aa) (3.11) 

7l(j Tlo 

It is understood that the foil is sufficiently thin so that (<rN Aa) 1. 
Therefore the number of a rays which are scattered twice is negligible in 
comparison with the number scattered only once. More briefly, only 
single scattering is considered here, not plural scattering (a few collisions 
per particle) nor multiple scattering (many collisions per particle). 

When the mass of the incident particle can be neglected in comparison 
with the mass of the target particle, then the reduced mass MQ becomes 
substantially equal to the mass of the (lighter) incident particle. Also, 
the deflection angle (r) in the center-of-mass coordinates becomes sub- 
stantially equal to the deflection angle tf in the laboratory coordinates. 
These simplified conditions do apply to the scattering of a particles by 
heavy nuclei such as gold. In these collisions the heavy target nucleus 
remains essentially stationary, or "clamped," during the collision. 

f. Experimental Verification of Rutherford's Nuclear Atom Model. 
In proposing that all the (positive) charge in the atom should be regarded 
as concentrated in a small central nucleus, Rutherford made use of 
experimental results which had been obtained by Geiger on the'angular 
distribution of the a rays scattered by a thin gold foil. These results 
were in sharp contrast with the predictions of the Thomson model of the 
atom, but they were in substantial agreement with the I/sin 4 (0/2) 
distribution of Eq. (.3.10) predicted by the nuclear model in which the 
central positive charge has such small dimensions that it is not reached 
by swift a rays even in head-on collisions (R43). 

Geiger and Marsden subsequently completed a beautiful series of 
experiments which completely verified Eqs. (3.10) and (3.11) point by 
point. Their original paper (G14) warrants reading by every serious 
student. The angle of deflection # in the laboratory coordinates was 
varied in small steps from 5 to 150; this brings about a variation in 
sin 4 (0/2) of more than 2r>0,000 to 1. Figure 3.3 shows the results for a 
particular gold foil. The collision diameter 6 was varied in two inde- 
pendent ways. First, the velocity V of the incident a rays was varied 
by interposing absorbers between the RaB + RaC source and the 
scattering foils; in this way the 1/7 4 term which enters all the cross sec- 
tions through 6 2 was varied in seven steps over a factor of about 9 to 1. 
Secondly, the nuclear charge Ze was varied by studying the scattering 
from gold, silver, copper, and aluminum foils. It was found that the 
intensity of the scattering per atom was approximately proportional to 
the square of the atomic weight. 



18 



The Atomic Nucleus 



[CH. 1 




This showed experimentally for the first time that the nuclear charge 
is approximately proportional to the atomic weight. The actual value 
of the nuclear charge was found to be about one-half the atomic weight, 
with an experimental uncertainty of about 20 per cent. These ex- 
periments by Geiger and Marsden completely verified Rutherford's 

concept of the atom as containing 
a small central nucleus in which all 
the charge of one sign is located. 

g. The Equivalence of Nuclear 
Charge and Atomic Number. It 
fell to van den Broek (B125) in 
1913 to collect the various types of 
evidence then available and to 
make the fertile suggestion that 
the charge on the atomic nucleus ?s 
actually equal to the atomic number. 
Bohr adopted this suggestion and 
developed his quantum theory of 
the structure of atoms and the ori- 
gin of spectra. This theory pre- 
dicted that the frequency of the 
X-ray lines in the K scries should 
increase with the square of the 
charge on the atomic nucleus, i.e., 
with Z 2 . Moscley's observations 
of these X-ray lines showed instead 
that the frequency v is substantially 
proportional to (Z I) 2 , if it be 
assumed that the charge Z on the 
nucleus equals the atomic number 
and that the atomic number of 
aluminum is 13. Mosclcy sug- 
gested correctly that the effective 
charge on the atomic nucleus, for 
K -series X rays, is about one unit 
less than the actual charge Z on the 
nucleus because of screening of the 
nuclear charge, especially by the 
one /v-shell electron which is pres- 
ent in the initial atomic state of 
any A"-series transition. Any doubt 
which may have persisted about 
this interpretation was later re- 
moved by Chadwick's direct 
measurement of the nuclear charge 
of Cu, Ag, and Ft by the a-ray- 
scattering method. 



30 60 90 120 150 180 
Mean angle of scattering tf 

Fig. 3.3 Differential cross section for the 
single scattering of a rays by a, thm foil of 
gold. The vertical scale represents the 
relative number of n rays scattered into n 
constant element of solid angle at the 
mean scattering angles tf which arc shown 
on the horizontal scale. The curve is 
proportional to I/sin 4 (d/2), as predicted 
hy the classical theory, and is fitted to 
the arbitrary vertical scale at tf = 135. 
The closed and open circles are the experi- 
mental data of Goiger and Marsden (G14) 
in two overlapping series of observations, 
one at small and one at large scattering 
angles. The agreement at all angles 
shows that, under the conditions of these 
experiments, the only force acting be- 
tween the incident a ra-ys and the gold 
nuclei is the inverse-square coulomb re- 
pulsion. The closest distance of ap- 
proach in these experiments was 30 X 
1(T 13 cm (for 150 scattering of the 7.68- 
Mev a rays from RaC'), and so the posi- 
tive charge in the gold atom is confined 
to a small central region which is defi- 
nitely smaller than this, or about JO" 4 of 
the atomic radius. 



3] Charge of Atomic Nuclei 19 

h. Absolute Determination of Nuclear Charge. Chadwick intro- 
duced an ingenious experimental arrangement which greatly increases 
the observable scattered intensity for any given angle, source, and thick- 
ness of scattering foil. The foil is arranged, as shown in Fig. 3.4, as an 
annular ring around an axis between the source of a rays and the scintilla- 
tion-screen detector. Precision a-ray-scattering experiments with this 
arrangement gave the absolute value of the nuclear charge of Cu, Ag, and 
Pi as 2J).3f, 4G.3<?, and 77 Ac, with an estimated uncertainty of 1 to 2 per 
cent (Oil). This is final confirmation of the atomic numbers 29, 47, and 
78 which had been assigned to these elements by Moseley. 




Scintillation detector 

Baffle to stop \Or/ X ^ for a rays sca ttered 

direct beam of ^^ between angles i\ and i> 2 

a rays ^Annular ring of 

scattering foil 

Fig. 3.4 Chud wirk's arrangement of sourer, scattoror, and detector for increasing 
the intensify of a rnys Hc.ultored between angles #1 and iJ 2 , as used for his direct 
measurement of the nuclear charge on Cu, Ag, and Pt. This annular geometry for 
the scattering body has subsequently been widely adapted to a variety of other scat- 
tering problems, e.g., the shadow scattering of fast neutrons by lead (Chap. 14). 

i. Limitations of the Classical Theory. It should be noted that the 
general wave-mechanical theory of the elastic scattering of charged par- 
ticles adds a number of terms to the simple cross sections -given in Eqs. 
(3.8) to (3.10), which are based only on classical mechanics. However, 
the wave mechanics (M(i3) and the classical mechanics give identical 
solutions for the limiting cases in which a heavy nucleus scatters an a ray 
of moderate energy. In general, the classical theory is valid when the 
rationalized de Broglie wavelength, \/2ir = X = h/M Q V, for the col- 
lision in the (/ coordinates is small compared with the collision diameter b. 
These conditions arc equivalent to b/\ - 2Zz/137ft 1, where ft = V/c, 
and are derived in Eqs. (83) and (100) of Appendix C. For the special 
case of the .scattering of identical particles (such as a rays by He nuclei, 
protons by H nuclei, and electrons by electrons), the wave-mechanical 
results [Chap. 10, Eq. (5.1); Chap. 19, Eq. (2.4)] are markedly different 
from those of the classical mechanics. The wave-mechanical theory is 
well supported by experiments. 

Problems 

1. A thin gold foil of thickness As cm has N atoms of gold per cubic centi- 
meter. Each atom has a nuclear cross section a cm 2 for scattering of incident a 
rays through more than some arbitrary angle @. The fraction of normally inci- 
dent a rays scattered through more than is n/n = <rN As. Show clearly what 



20 The Atomic Nucleus [CH. 1 

fraction of the incident a rays is scattered through more than if the a rays are 
incident at an angle \l/ with the normal to the foil. 

2. Starting with any of the general equations for Rutherford scattering, 
derive an expression for the cross section for backscattering in the laboratory 
coordinates (that is, # > 90) , and show that your equation will reduce to 

(r(backscatter) = TT 



where, as usual, the incident particle has charge ze, mass MI, .and velocity V, and 
the target particle has charge Ze and mass M 2 and is initially stationary in the 
laboratory coordinates. 

3. In an a-ray-scattering experiment, a collirnated beam of polonium a rays 
(5.30 Mev) strikes a thin foil of nickel, at normal incidence. The number of 
a rays scattered through a laboratory angle greater than 90 (i.e., reflected by the 
foil) is measured. Then the nickel foil is replaced by a chromium foil, and the 
measurements are repeated. It is found that the chromium foil reflects 0.83 
times as many a rays as the nickel foil. The foils are of such thickness that each 
weighs 0.4 mg/cm 2 . 

(a) Use the results of this reflection experiment to determine the nuclear 
charge for chromium, if the atomic weight of chromium is W = 52.0, while for 
nickel Z = 28 and W = 58.7. 

(6) Show whether classical theory should be valid for these collisions between 
5. 30- Mev at rays and chromium nuclei. 

4, Consider the classical (Rutherford) scattering of l.02-Mev a rays by 
aluminum nuclei. For the particular collisions in which the impact parameter is 
just equal to the collision diameter, determine the following details: 

(a) Velocity of the center of mass in the L coordinates. 

(6) Reduced mass of the system, in amu (atomic mass unity). 

(c) Kinetic energy in the C coordinates, in Mev. 

(d) Collision diameter, in 10~ 13 cm. 

(e) Scattering angle in C coordinates. 

(/) Deflection angle of the a ray in L coordinates. 

(g) Deflection angle of the Al nucleus in L coordinates. 

(h) Minimum distance of approach between the a ray and the Al nucleus dur- 
ing the collision, in 10~ 13 cm. 

(i) Minimum distance of approach of the a ray to the center of mass during 
the collision. 

0) Approximate nuclear radius of Al, if R = 1.5 X 10" 13 A* cm. 

(k) The angular momentum of the colliding system, about the center of mass, 
in units of h/2w. 

(I) The nuclear cross section for deflections larger than those found in (c) or 
(/) above, in barns per nucleus. 

(m) The fraction of 1.02-Mev a rays, incident normally on an Al foil 0.01 mg/ 
cm 2 thick, which are deflected through more than the angles found in (e) or (/). 

(n) From the same foil, the fraction of the normally incident a rays which 
would strike a 1-mm-square screen placed 3 cm away from the scattering foil and 
normal to the mean scattering angle found in (/). 

(o) Sketch the trajectories of both particles during the collision, in C coordi- 
nates and also in L coordinates. Are the paths hyperbolas in L coordinates? 

(p) De Broglie wavelength for the collision in the C coordinates, in 10~ 13 cm. 

(?) Same as (p) but for an incident a-ray energy of 10.2 Mev. Would classi- 
cal theory be valid for such a collision? Why? 



4] Charge of Alomic Nuclei 21 

4. Frequency of K- and L-series X Hays 

a. Bohr Theory. Following the proof of the existence of atomic 
nuclei hy the a-ray-scattering experiments, Bohr (B92) assigned the 
principal part of the atomic mass to nuclei and introduced his quantum 
theory of the origin of atomic spectra. To the extent that the simple 
theory is valid, the energy hv of characteristic X-ray quanta would be 
expected to be given by 



(4 - 2) 

where ?M and n 2 are the principal quantum numbers for the initial and 
final electron vacancies (HI = 1, n z = 2, for the K a series; n\ = 2, n z = 3, 
for the L a series), a is the fine-structure constant (a ^TTT; 2 wi c 2 /2 = 13.6 
ev), and all other symbols have their customary meaning and the numer- 
ical values given in Appendix E. 

b. Screening of Nuclear Charge by Atomic Electrons. However, the 
effective nuclear charge is actually somewhat less than Ze because of 
screening of the nuclear field by the potential due to the other K , L, . . . 
electrons present in the ionized atom. The screening in the initial state 
will be less than the screening in the final state of an X-ray transition, and 
separate screening corrections can be introduced for each electron level 
in the atom if desired (C37). 

Moseley applied the then new principles of Bragg reflection to the 
study of X-ray lines and thereby introduced a new era of X-ray spectros- 
copy. In two monumental papers (M60) he showed the existence of a 
linear relationship between the atomic numbers of the light elements, as 
previously assigned from chemical data, and v* for the characteristic K a 
and L a X-ray lines. 

Motley's data are shown in Fig. 4.1. The plot of atomic number 
against v* for the K a series does not pass through the origin but has an 
intercept of about unity on the atomic number axis. If the nuclear 
charge Z is assumed to be the same as the atomic number, then Moseley's 
data on the K a series have the form 

y* = const X (Z - 1) (4.3) 

and an effective value; of the screening constant for the over-all transition 
can be taken as about unity. Similarly, Moseley 's data on the L a series 
exhibit a substantially linear relationship given by 

F* = const X (Z - 7.4) (4.4) 

Under the same interpretation, this would suggest an over-all or effec- 
tive screening constant of about 7.4, as seen from the L shell. Both 
these effective screening constants are physically reasonable. 

It is concluded that the atomic number is equal to the charge on the 



22 



The Atomic Nucleus 



[CH. 1 



atomic nucleus and hence also to the number of atomic electrons in the 
neutral atom. 

c. Atomic Numbers for Heavy Elements. The original method of 
assigning atomic numbers on a basis of increasing atomic weight and the 
periodicity of chemical properties was applicable only up to Z = 57. 
Beginning at Z = 57, the group of 15 rare-earth elements all exhibit 
similar chemical properties and stand in the same column of a Mendeleev 
periodic, table. The total number of rare-earth elements was unknown 

in 1912, Therefore it was impossible 
to assign correct atomic numbers to 
the elements which are heavier than 
the rare earths. For example, it was 
convention al to assume the value 
Z = 100 for uranium, which is now 
known to be Z = 92. Moseley's 
work was the first to show that a 
total of 15 places (Z = 57 to 71) had 
to be reserved for the rare earths. 

Moseley examined the K a X rays 
of 21 elements from i 3 Al to 4?Ag, and 
also the L a X rays of 24 elements 
from 4iZr to ygAu. The overlap, be- 
tween 40 Zr and 4?Ag, oriented the L 
series and permitted its use for 
bridging over the rare-earth group of 
elements in order to establish for the 
first time the atomic- numbers in the 
upper part of the periodic table. 

The fundamental significance of 
atomic number was firmly estab- 
lished by Moseley's data. Cobalt 
was shown to be atomic number 27 and Ni to be 28, as had been suspected 
from their chemical properties. It may be noted that the ratio of atomic 
weight, or more accurately the mass number A, to the atomic number Z 
is nearly constant and has the value 



J 
20 

2 10 







/ 








X 




j/f 








y 


f 




/ 




/ 
















) 20 40 60 80 101 
Atomic number 



Fig. 4.1 Mosclcy's original data (1 91 4) 
.shu\\'mp tho frequency v of the K a and 
L a X-ray lines of all available elements 
and (fie uniform variation of v^ with 
integers % assignable as atomic num- 
bers to the 38 elements tested. Each 
A" and L a line is actually a close dou- 
hlft; none of these had been resolved 
:tt Mosul ey's time. 



2.0 <~ < 2.6 

& 

for all stable* nuclei, except H 1 and He 3 . 

d. The Identification of New Elements. There have been a number 
of new elements produced by transmutation processes in recent years. 
These elements (Z = 43, 61, 85, 87, 93, 94, 95, . . .) have no stable 
isotopes, but each does have at least one isotope whose radioactive half- 
period is sufficiently long to permit the accumulation of milligram quanti- 
ties of the isotope. In every case, the atomic number has been assigned 
first \yy combining chemical evidence and transmutation data, at a time 
when the total available amount of the isotope was perhaps of the order 
of JO" 11 ' g. Confirmation of most of these assignments of atomic number 
liiis been made by measurement of the K- and L-series X rays, excited in 



4] Charge of Atomic Nuclei 23 

the conventional way by electron bombardment of milligram amounts 
of the isotope. [See (B143) for Z = 43, (B144) and (P13) for Z = 61.] 
Such measurements are regarded as conclusive in the identification of 
any new element. 

e. Characteristic X Rays from Radioactive Substances. Whenever 
any process results in the production of a vacancy in the K or L shell 
of atomic electrons, the ensuing rearrangement of the remaining electrons 
is accompanied by the emission of one or more X-ray quanta of the K or 
L series, or by Auger electrons, or both. There are two general types of 
radioactive transformation in which vacancies are produced in the inner 
electron shells of atomic electrons. Any radioactive substance whose 
decay involves either electron capture or internal conversion is found to 
be a source of an entire line spectrum of X rays. Full discussions of 
internal conversion will be found in Chap. 6i, Sec. 5, and of electron 
capture in Chap. 17, Sec. 3. Here we focus our attention only on the 
determination of atomic number by means of the X rays which are 
invariably associated with these transitions. 

Electron Capture. The capture of an atomic, electron by a nucleus is 
an important mode of radioactive decay, which generally competes with 
all cases of positron -ray decay. Several radioactive substances are 
known in which the transition energy is insufficient to allow positron 
jS-ray emission, and in which all radioactive transitions proceed by elec- 
tron capture (for example, 4 Be 7 , 2 4Cr 51 , a,Ga 67 , 4 9 In ul ). It is generally 
more probable that a K electron will be in the vicinity of the nucleus 
and will be captured than that an L, M, . . . electron will be captured. 
The majority of the vacancies are therefore produced in the K shell. If 
Z is the atomic number of the parent radioactive substance, then (Z 1) 
is the atomic number of the daughter substance in which the electron 
vacancy exists and from which the X rays are emitted. The existence 
of the electron-capture mode of radioactive decay was first established 
by Alvarez's observation (A22) of relatively intense K a X rays of titanium 
(Z = 22) among the radiations emitted in the radioactive decay of the 
16-day isotope of vanadium, 2 .iV 48 . More rigorous experimental proof 
was subsequently obtained from absorption curves (A23) and from Abel- 
son's bent-crystal spectrometer studies (Al) of the X rays of zinc (Z = 30) 
which are emitted in the pure electron-capture decay of 3 iGa 67 . Several 
isotopes of technetium (Z = 43) decay predominantly by electron cap- 
ture, and the early identification of element 43 was aided by the obser- 
vation of the molybdenum (Z = 42) X rays which are emitted in the 
decay of these technetium isotopes. 

Internal Conversion. The second general class of nuclear transitions 
which invariably result in X-ray-emission spectra is the internal-con- 
version transitions. There are numerous methods for producing nuclei 
in excited energy levels. Perhaps half the daughter nuclei which are 
produced by a decay or ft decay are formed in excited levels rather than 
in their ground levels. Generally the deexcitation of these nuclei pro- 
ceeds by the emission of 7 rays. Internal conversion is an alternative 
mode of deexcitation which always competes with 7-ray emission and 
which often predominates over 7-ray emission if the nuclear excita- 



24 



The Atomic Nucleus 



[CH. 1 



tion energy is small and the angular-momentum change is large (Chap. 
6, Sec. 5). The nuclear excitation energy is transferred directly to a 
penetrating atomic electron, and this additional energy allows the elec- 
tron to overcome its atomic binding energy and to escape, or indeed to 
be expelled, from the atom. In the most common cases, internal con- 
version is more likely to expel a K electron than an L, M , . . . electron 
from the atom. Thus the majority of the vacancies are produced in the 
K shell of atomic electrons. 

Internal-conversion transitions are therefore accompanied by X-ray- 
emission spectra. Neither internal conversion nor 7-ray emission 
involves any change in the nuclear charge, so that the X-ray spectra are 
characteristic of the element in which the actual nuclear transition took 
place. For example, the 0-ray decay of 7 9Au 198 results in the production 
of the daughter nucleus 8 oHg 198 in an excited level which is 0.41 Mev above 
the ground level of Hg 198 . About 95 per cent of these excited nuclei 
go to ground level by emitting a 0.41-Mev 7 ray. The others go to 
ground level by internal conversion, 3 per cent in the K shell, 1 per cent 
in the L shell, and 0.3 per cent in the M shell. The X-ray-emission 
spectra are characteristic of mercury (Z = 80), not gold. 

The chemical identification of a number of radioactive nuclides 
among the transuranium elements has been made or confirmed by obser- 
vations of the L-series X rays of Th, 9 iPa, 92!!, 9 aNp, 9 4Pu, g B Am, and 
96 Cm (B18). 

Nuclear Isomers. Nuclear isomers are long-lived excited levels of 
nuclei, in which the decay by internal conversion and 7-ray emission to 
the ground level is measurably delayed (Chap. 6, Sec. 6). Many nuclear 
isomers are sufficiently long-lived to permit them to be isolated chem- 
ically and to be dealt with as a parent radioactive substance. The iso- 
meric transition to the ground level involves, no change in Z. Conse- 
quently, the X rays which are associated with the isomeric transition by 
internal conversion will be characteristic of the Z of the parent radio- 
active element, even if its ground level is a /3-ray emitter (for example, 
siSb 122 ). This X-ray-emission property is useful in identifying nuclear 
isomers, especially in those cases in which isomeric transitions are in 
competition with 0-ray emission from the excited level (for example, 



Problems 

1. The wavelengths of the K a \ line and of the K edge (for ionization of the 
K shell) are given below, in angstrom units (A), for a number of elements. 



Element 


cC 


iAl 


2 ,Cu 


4 2 Mo 


7 3 Ta 


u 


K ah A .... 


44 54 


8.3205 


1 5Ii74 


0.7078 


2149 


0.12640 


A. n d|e, A . . . . 


43.5 


7.9356 


1 . 3774 


0.6197 


1836 


0.10658 



(a) Make a new table, expressing K ai and K 9d9 energy in kev. 

(6) Test the simple Bohr theory: (hv) K .^ = m&*(a*/2)Z* = 0.0136Z 2 kev, 



5] Charge of Atomic Nuclei 25 

and (M*.! - 0.0136Z* 1 key, for these elements. Do the experimental values 
approach the theoretical values for large Z or for small Z? What is the physical 
reason for this? 

(c) Does the ratio of K 9df9 to K a i energy approach the theoretical value of 
for small Z or for large Z? What physical reason is there for this behavior? 

2. The wavelength of the L a X rays of Ag, I, and Pt are 4.1456, 3.1417, and 
1.3103 A. Taking the atomic numbers of Ag and I as known (47 and 53), deter- 
mine the atomic number of Pt. 

3. A source of aoZn" emits a continuous negatron 0-ray spectrum, a single 
7 ray of about 0.44 Mev, and a line spectrum of conversion electrons as shown at 
the left. The decay scheme is one of the two shown below. 



8j64 kev 




= energy of conversion 
electrons 

The X-ray energies for various lines of zsCu and 3 2Ge are 




Element 


Z 


K a , kev 


L a , kev 


Cu 


29 


8.06 


0.93 


Ge 


32 


9 89 


1 19 



Determine, with the aid of Moseley's law, which of the two possible decay schemes 
is actually followed. 

5. The Displacement Law 

Comparative studies of the chemical properties of the radioactive 
decay products of uranium and thorium first led Soddy (S58) to enunci- 
ate his so-called displacement law in 1914. In its original form the dis- 
placement law simply stated that any element which is the product of an 
a-ray disintegration is found in the Mendeleev periodic table two columns 
to the left of the parent radioactive element, while the product of a 
/3-ray disintegration is found one column to the right of its parent. For 
example, thorium is found in group IV of the periodic table (Appendix F), 
while the product of its a-ray decay has chemical properties which are 
indistinguishable from those of radium, in group II. This product, 
mesothorium-1, happens to be a 0-ray emitter, and so is its daughter 
product, mesothorium-2. The product of these two successive trans- 
formations is radiothorium, which has chemical properties which put it 
again in group IV. In series-decay notation, we have simply :f 

goTh" 2 A BaMsTh? 28 A 89 MsThr 8 -^ 9oRdTh 8 A 

t It was, of course, the fact that Th and RdTh differ in atomic weight by four 



26 



T/tc Atomic Nucleus 



[CH. 1 



Since Soddy's day several other types of radioactive decay have been 
discovered. These are summarized in Table 5.1, with their character- 
istic shifts in atomic number. 

TABLE 5.1. THE SHIFT IN ATOMIC NUMBER ASSOCIATED WITH VARIOUS 
TYPES OF SPONTANEOUS NUCLEAR TRANSFORMATIONS 



Type of radioactive 
transformation 


Usual 
ay mbol 


Atomic number 
of initial state, 
or parent 


Atomic number 
of final state, 
or daughter 


Alpha decay 


a 


Z 


Z - 2 


Positron beta decay 


0+ 


z 


Z - 1 


Electron rapture . ... 


EC 


Z 


Z - 1 


Gamma ray . . 
Internal conversion . 


y 
e~ 


z 
z 


Z 
Z 


Isomeric transition 
Neutron emission . ... 


IT 
n 


z 
z 


Z 
Z 


Negatron beta decay . . . 


ft- 


z 


2+1 



A self-evident extension of Soddy's displacement law applies to all 
types of nuclear reactions. Thus if boron (Z = 5) captures an a ray 
and emits a" neutron, the product of the reaction has to have a nuclear 
charge of Z + 2 = 7, and it therefore must be an isotope of nitrogen. 
This reaction is written more compactly as B(a,n)N. A few of the 
best-known nuclear type reactions, such as the (a,n) reaction, are listed 
in Table 5.2 with the change in atomic number which they produce. 

TABLE 5.2. THE SHIFT IN ATOMIC NUMBER ASSOCIATED WITH SOME COMMON 

NUCLEAR TYPE REACTIONS 
(a -* alpha, n neutron, p proton, d deuteron, y = gamma ray) 



Type of nuclear 


Atomic number 


Atomic number 


reaction 


of target 


of product 


(,n) 


Z 


Z + 2 


',?) W,n) 


Z 


Z + l 


f d,p) (n, T ) 


Z 


Z 


d,a) (n,p) 


z 


Z - 1 


(n,) 


z 


Z - 2 



The atomic number Z for many artificially produced radioactive 
substances has been determined by applications of the displacement law. 
For example, neptunium ( 93 Np) and plutoniuin (94Pu) were first assigned 
their atomic numbers from studies of the negatron ft decay of &2U 239 which 
was formed in the reaction U 288 (^,T)U 239 . A part of this series is 

units (because of the one a decay in the chain), but have identical chemical properties, 
which formed the type of evidence on which Soddy based his suggestion of the exist- 
ence of isotopes. 



5] Charge of Atomic Nuclei 27 




Similarly, atomic-number assignments were first made for americium 
( 96 Am), curium ( 9 eCm), berkelium (9?Bk), and californium (9sCf) from 
applications of the displacement law. All these have been confirmed 
subsequently by observations of their L-series X-ray spectra as excited 
by internal conversion or by electron-capture transitions. 

Problems 

1. A uranium target is bombarded with high-energy a rays, and then at some 
later time the following three chemically distinct radioactive elements are 
separated from the target. 



Element 


Principal radiations 


Half -period 


1 
2 
3 


a, 7, X ray 
ftr, 7, X ray 
X ray (no 0~, /3 + , or 7 ray) 


490 yr 
6.6d 
40 d 



Each of these three elements emits the same line spectrum of X rays, which is 
characteristic of a certain atomic number Z. The L aZ line of this spectrum has 
a quantum energy of 13.79 kev. It is known that the L aZ line of ai.C'ru (curium) 
has a quantum energy of 14.78 kev, while the L a2 line of 9 oTh (thorium) has an 
energy of 12.84 kev. 

(a) From the X-ray data, determine the atomic number Z of the atoms which 
emit the 13.79-kev L a - 2 line. 

(b) Determine the atomic number of element 1, and state what physical 
process gives rise to the X rays, accompanying its radioactive decay. 

(c) Same as (6) for element 2. 

(d) Same as (6) for element 3. 

2. In the series decaj* of g 2 U 235 to its final stable product, seven a particles and 
four negatron ft rays are emitted. 

(a) Deduce the nuclear charge and mass number of the final product of this 
decay series. 

(6) If the wavelength of the K al line of 92 U 23B is 0.1267 A, calculate the wave- 
length to be expected for the K ai line of the stable atoms formed in (a). 

(c) The observed value for the K ai line of Pb is 0.165 A. Assuming the dis- 
crepancy to be due to the assignment of a value of unity to the screening constant, 
what value of the screening constant would be required to make Moseley's law 
check with experiment? Is this value reasonable? If not, are there any other 
factors which would cause a departure from Moseley's law? 

(d) Give an approximate expression for the ratio of the volume of the nucleus 
of 9 2U 23B to that of the nucleus of the nuclide formed in (a). 

3. Mention several types of experimental evidence which show that the atomic 
numbers of H, He, and Li are 1, 2, and 3 and are not, for example, 2, 3, and 4. 
How many of these observations depend, for their interpretation, on theories 
which have been convincingly verified by independent experiments? 



CHAPTER 2 

Radius of Nuclei 



We now turn our attention to the experimental and theoretical evi- 
dence concerning the size and the shape of atomic nuclei. 

The a-ray-scatteriiig experiments, which we have reviewed in Chap. 1, 
first showed that the positive charge in each atom is confined to a very 
small region within the atom. On grounds of symmetry, this positive 
region was thought of as being spherical in shape and as being located 
in the center of the atom. It was therefore called the nucleus. The 
original observations on a-ray scattering showed only that the nucleus 
was not reached by a rays whose closest distance of approach to the 
center of the atom is about 30 X 10~ 18 cm for the case of gold (Chap. 1, 
Fig. 3.3) and several other heavy elements. 

Bohr's theory of the origin of atomic spectra met with sufficient 
initial success in 1913 to constitute an acceptable confirmation of his 
assumption that the principal part of the atomic mass is also located 
within this small, positively charged, central nucleus. 

Experimental studies of the spatial distribution of nuclear charge 
and mass involve a wide variety of nuclear and atomic phenomena. The 
finite size of the nucleus acts only as a minor perturbation in some 
phenomena, e.g., in the fine-structure splitting of X-ray levels in heavy 
atoms. At the opposite extreme, there are phenomena in which the 
nuclear radius plays the predominant role, such as in the elastic scattering 
of fast neutrons. In this chapter we shall review and correlate a number 
of different types of evidence which have been brought to bear upon the 
question of nuclear radius. 

1. The Growth of Concepts Concerning the Size of Nuclei 

By 1919, Rutherford (R45) himself had shown that deviations from 
the scattering which would be produced by a pure coulomb field are 
experimentally evident when a rays are scattered by the lightest ele- 
ments. In these light elements, the closest distance of approach, for 
the energy of a ray used, was of the order of 5 X 10~ 18 cm. The non- 
coulomb scattering observed at these close distances became known as 
anomalous scattering. The distance of closest approach at which anom- 
alous scattering begins was identified as the first measure of the nuclear 
radius. 

28 



1] Radius of Nuclei 29 

We shall discuss the contemporary interpretation of the experiments 
on anomalous scattering later, in Sec. 7. Here it is worth noting that the 
early efforts to interpret these results, in terms of collisions which could 
be described by classical mechanics, led to the introduction of a number 
of ad hoc, if not bizarre, models of the inner structure of atomic nuclei. 
Some of these models had to stay in vogue for over a decade because no 
more acceptable model could then be found. These included Chad- 
wick's (CIS) "platelike a particle' 7 and Rutherford's (R49) "core-and- 
neutral-satcllite " nucleus which contained a small positively charged 
core, surrounded by other nuclear matter in the form of heavy but 
uncharged satellites moving in quantized orbits, under a central 1/r 6 law 
of attraction which was attributed to polarization of the neutral satellites. 
The early speculations on the idea of neutrons are visible in this model. 

The gradual development of the wave mechanics, in the latter 1920s, 
provided the first basis for scrapping many of these classical ad hoc models 
of the structure of nuclei. A wide variety of nuclear phenomena can 
now be interpreted on a basis of wave mechanics, as it is applied to a few 
newer nuclear models which are reasonably self -consistent. Much prog- 
ress has been made, but much remains to be done. 

A variety of experimental evidence (Chap. 8) now is consistent with the 
concept that nuclei are composed of only protons and neutrons and that 
these two forms of the "nuclcon," or heavy nuclear particle, are bound 
together by very strong short-range forces. The shape of the nucleus 
is taken as being substantially spherical, because for a given volume this 
shape possesses the least surface area and will therefore provide maximum 
effectiveness for the short-range binding forces between the nucleons in 
the nucleus. 

The existing experimental evidence also supports the view that within 
the nucleus the spatial distribution of positive charge tends to be sub- 
stantially uniform; thus the protons are not appreciably concentrated 
at the center, the surface, the poles, or the equator of the nucleus. 
Small asymmetries of the distribution of positive charge are present in 
some nuclei, as is known from the fact that many nuclei have measurable 
electric quadrupole moments. These charge asymmetries are discussed 
in Chap. 4; here we note that, if the positive charge in a nucleus is 
regarded as uniformly distributed within an ellipsoid of revolution, then 
the largest known nuclear quadrupole moment (of Lu 176 ) corresponds to 
a major axis which is only 20 per cent greater than the minor axis of the 
assumed ellipsoid. In most nuclei the corresponding ellipticity is only of 
the order of 1 per cent. Therefore we may regard most nuclei as having 
nearly uniform and spherical internal distributions of positive charge. 

In the succeeding sections of this chapter we shall discuss nine varied 
types of experimental evidence, which lead to the conclusion that the. 
nuclear volume is substantially proportional to the number of nucleons in a 
given nucleus. This means that nuclear matter is essentially incompress- 
ible and has a constant density for all nuclei. The variations from 
constant density, due to nuclear compressibility, appear to be only of 
the order of 10 per cent (P30, F17). 



30 The Atomic Nucleus [CH. 2 

The number of nucleons in a nucleus is equal to the mass number A ; 
hence in the constant-density model, the nuclear radius R is given by 

R = fl A* (1-1) 

where the nuclear unit radius RQ probably varies slightly from one nucleus 
to another but is roughly constant for A greater than about 10 or 20. 

There is no single, precise definition of nuclear radius which can be 
applied conveniently to all nuclear situations, The nuclear surface can- 
not be defined accurately but is always a surface outside of which there 
is a negligible probability of rinding any of the nuclear constituents. In 
the following sections, we shall see that there are several specific defini- 
tions of nuclear radius, each applying to the particular experimental 
situation used for evaluating the radius. Even with this vagueness, the 
nuclear radius can usually be specified within 1 X 10~ 18 cm or less, or the 
order of 10 or 20 per cent. Thus nuclear radii are actually known with 
much greater accuracy than the radii of the corresponding whole atoms. 

The trend of present experimental results is toward a nuclear unit 
radius in the domain of 

#o = (1.5 0.1) X 10- 13 cm (1.2) 

for phenomena which depend primarily on the "specifically nuclear" 
forces between nucleons. Such radii are called nuclear-force radii, and 
they serve to describe phenomena in which coulomb effects are minor or 
absent, such as the cross section for elastic scattering of fast neutrons by 
nuclei. 

All other common experimental methods involve the use of some 
charged particle as a probe of the nuclear interior. These phenomena 
therefore depend partly upon coulomb effects and also on any non- 
coulomb interactions which may exist between the probing particle (pro- 
ton, electron, \i meson, etc.) and nuclear matter. For phenomena which 
depend primarily upon the spatial distribution of the nuclear charge, the 
trend of present experimental results is toward a different and smaller 
nuclear unit radius, in the domain of 

R Q = (1.2 0.1) X 10- 13 cm (1.3) 

This smaller radius is closely related to the radius of the "proton- 
occupied volume/' and it is now commonly called the electromagnetic 
radius of the nucleus. 

As the mass numbers of all nuclei run from A = 1 to about 260, we 
see from Eq. (1.1) that nuclear radii can be expected to extend from about 
2 X 10~ 18 to 10 X 10~ 13 cm. Aluminum, for which A = 27, has a 
nuclear radius of about 1.4 X 27* X 10~ 13 = 4.2 X 10~ 13 cm, and a 
nuclear volume of Tpr(4.2 X 10- 13 ) 8 cm 3 = 3.1 X 10~ 87 cm 3 . In alumi- 
num there are 

(2.7 g /cm')(6X 10" atoms/mole) . 1Q22 atoms/cm , 
27 g/mole 

and the total volume of their nuclei is 2 X 10~ 14 cm 3 . Thus the nuclei 
occupy only about 2 parts in 10 14 of the volume of the solid material. 
The density of nuclear matter is then of the order of 10 14 g/cm 8 . 



2] 



Radius of Nuclei 



31 



It is useful to classify the types of nuclear experiments through which 
nuclear radii are measured, according to the physical principles involved 
in each method. This is done in Table 1.1. It will be noted that only 
one of the methods can be interpreted clearly by classical electrodynam- 
ics. The other types of experiments give results which are sometimes 
in direct violation of the predictions of classical mechanics. Many of 

TABLE 1.1. CLASSIFICATION OF NINE PRINCIPAL METHODS FOB MEASURING 
THE RADII OF NUCLEI 



Experimental phenomenon 
which depends on 
nuclear radius 


Basic physical 
principles on which 
the method rests 


Type of mechanics 
which can provide 
an interpretation 
of the observations 


1. Energy of radioactive jS-ray decay 
(coulomb-energy difference between 
isobars) 


Coulomb energy of a 
sphere of charge 


Classical 


2. Isotope shift in line spectra 
3. Elastic scattering of fast electrons 
by nuclei 
4. Characteristic electromagnetic radi- 
ations from /u-mesonic atoms 
5. Fine-structure splitting of ordinary 
electronic X-ray levels in heavy 
atoms 


Coulomb potential in- 
side a sphere of charge 


Wave 


6. Lifetime of -ray emitters 
7. Anomalous scattering of rays 
8. Cross section for nuclear reactions 
produced by charged particles, such 
as (,n), (a,2n), (p,7i), etc. 


Penetration of nuclear 
potential barriers by 
charged particles 


Wave 


9. Elastic scattering of fast neutrons 
by nuclei 


Diffraction of un- 
charged matter waves 


W T ave 



these are historically important experiments which first showed the 
limitations of classical mechanics. In each case, the wave mechanics has 
provided an acceptable interpretation of the observations. 

It should be pointed out that only the ninth method (scattering of 
fast neutrons) gives experimental results which are independent of 
nuclear charge. The other eight methods all involve the combined 
effects of nuclear charge and nuclear size. 

2. Coulomb-energy Difference between Isobars 

The electrostatic energy of a charge q which is uniformly distributed 
throughout a sphere of radius R is 



w - 
W -* - 



(2.1) 



32 The Atomic Nucleus [GH. 2 

If the nuclear charge Ze is considered as smeared out throughout the 
nuclear volume, then the coulomb energy W^i of a nucleus is 

*--!* (2 - 2) 

If, on the other hand, each proton remains an aloof and discrete entity 
inside the nucleus and interacts electrostatically with all other protons, 
but not with itself, then the coulomb energy would be 

TF,-|^Z(Z-l) (2.3) 

The difference between Eqs. (2.2) and (2.3) depends simply on the model 
chosen and becomes smaller as Z increases. 

a. Classical Theory of the Coulomb -energy Radius. The coulomb 
energy is a measurable quantity in some nuclei which undergo radioac- 
tive ft decay. For such nuclei Eq. (2.2) constitutes one of our definitions 
of nuclear radius. This particular radius is often called the coulomb- 
energy radius R^i whenever it is necessary to distinguish it from other 
definitions of the size of the same nucleus. 

In ft decay, the mass number A does not change, and therefore R does 
not change, at least not within the domain of validity of the constant- 
density model R = RoA*. In positron decay, the nuclear charge Z of 
the parent decreases to Z 1 for the decay product. Therefore, in 
positron ft decay, a decrease in nuclear coulomb energy occurs, and this 
energy is a part of the total disintegration energy. Conversely, in nega- 
tron ft decay, in which Z changes to Z + 1, the corresponding increase 
in coulomb energy detracts from the transition energy which would be 
available otherwise. For positron ft decay the decrease in coulomb 
energy is, using Eq. (2.2), 

ATP- = 1 I* [Z* - (Z - 1)*] = | ^ (2Z - 1) (2.4) 

where Z is the atomic number of the parent nucleus. 

In positron ft decay, one proton in the parent nucleus changes into a 
neutron in the product nucleus. Simultaneously a neutrino and a posi- 
tron (the ft ray) are expelled from the nucleus. The total energy of the 
nuclear transition [Chap. 3, Eq. (4.23)] is seen as the total kinetic energy 
of the neutrino and positron (equal to the maximum kinetic eneigy E mu[ 
of the positron 0-ray spectrum) plus the rest energy of the positron 
(m c 2 ) and of the neutrino (zero) plus the recoil energy of the residual 
nucleus (negligible for ft decay). Thus the total nuclear disintegration 
energy can be written as 

E^ + m c 2 (2.5) 

This energy is supplied by and is equal to the change of total mass energy 
between the parent and the product nucleus. In the particular positron 
j9 decay transitions which we shall consider here, the dominant contribu- 



2] Radius of Nuclei 33 

tion to the transition energy comes from the change in nuclear coulomb 
energy. The remaining contributions include any difference in the 
nucleon binding energies but come almost entirely from the difference 
between the rest mass of the parent proton M p and the product neutron 
M n . We shall derive general expressions for all these contributions in 
Chap. 11, but we need not await those generalizations in order to estab- 
lish the one special case with which we are concerned here. 

Fowler et al. (F61, B130) first drew attention to a group of nuclei, 
which undergo ft decay, in which the binding energy due to short-range 
nuclear forces between the nucleons is substantially the same in both 
parent and product. These nuclei constitute a series of so-called mirror 
nuclei, one example of which is the isobaric pair, O JB and N 1B , in which O 16 
undergoes positron ft decay to stable N 1B , according to 

8 O 16 - 7 N 15 + ft + (positron) + v (neutrino) 

Any pair of nuclei which can be made from each other by interchanging ail 
protons and neutrons are called mirror nuclei. A number of known posi- 
tron emitters from 6 C n to ziSc 41 have just one more proton than the 
number of neutrons. Their stable decay products each contain just one 
more neutron than the number of protons; hence each of these particular 
pairs of parent and product are mirror nuclei. In each of these nuclei, 
the mass number A is 

A = 2Z - 1 (2.6) 

where Z is the atomic number of the parent positron 0-decaying nucleus. 

With respect to the specifically nuclear attractive binding forces 
between nucleons, there is good experimental evidence that the nuclear 
binding between two neutrons is the same as that between two protons, 
if the classical coulomb repulsion between the protons is not included as 
a "specifically nuclear force." The analysis of the mirror nuclei, carried 
out below, supports other evidence (Chap. 10) that the nuclear forces are 
symmetrical in neutrons and protons. 

As an example, consider the isobaric pair 8 1B and 7 N 1B as composed 
of some kind of core or central nuclear structure containing seven neu- 
trons and seven protons and thus corresponding in this case to yN 14 . 
Adding one proton to this structure gives us O 1B , whereas adding one 
neutron gives us N 1B . We can express the mass of the O 1B and N 16 nuclei 
as the mass of their constituent protons and neutrons, diminished by the 
net binding energy resulting from the short-range attractive nucleon 
forces and from repulsive coulomb forces. Then the mass of the nuclei 
of O 1B and of N 1B can be expressed as 



[(7M p + 7Mn) + M p ] - [(nucleon binding energy) - W^J (2.7) 

1B ) 

[(7M p + 7Mn) + M n ] - [(nucleon binding energy) - W ml ] (2.8) 

With respect to the total binding energy given in the second square 
brackets, note that, because the coulomb force is a repulsive one, the 



34 The Atomic Nucleus [CH. 2 

coulomb energy is a negative term and is deducted from the binding 
energy term which describes the attractive nuclear forces between the 
nucleons. The difference between the nuclear mass of parent and prod- 
uct is then the difference between Eqs. (2.7) and (2.8) and can be written 



M p - M n - ATP nuc + AH^ (2.9) 

in which AlF coll i has been defined and evaluated in Eq. (2.4), while ATV r nuo 
is the difference between the nucleon binding energies in the pair of mirror 
nuclei. 

We now equate the two expressions (2.5) and (2.9) for the nuclear 
disintegration energy, obtaining 



- (M n - M p ) - AJF nllc (2.10) 



which, on substitution of the measured values, m^ 2 = 0.51 Mev and 
(M n Mp) = 1.29 Mev, becomes 

E m = AJFeou! - 1.80 Mev - ATF nuo (2.11) 

We wish to compare this equation with the experimental data on the 
& decay energy of the mirror nuclei for which A = 2Z 1 , to see whether 
their radii R are consistent with the constant- density model R = /SV4*. 
Then Eq. (2.4) becomes 



Substituting this into Eq. (2.11), we have as the theoretical connection 
between ., A, and R 

E mn = |- A* - 1.80 Mev - ATF nuo (2.13) 

5 /to 

Table 2.1 lists the current (H61) experimental values of the maximum 
positron energy -B max for this series of mirror nuclei . None of these emits 
any 7 rays; hence the total kinetic energy of the decay process is simply 
#ma*. These values of max are plotted in Fig. 2.1 against A*. 

It will be noted that the best straight line through the data intersects 
the A = axis at -1.80 Mev. Thus in Eq. (2.13) we find AJF nuo = 0, 
providing independent evidence for the general symmetry of nuclear 
forces in protons and neutrons. The slope of the best straight line is a 
measure of the nuclear unit radius R Q and corresponds in Fig. 2.1 to 
about 

R Q ~ 1.45 X 10~ 13 cm (2.14) 

Dotted lines for R = 1.4 and 1.6 X 10~ 18 cm are shown; they clearly 
bracket the probable value of /? for the coulomb-energy unit radius. 



2] 



Radius of Nuclei 



35 



TABLE 2.1. MEASURED VALUES (H61, R24) OF THE MAXIMUM KINETIC 

ENERGY E m ^ OF THE POSITRON fl DECAY SPECTRA IN THE MIRROR NUCLEI 

FOR WHICH A = 2Z - ] 



Z Element A 


#max, MCV 


Z Element A 


Em**, Mev 


5B 9 




14 Si 27 


3.48 


6C 11 


0.99 


15 P 29 


3.94 


7 N 13 


1.24 


16 S 31 


3.9 


8O 15 


1.68 


17 Cl 33 


4.1 


9F 17 


1.72 


ISA 35 


4 4 


10 Nc 19 


2.18 


19 K 37 


4.6 


]1 Na 21 


2.50 


20 Ca 39 


5.1 


12 Mg23 


2.99 


21 Sc 41 


4.94 


J3A1 25 




22 Ti 43 






-2 



14 



Fig. 2.1 Positron tf-ray rnrrpy vs. the two-thirds power of mass number A for the 
mirror nuclei A = 2Z 1. The intercept of 1.80 Mev on the energy axis shows 
that the nuclear forces in these nuclei are essentially symmetric in neutrons and pro- 
tons. The fact that the experimental values tend to lie on a straight line indicates 
that these nuclei have coulomb-energy radii which correspond to a constant-density 
model fieoui = RoA*, with the slope of the data giving the particular value .Ro 1.45 X 
10~ 1B cm for the nuclear unit radius. 



36 The Atomic Nucleus [CH. 2 

If we had used the discrete-proton model, Eq. (2.3), then we would 
have had 

Atf - ~ (2Z - 2) (2.15) 

5 K 



- 1.80 Mev - ATT... (2.16) 

5 /to 

When the experimental data on # m are plotted against the quantity 
(A* A"*), the fit is about the same as that in Fig. 2.1, and the nuclear 
unit radius is again about R Q ^1.45 X 10~ 18 cm. Future improvements 
of the data in Table 2.1 should be watched. In the meantime, the data 
fit the smeared-proton model of Eqs. (2.2) and (2.13) and the discrete- 
proton model of Eqs. (2.3) and (2.16) about equally well. 

The coulomb-energy unit radius B 1.45 X 10~ ia cm obtained 
from Fig. 2.1 is in good agreement with the nuclear radii obtained for 
these same nuclei by other methods. We may note that this constitutes 
some degree of verification of the factor in Eq. (2.2). Physically, the 
factor 7 is due to the assumed uniform distribution of charge throughout 
the volume of the nucleus. For example, if all the charge were on the 
nuclear surface, this factor would be instead of f , and the coulomb unit 
radius would be only about 1.2 X 10~ 13 cm. 

We conclude from the classical analysis of the ft decay energies that 

1. Nuclear charge behaves as though uniformly distributed through- 
out a spherical nuclear volume. 

2. The coulomb-energy radii of nuclei having A < 41 follow the 
constant-density model R mu ^ = R A* and have a unit radius of R Q ~ 1.45 
X 10~ 18 cm. 

3. The specifically nuclear binding forces between nucleons are sub- 
stantially symmetrical in neutrons and protons. [AW nuo = 0, in Eq. 
(2.13).] 

b. Electromagnetic Radius Deduced from the Coulomb -energy 
Radius. The coulomb-energy unit radius is a purely classical quantity, 
defined by Eq. (2.12). Some other types of experiments, which depend 
upon coulomb potentials within the nuclear volume, and which require 
a wave-mechanical interpretation, lead to "electromagnetic radii' 7 which 
are about 20 per cent smaller than these classical "coulomb-energy 
radii," for the same nuclei. These differences can be reconciled, at least 
qualitatively, when wave-mechanical refinements are invoked in the 
interpretation of the experimental data. 

When the protons in the nucleus are represented by equivalent central 
potential wave functions, the integral of the coulomb energy throughout 
the nuclear volume reduces, in the case of A = 2Z, to (B48, C42) 

W ml =* \ (Z(Z - 1) - 0.77Z*] (2.17) 

o ri 

instead of the classical expression of Eq. (2.2). The correction term 
0.77Z*(3e 2 /5#) arises from the antisymmetry of the proton wave func- 



2] Radios of Nuclei 37 

tions and is called the coulomb exchange energy. For Z ~ 15, the square 
bracket in Eq. (2.17) is roughly 10 per cent smaller than the correspond- 
ing classical expression. Consequently, the experimental values of W^ 
lead, on this model, to nuclear radii R which are roughly 10 per cent 
smaller than the classical coulomb-energy radii. 

A second wave-mechanical correction arises when a more detailed 
model is assumed for the interior of the nucleus. When individual 
quantum numbers are assigned to each of the nucleons in the nucleus, in 
accord with the shell model of nuclei (Chap. 11), it can be presumed that 
in many of the mirror nuclei the transforming nucleon is initially in a 
state of greater orbital angular momentum than most of the other pro- 
tons. As a consequence of its greater angular momentum, the individual 
ft transforming proton would not correspond to a uniformly distributed 
charge, but its radial distribution would tend to be concentrated near the 
nuclear surface. If this is so, then the ft transforming proton is one 
whose contribution to the total coulomb energy is less than that for a 
uniformly distributed proton, because its charge distribution is concen- 
trated near the surface of the nucleus, where the coulomb potential due 
to the rest of the nucleus is smallest. The over-all distribution of charge 
within the nucleus is still regarded as uniform. When this concept is 
quantified, the presumed reduction in coulomb effectiveness of the indi- 
vidual ft transforming proton requires a corresponding decrease in the 
effective radius of the nuclear charge distribution, in order to match the 
experimental values of W^. In this way, the observed coulomb-energy 
differences for mirror isobars (Fig. 2.1) can be reconciled with an effective, 
or electromagnetic, radius, whose unit value is as small as (C42) 

flo~ 1.2 X 10~ 13 cm (2.18) 

The distinction between this wave-mechanical "electromagnetic unit 
radius" and the classical "coulomb-energy unit radius" (Ro~lA5 
X 10~ 18 cm) lies entirely in the nuclear models which are used for the 
theoretical interpretation of the experimental data. If the transforming 
proton is thought of as a probe for studying the coulomb potential in the 
interior of the nucleus, then the wave-mechanical interpretation repre- 
sents a means of correcting the observed coulomb energies for the spe- 
cifically nuclear (noncoulomb) effects between the transforming proton 
and the other nucleons in the nucleus. 

Problems 

1. Derive Eq. (2.2) for the total coulomb energy of a homogeneous distribu- 
tion of charge Ze occupying a sphere of radius R. 

2. Calculate a predicted value for the maximum kinetic energy of the positron 
ft rays emitted by (a) 12 Mg" -> ft+ + nNa 23 and (6) 13 A1 -> ft+ + 12 Mg using 
the constant-density nuclear model, with R = 1.45 X 10~ 13 A*. 

(c) Compare with observed values found in tables. 

3. Prepare a graph similar to Fig. 2.1 but based on the classical discrete-proton 
model, and compare the correlation between ^ max and A for the discrete-proton 
model and the uniformly distributed proton (classical) model. 



38 The Atomic Nucleus [CH. 2 

3. Coulomb Potential inside a Nucleus 

a. Isotope Shift in Line Spectra. The size of nuclei and the distri- 
bution of the charge within nuclei produce small but observable effects, 
known as isotope shift, in certain atomic spectra. The origin of these 
effects can be understood on classical grounds, but their quantitative 
interpretation requires evaluation of the wave functions for atomic elec- 
trons near, and indeed inside, the nucleus. To the extent that these 
electron wave functions are known, the observations on tho isotope shift 
in the line spectra of heavy elements can be interpreted in terms of the 
classical size and charge density of the "proton-occupied volume" 
within the atomic nucleus. In the present section we shall examine only 
those aspects of the isotope-shift phenomena which shed light on the 
questions of nuclear size and charge distribution. Chapter 7 contains a 
discussion of isotope shift and its implications with respect to nuclear 
mass and nuclear moments. 

Most of the quantitative aspects of atomic spectroscopy are deter- 
mined in one way or another by the total charge Ze of the atomic nucleus. 
Thus the energy of an electronic state depends upon the energy of the 
atomic electron in the central coulomb potential U(r) provided by the 
nucleus. The s electrons have a finite probability of being at and near 
the origin (r = 0) of this central field and hence of being actually inside 
the nuclear radius (r = R}. If $*(r) represents the probability density of 
the electron being at distance r from the center of the nucleus, then the 
potential energy of this electron c in the central field U(r) could be written 
as 

e JT" iP(r)Z7(r)4irr 2 dr (3.1) 

If the nucleus had no finite size,, then the potential U(r) would have its 
simple coulomb value 

U(r) = ^ (3.2) 

for all values of r. For a nucleus of finite size, Eq. (3.2) is valid only 
outside the nucleus, i.e., for r > 7?, where R is the -nuclear radius. If 
the nuclear charge Ze is spread in a uniform layer on the outer surface 
only of the nucleus, then the potential U A (r) everywhere inside this simple 
shell of charge would be the same as the value at the surface, which is 
the constant value 

U.(r) = | 6 (3.3) 

On the other hand, if the nuclear charge is distributed uniformly through- 
out the nuclear volume, then the internal potential U v (r) at distance r 
from the center can be shown to be 



3] 



Radius of Nuclei 



39 



The decrease of the atomic binding energy of an s electron, because of 
the finite size of the nuclear charge, can then be calculated (B123, B83), 
using the potential U(r) for the surface-charged nucleus, or U v (r) for 
the volume-charged nucleus. The decrease AFF in electron binding 
energy can then be represented, for the volume-charged nuclear model, as 



= e 



t 2 (r)(V v - U)4*r* dr 



(3-5) 



where V = Ze/r for the point nucleus and the integration extends only 
throughout the nuclear volume < r < R. 

The three potentials, V(r) for a point nurleus, U s (r) for a surface- 
charged nucleus, and U 1t (r) for a uniform volume-charged nucleus, are 
compared graphically in Fig. 3.1, from which a qualitative idea of the 

Distance from center of nucleus r >- 
r-Nuclear radius R 



o 



o 

I 




Ze 



^Extranuclear region 



Fig. 3.1 Comparison of the electrostatic potential inside nuclei, on three models. 
Curve 1 is for a point nucleus of zero radius, Eq. (3.2). Curve 2 is for a nucleus 
having all its charge on its surface, Eq. (3.3). Curve 3 is for a nucleus in which the 
charge is uniformly distributed throughout the nuclear volume, Eq. (3.-0. 

direction and relative magnitude of the resulting energy changes AW 
can be obtained. 

It is foiind experimentally that there are a number of elements having 
two or more stable isotopes which differ in mass number by two units. 
Examples include H2 Pb 2 " 4 , Pb 2 " 6 , Pb 208 ; 8 ,>Hg' ! ' fi , Hg 19 , Ilg 200 , Hg 202 , Hg 204 , 
etc. Under high resolution, certain lines in the emission spectra of these 
elements will be found to consist of a number of closely spaced com- 
ponents, one for each isotope of even mass number. These are the 
isotope-shifted components in which we are interested here. Each of 
these components is itself single, i.e., it is not further split into a group 
of hyperfinc-structure components, because the nuclear moments are 
zero-valued. [The actual spectral "line" will generally contain other 
components which arise from one or more stable isotopes whose mass 
numbers are odd, for example, Pb 207 , Hg 201 , etc. These components from 



40 The Atomic Nucleus [CH. 2 

odd isotopes will be further split by hyperfinc structure (Chap. 5) because 
of their finite nuclear moments.] 

The largest isotope shifts are usually found in transitions between 
atomic configurations containing different numbers of s electrons, espe- 
cially the deeply penetrating 6s electrons, as, for example, in the transi- 
tion 5d n 6p 5d n 6s. The isotope shift is seen to represent the energy 
difference AWi AWz between two evaluations of Eq. (3.5), once for 
each of the two isotopes concerned. That these differences are finite 
shows at once that the nuclear radii RI and R z arc different for the two 
isotopes. More exactly, the "electromagnetic radius," which is the true 
meaning of 7? in Eq. (3.5), is found to be larger in the heavier isotope. Of 
course, the nuclei of both isotopes contain the same number of protons 
and have the same total charge Zc. If the heavier isotope were formed 
from the lighter isotope by merely adding two extra neutrons to the 
outside of the lighter nucleus, and not also increasing the proton-occupied 
volume, there would be no isotope shift. Thus the very existence of 
the isotope shift shows that the protons in the nuclei of both isotopes move in 
regions of different size. The penetrating s electron serves as a useful 
probe because it spends a part of its time actually within the nuclear 
volume, and its noncoulomb interactions with protons and neutrons are 
negligible. 

In principle, we should be able tt> determine how the nuclear charge is 
distributed inside the nucleus by appropriate application of Eq. (3.5), 
and its corollaries, to suitable speotroscopic data. This cannot yet be 
done with high accuracy because of both theoretical and experimental 
inadequacies. The existing status has been ably summarized, especially 
by Brix and Kopfermaim (BP23), Foster (F59), and Bitter and Feshbach 
(BG1). In general, it is found that the data are in better agreement 
with theory when the nuclear charge is assumed to be uniformly dis- 
tributed throughout the nucleus than when the charge is assumed to lie 
only on the nuclear surface. This same conclusion was reached as early 
as 1932 by Breit (Bill) in his excellent pioneer work on the theoretical 
explanation of isotope shift in heavy elements as an effect due to the 
finite extension of the nuclear volume. 

Figure 3.2 summarizes (B123, F59, B61) the present experimental 
data on 19 elements in a form which allows comparison with the predic- 
tions of existing theory. It will be rioted that the observed isotope 
shifts are about one-half as large as the calculated shifts if the nuclear 
unit radius is taken as R = 1.5 X 10~ 13 cm. Although the variations 
are large, the data are not in disagreement with an electromagnetic 
nuclear unit radius as small as 7t! = 1.1 X 10~ 13 cm. Important 
improvements in the use of isotope shift as a means of studying the inner 
structure of nuclei (C52) can be expected as spectroscopic investigations 
are extended to enriched or separated isotopes and as advances are made 
in the theory (B83), especially with regard to the evaluation of the elec- 
tronic wave functions. 

b. Elastic Scattering of Fast Electrons by Nuclei. Nuclei are essen- 
tially transparent to electrons, and their mutual interactions are confined 



3] 



Radius of Nuclei 



41 



to the long-range coulomb force. Bombardment of nuclei by high-energy 
electrons (say, > 10 Mev) therefore provides an opportunity for probing 
the coulomb field in the interior of the nucleus, with a minimum of inter- 
ference from noncoulomb effects. Classically, the collision diameter 
between an incident 10-Mev electron and a Cu nucleus (Z = 29) is 
6 ~ 4 X 10~ 13 cm. The rationalized cle Broglie wavelength for the same 
1000 




100 



Fig. 3.2 Comparison of observed and theoretical values of the isotope shift in It) ele- 
ments, in the form developed by Brix and Kopfermaim (B123, K59). The "iaotope- 
shift constant," shown on the vertical scale, is proportional to the absolute term 
difference, which contains several other para.met.ers of the optical transition. The 
isotope-shift constant depends strongly on Z and only weakly on the mass number A\ 
hence the data for each element are plotted against Z, using an average value of the 
mass. The solid lines are the predicted values for nuclei containing a uniform dis- 
tribution of charge within a sphere whose electromagnetic radius is R 1.5 X 10~ 13 A* 
cm, or H = 1.1 X llr~ 18 A J cin. All theoretical and experimental values correspond 
to the shifts when A/2 corresponds to AA = 2. [From Bitter and Feshback (B61).l 

electron is X c^ 20 X 10~ J3 cm. Therefore, by the criteria noted in 
Chap. 1, classical collision theory is invalid bucause b/X < 1. For 
incident electron energies below about 2 Mev the nucleus can be con- 
sidered as a point charge. Then the relativistic wave-mechanical theory 
of electron scattering developed by Mott gives good agreement with 
experiments (Chap. 19). At higher energies, and for nuclei of finite 
size, the incident electron may be considered as penetrating into the 
nucleus and thereby experiencing a smaller coulomb potential (Fig. 3.1). 
The cross sections for elastic scattering of swift electrons are therefore 
diminished, especially at large scattering angles. 



42 The Atomic Nucleus [CH. 2 

Physically, high-energy electron scattering is closely related to isotope 
shift, and both can be shown to depend primarily upon the volume 
integral of the potential taken throughout the nucleus (F43, B61, B83). 
Experimentally, marked deviations from the scattering which would be 
expected from a point nucleus have already been observed for a variety 
of elements, with electrons of 15.7 Mev (L37), 30 to 45 Mev (P22), and 
125 to 150 Mev (H58). Present interpretations (B61) of these experi- 
ments give reasonable agreement with a uniformly charged nucleus having 
an electromagnetic unit radius in the domtain of 

tfo -^ (1.1 0.1) X 10- 13 cm (3.6) 

Both the theory and the experiments are difficult, but the importance of 
the results suggests that marked improvements can be forecast. 

c. Characteristic Electromagnetic Radiations from ^-Me sonic Atoms. 
The properties and behavior of w mesons and /i mesons are now rather 
well understood (M14, M15, P29, Bf>3, T25). Bombardment of nuclei 
by high-energy particles or photons (*> 150 Mev) (Bll) can evoke the 
emission of positive or negative TT mesons from the target nuclei. Because 
of their positive charge, the T f mesons are repelled by nuclei. They 
decay with a mean life of about 02 /xsec into IL+ mesons, which in turn 
decay into positive electrons, according to 



n+ + v T ~ 0.02 /zscc (3.7) 

e + + v + v T ~ 2.15 MSCC (3.8) 



where v and v represent a neutrino and aiitineutrino. (In terms of the rest 
mass ra of the electron or positron, the rest masses of the TT meson and \i 
meson are close to M r ~ 273m , M M ~ 207rao, for both the positive and 
negative varieties.") 

The negative IT mesons arc especially interesting. If they are not 
captured by a nucleus, they decay into a /i~ meson in a manner analogous 
to Eq. (3.7). The resulting n~ meson has the opportunity of being 
slowed down by ionizing collisions to a substantially thermal velocity 
and then of being captured by a nucleus. This capture process is 
thought to proceed somewhat as follows: A fjr meson, having the ame 
spin and charge as an atomic electron, may be expected to fall into a 
hydrogenlike "Bohr orbit 7 * around the nucleus. This atomic energy 
level should be similar to an energy level for an atomic electron, except 
that the "Bohr radius " around a point nucleus, (n^) 2 /Zr? 2 m , will be 
about 200 times smaller than the corresponding radius for an electron, 
because of the larger rest mass of the i~ meson. As all the mesonic 
" atomic states" are unoccupied, the \r meson will fall to states of lower 
energy, the transitions being accompanied by the emission of character- 
istic electromagnetic radiation, or of Auger electrons, and taking place 
within a time of the order of 10~ 13 sec. In the " K shell," the /*" meson 
will be some 200 times nearer the nucleus than is a 7if -shell electron, and 
the p~ meson will therefore spend an appreciable fraction of the time 
within the nucleus itself. The life of the individual uT meson mav 



3] Radius of Nuclei 43 

terminate by a charge-exchange reaction with a proton in the nucleus 

M~ + P -> n + v r ~ 10~ 7 (-* V sec (3.9) 

or by radioactive decay into an energetic electron and a neutrino-anti- 
neutrino pair. 

/x - - f- + v -+ v 7 ~ 2.15 ^ec (3.10) 

In Pb, the radius of the ^-mesonic K shell for a point nucleus (R Q 0) 
would be only about 3 X 10" l3 cm, while the L shell would have a radius 
of about 12 X 1()~ 13 cm. Transitions between the 2pj and Is states, 
which correspond to the A'ai X ray in the ordinary electronic case, would 
be expected to have an energy release of 16.4 Mev. When p mesons are 
captured into Pb atoms, the 2p> > l,v electromagnetic radiation is 
observed, but it has a quantum energy of only t>.02 Mev (F52). 

This enormous shift from the transition energy expected for a point 
nucleus is identical in principle with the isotope shift in ordinary elec- 
tronic line spectra, but it is greatly exaggerated by the smallness of the 
u~-mcsonic Bohr radii. For light elements the theoretical shift in 
the energy of the Is level (K shell) is approximately proportional to 
R 2 Z*, where R R^A* is the radius of an assumed uiuform distribution 
of charge within the finite volume of the nucleus (F52, C42, W33), The 
shift is greatest for the Is level, much less for the 2pj level (L\\ level in 
X-ray notation), and still smaller for the 2^,. level (L u \ level). Figure 
3.3 summarizes the experimental measurements hy Fitch and Kaiii \vatur 
(F52) of the quantum energies for the characteristic, /i~-mesonic "X rays " 
arising from the 2p 3 > Is transition in nine elements. Comparison with 
the calculated values for a point nucleus (7? 0) and for a homo- 
geneously charged nucleus having a unit radius of R Q = 1 .3 X 10~ 13 cm 
is shown by the two curves in Fig. 3.3. Clearly, the measured transition 
energies in M~-mesonic atoms correspond to a unit radius which is slightly 
smaller than 1.3 X 10~ 13 cm. When computed numerically, the data 
for Ti, Cu, Sb, and Pb give nuclear electromagnetic unit radii, R Q = R/A^ 
which fall in the domain 

R Q = (1.20 + 0.03) X 10- "cm (3.11) 

if the distribution of charge is assumed to be homogeneous within bho 
nucleus. 

The nearly ideal character of the ^ meson (or the electron) as a 
probe for the distribution of nuclear charge arises from its exceedingly 
weak interaction with nuclcons, as well a,s from its large mass, as wa? 
first pointed out by Wheeler (W32, W33). The role of the \L~ meson 
has been beautifully pictorialized by Wheeler (W33): 

To it (the \L~ meson), the nucleus appears as a transparent doud of electricity. 
The degree of transparency is remarkable, in view of the density of nuclear mat- 
ter, 1 or 2 X 10 5 tons/mm 3 . Thus a meson moving in the K orbit of lead spends 
roughly half of its time within the nucleus, and in this period of ~4 X 1 0~ 8 sec 
traverses about 5 meters of nuclear matter, or ^lO 17 g/cm 2 . This circumstance 



44 



The Atomic Nucleus 



[CH. 2 



means that the major features of the nuclear electric field uniquely determine 
the mesonic energy level diagram. Conversely, these features can be deter- 
mined by the position of the mesonic states. 

The fine-structure splitting of the ^-mesonic "X-ray 11 spectra appears 
to have been resolved in the experiments by Fitch and Rainwater (F52), 
Tn electronic X-ray spectra, the electron spin of T gives rise to the two 
fine-structure levels 2p (or Ln) and 2p a (or Lin). Transitions from these 
to the ls$ (or K) level constitute the K a * and K ai lines, respectively, and 
the Kai energy slightly exceeds the K a z energy. For the /z-mesonic levels 
in Pb, a fine-structure splitting of about 0.2 Mev is observed for the 




0.2, 



20 



50 60 70 80 100 



30 40 
Atomic number Z 

Fig. 3.8 Energies of the ji-mesonic transition 2pj > Is, which corresponds to the 
K a \ line in electronic X-ray spectra. Calculated values for point nuclei (flo = 0) 
and for homogeneously charged nuclei with /2 =* 1.3 X 10~ n cm are shown by the 
two'curves. The experimental values obtained by Fitch and Rainwater (F52) are 
shown as open circles. 

corresponding fine-structure doublet. When higher accuracy becomes 
available, observations of this type may be useful for measuring the 
magnetic moment of the p meson, as well as verifying the values of 
nuclear electromagnetic radius. 

In marked contrast with p mesons, ir mesons have a very strong inter- 
action with nucleons. Hence ?r~-mesonic atoms are not useful for study- 
ing nuclear radii. In elements of low Z (Be, C, O), the ir-mesonic /C-shell 
radius lies outside the nuclear volume, and for these cases the 2p > Is 
transition in ir~-mesonic atoms has been observed also (C4, S68). 

d. Fine-structure Splitting of Electronic X-ray Levels in Heavy 
Atoms. Each ordinary electronic X-ray level is also reduced slightly in 
energy because of the finite size of the nucleus, but of course the effect 



4] Radius of Nuclei 45 

is minute when compared with the shifts in /i--mesonic atoms. Schawlow 
and Townes (87) have summarized the pertinent theoretical and experi- 
mental material, showing that a homogeneously charged nucleus, whose 
electromagnetic unit radius is fi ^ 1.5 X 10~ 18 cm, should produce a 
change of only 0.3 per cent in the fine-structure separation of the 2p and 
the 2p} levels for Z = 90. The effect diminishes rapidly for smaller Z. 
Schawlow and Townes found that the existing data on the K a i-K a2 X-ray 
fine-structure separations for Z = 70 to 90 appear to be in agreement 
with an electromagnetic unit radius of R ~ 1.5 X 10~ 18 cm. Further 
improvements in X-ray energy measurements and in the theory of X-ray 
fine structure would be required in order to improve the accuracy of this 
estimate of R . 

Problems 

1. Show that the electrostatic potential U(r) at distance r from the center of 
a sphere containing a uniform density of positive charge is 



if q is the total charge in a sphere whose radius is R. 

2. In the sphere containing a uniform density of positive charge, evaluate the 
electric field strength for all values of r and show that the field strength is con- 
tinuous at the boundary r - R. 

4. The Nuclear Potential Barrier 

a. Coulomb Barrier with Rectangular Well. Imagine that originally 
we have a mercury nucleus, whose charge is Ze = 80e, fixed with its 
center at the origin of coordinates in Fig. 4.1, and let r be the distance 
between this center and the center of a stationary a particle whose charge 
is ze = 2e. We will call the potential energy zero when the separation 
between these two nuclei is very large. Imagine that we can, by some 
means, push on the a particle and force it closer to the mercury nucleus. 
Then for any large separation distance r, the work done will equal the 
electrostatic potential energy (Ze)(ze)/r between the charges. 

As we decrease r, we finally come to some small distance which is of 
the order of the nuclear radius of mercury. Here the short-range 
attractive nuclear force begins to be felt, and as we continue to decrease r 
this attractive force increases until it just equals the coulomb repulsive 
force, leaving zero net force between the two particles. On decreasing r 
still further, the attractive force dominates, and the two nuclei coalesce. 
If the original nucleus was eoHg 204 , then the addition of an a particle 
( 2 He 4 ) forms M Pb 208 . Now B aPb 208 is a stable nucleus. It does not 
spontaneously emit a rays. Therefore its total energy may be taken 
tentatively as less than that of the original system of widely separated 
Hg 204 and He 4 nuclei. 

Figure 4.1 is the usual schematic illustration of the potential energy 



46 



The Atomic Nucleus 



(en. 2 



U as a function of distance r for such a system. The simplest model is 
the so-called square-well model, in which the potential energy of the bound 
system is taken as constant and equal to f/ for r = to r =* R } while 
at r R the potential energy increases discontinuously to the coulomb 
value (Ze)(ze)/R. For r > R the potential energy consists only of the 
coulomb energy (Ze)(ze)/r. In the square-well model, R is called the 
nuclear radius. 



Zzv 2 

Barrier height, B= ~- 

K 




-B 



R u 

r 

Fig. 4.1 Schematic diagram of the nuclear potential barrier between a nucleus of 
charge Ze and a particle of charge zc at a center-to-centcr distance r. 

The coulomb region from r = R to r = & is called the coulomb 
potential barrier, while the entire curve of L T against r is called the nuckar 
potential barrier. The so-called height B of the barrier is its maximum 
value, which occurs at the nuclear radius, and is 






(4.1) 



Note that the height of the barrier depends on the incident particle's 
charge ze. 

b. Modifications Due to Short-range Forces. Clearly the discon- 
tinuities in this square-well model are unrealistic. The simplest refine- 
ment is to replace the infinite potential slope at r = R by a finite but 
very steep slope and to round off the bottom and top of the potential 
well, as indicated by the dotted potential curve in Fig. 4.1. When this 
is done, the definition of nuclear radius requires reconsideration; it will 
generally be some parameter entering the analytical functions which are 
chosen to describe the new potential well. In some such models the 
nuclear radius remains defined as the position of the top of the rounded-off 
barrier, i.e., the distance for zero force. In other models the nuclear 
radius may signify the point of maximum slope within the potential well. 
Moreover, the distance r signifies only the separation between the centers 
of the two particles, Zc and ze. Each particle has an assignable radius 
of its own, and the radius of the ze particle will obviously depend on 
whether ze is a proton, an a particle, or even some larger nucleus such as, 
say, O 16 . If Ze and ze are regarded as uniformly charged spheres, it is 
well known that their external electrostatic fields are the same as though 



4] Radius of Nuclei 47 

their entire charges were located at their geometrical centers. There is, 
therefore , no ambiguity in the coulomb potential (Zc)(zv)/r, as long as r 
is larger than the Mini of the radii of the two particles. 

In contrast to coulomb forces, the attractive nuclear forces between 
nucleons are short-range forc.es and are significant only when the distance 
between two nucleons is of the order of 2 X 10~ 13 cm or less, or pictorially 
when the two nucleons are practically in contact with each other. Then, 
when r is essentially equal to the sum of the radii of Ze and ze, the nuclear 
attractive forces depend on the separation between the surfaces of the two 
particles, while the coulomb forces are still dependent on the separation 
of the centers of the two particles. 

This marked difference in behavior between short-range and long- 
range forces has to be recognized in those models in which nuclear radius 
signifies some particular point along the mutual potential energy curves 
of Fig. 4.1, such us the top of the barrier. For example, if Ze and ze are 
spheres having radii R z and R z , then their surfaces first make contact when 
the centers are separated by r = K 7i + R z . For smaller values of r the two 
nuclei begin to merge, and the attractive nuclear forces become stronger 
because of the overlap. At some separation r < (R z + R z ) the nuclear 
attractive forces will just balance the coulomb repulsion. This is the 
"top of the barrier," and it corresponds to some separation r lying 
between the radius of the huger nucleus and the sum of the radii. When 
the joint action of long-range and short-range forces is included in the 
model, a more realistic definition of barrier height #, in terms of nuclear 
radii R z and R ZJ would be 



where R z < r < (R z + R z ) (4.3) 

Although there are many alternative choices for the parameter 
called the nuclear radius, the actual absolute difference between them is 
usually less than about 10 to 20 per cent. In nonspecialized discussions, 
the terms nuclear radius and coulomb barrier height generally con- 
note the simpler and approximate relationships of Eq. (4.1), that is, 
B = Zze*/X, with R = 7tVl'. 

c. Inability of Classical Mechanics to Reconcile a-Ray Scattering and 
Radioactive a Decay. According to classical electrodynamics, an a par- 
ticle which is released with no initial velocity from the surface of a radio- 
active nucleus, such as uranium, will be accelerated away from the 
residual nucleus whose charge is Ze. When the a particle and residual 
nucleus have become widely separated, the total kinetic energy gained 
must be just equal to their initial electrostatic potential energy (Ze)(ze)/r, 
where r was their initial separation when the a particle was released. 
Classically, we would require that r be substantially equal to the nuclear 
radius. 

In the particular case of the radioactive decay of 9 2 U 238 , a. rays are 
spontaneously emitted for which the kinetic energy of disintegration is 
4.2 Mev. Equating this to an initial potential energy between the a 



48 



The Atomic Nucleus 



[CH. 2 



particle (z = 2) and the residual nucleus (Z = 90) gives r = 61 X 10-" 
cm as the apparent initial separation and hence as a classical measure 
of the radius of the decay product 90 Th 234 . 

We ha\re noted in Chap. 1 that the a-ray-scattering experiments 
had shown the presence of only a pure coulomb field down to much smaller 
distances than this, at least for the case of gold. Rutherford first over- 
came the technical difficulties of preparing and studying thin scattering 
foils of uranium and showed (R49) that the 8.57-Mev a rays which are 
emitted by a source of ThC' are scattered classically by uranium nuclei. 
In central collisions, these 8.57-Mev a rays can approach to within 
30 X 10~ 13 cm of the center of the uranium nucleus. Therefore the 
potential is surely purely coulomb down to this distance, as shown in 
Fig. 4.2. 



30 



I" 

20 



15 

10 

5 



^-B-28 Mev 






Coulomb potential 
demonstrated by 
scattering of a rays 
by uranium N 



tr 




10 20 



50 



60 70 80 



30 40 

r in 10" 13 cm 

Fig. 4.2 The coulomb barrier to particles (z = 2) for Z about 90 or 92. The 
region of the solid curve beyond r = 30 X 10~ 13 cm is verified by direct a-ray-scatter- 
ing experiments. If the 4.2-Mev a rays of U 238 were emitted classically, i.e., over the 
top of the barrier, the coulomb potential would have to atop at about r 60 X 10~ 13 
cm. Classical mechanics is therefore unable to provide a simple, single model which 
can account for both observations. 

This observation marked the complete breakdown of classical mechan- 
ics in dealing with nuclear interactions. The a rays from uranium could 
not have been emitted from the top of a potential barrier of 4.2-Mev 
height at a distance of 61 X 10~ 13 cm if the coulomb potential actually 
extends in to 30 X 10~ 13 cm or less. 

The subsequent development of the wave-mechanical treatment of 
the interaction of charged particles with potential barriers provided a 
satisfactory description of a wave mechanism whereby particles can pene- 
trate through potential barriers, instead of being required to surmount 
them as they must in classical mechanics. In the case of the uranium 
decay, the evidence is now that the radius of the residual nucleus is about 
9.3 X 10~ 1! cm and the barrier height about 28 Mev. The 4.2-Mev 
uranium a ray has a probability of only 10~ 19 of penetrating this barrier 
in a single collision, either from outside or inside the nucleus, but this is 
sufficient to account for the known radioactive half-period of U 28B . We 



5] Radius of Nuclei 49 

shall review the wave-mochanical principles of the transmission of mate- 
rial particles through potential barriers in the next section. 

Problems 

1. Consider the details of the collision of a 5.3-Mev a particle with a nucleus 
of chromium ( 24 Cr 62 ). Calculate the following parameters, and locate them on a 
plot of the coulomb barrier. 

(a) The approximate radius R of the chromium nucleus. 
(6) The barrier height to a rays. 

(c) The initial kinetic energy in C coordinates. 

(d) The de Broglie wavelength of the relative motion in C coordinates. 

(e) The collision diameter ft, or distance of closest approach, for a head-on 
collision. 

2. Show that an approximate expression for the height of any nuclear coulomb 
barrier is 

B = 0.76zZ* Mev 
if Ro = 1.5 X 10- 13 cm. 

3. The a rays emitted by U 23H have a kinetic energj' of 4.180 Mev. 

(a) Compute the total kinetic energy of the disintegration by evaluating and 
adding in the kinetic energy of the residual recoil nucleus. 

(6) At what distance from the center of a U 23M nucleus would this a ray have 
been released with zero velocity, if it acquires its final velocity by classical 
coulomb repulsion from the residual nucleus? 

4. What is the distance of closest approach between a U 23B nucleus and 
an incident 8.57-Mev a ray for the case of 160 scattering in the laboratory 
coordinates? 

5. Wave Mechanics and the Penetration of Potential Barriers 

The introduction of wave mechanics brought tremendous improve- 
ments in the theoretical description of the interaction of atomic particles. 
Classical mechanics was then recognized as a special case of the more 
general wave mechanics. Classical mechanics is the limit approached 
by the wave mechanics when very large quantum numbers are involved. 
In describing atomic interactions the quantum numbers are commonly 
small; therefore classical mechanics can usually give only approximate 
solutions, and wave mechanics is required for the more accurate solutions. 

The inherent stability and reproducibility of atomic and nuclear sys- 
tems are to be attributed to the existence of discrete quantized states of 
internal motion, which are the only states in which the system can exist. 

a. Particles and Waves. The original quantum concepts of Planck 
(1901) introduced the quantum of action h into the theory of electro- 
magnetic waves. Thus the frequency of oscillation v when multiplied 
by h was recognized as representing the quantum of energy hv in electro- 
magnetic radiation. A number of physical phenomena involving light 
were soon found to be best understood by descriptions in terms of these 
photons. The simplest classical properties of electromagnetic waves in 
free space are the frequency v y wavelength X, and the phase, or wave, 
velocity c, connected by the relationship \v = c. The introduction of 



50 The Atomic Nucleus [CH. 2 

Planck's constant h has the effect of introducing the characteristic cor- 
puscular properties of energy, W = hv\ of momentum, p = W/c = hv/c; 
and of relativistic mass, M = W/c 2 = hv/c 2 , into the description of the 
physical behavior of these waves. This "dual" approach has been fruit- 
ful in the theoretical description of black-body radiation, of the photo- 
electric effect, of the Comptoii effect, and of many other phenomena. 
Corpuscular properties arc conferred on waves by the introduction of h. 

The "new quantum theory," or "quantum mechanics/' or "wave 
mechanics/' confers wave properties on corpuscles, also by the introduc- 
tion of h. The wavelength of a photon can be expressed in terms of its 
momentum and Planck's constant as X = c/v = h/(hv/c) = hfp. De 
Broglie (1924) first proposed the extension of this "definition" of wave- 
length to a description of corpuscles. Thus an electron, proton, neutron, 
or any other material particle whose momentum is p is said to have a 
de Broglie wavelength of 

X = - (5.1) 

P 

Due to the smallness of A, these wavelengths of material particles are 
usually of the order of atomic or of nuclear dimensions. As in the case 
of visible light or any other wave motion, phenomena in which the wave- 
length plays a role are confined to interactions involving obstacles whoso 
linear dimensions are at least roughly comparable with the wavelength. 
In such interactions, wave properties are conferred on corpuscles by the 
introduction of h. 

In response to the question "Is an electron a wave or a particle?" 
the late E. J. Williams said, "It is, of course, a particle. The wave 
properties are not properties of the electron but properties of quantum 
mechanics." 

Experimentally, there is abundant evidence that electrons, protons, 
neutrons, arid other particles exhibit diffraction phenomena (and hence 
can be described by waves) in their collisions with atoms and nuclei. 
Thus, as was first shown by Davisson and Gernicr (1927), the regularly 
arranged atoms in a crystal of zinc act as a diffraction grating for incident 
monoenergetic electrons whose energy is of the order of 100 ev and whose 
corresponding de Broglie wavelength is comparable with the distance 
between successive planes of zinc atoms in the crystal. In addition to 
exhibiting diffraction maxima and minima in the reflected beam, the 
electrons could be shown to suffer refraction on entering the zinc crystal 
at an angle with the normal. f Similarly G. P. Thomson (1928) first 
obtained electron diffraction patterns by passing an electron beam 
through a thin film of metal composed of randomly oriented crystals. 
These diffraction patterns are similar in appearance to the powder diffrac- 
tion patterns obtained with X rays. 

b. Refractive Index. The experimental evidence by Davisson and 
Germer that a beam of electrons suffers refraction when entering a metal- 

fAn excellent summary of these and related experiments has been given, for 
example, by Richtrnyer and Kennard (pp. 248-259 of R18). 



5] Radius of Nuclei 51 

lie single crystal at an angle with the normal suggests that a refractive 
index M can be formulated for matter waves. An impinging free electron 
is attracted by the surface of a metal and, in the case of nickel, experiences 
a drop of about 18 volts in potential energy, and a corresponding increase 
of about 18 ev in its kinetic energy, as it passes through the surface into 
the metal. We note that the electron has a different velocity, momen- 
tum, and de Broglie wavelength outside and inside the metal. We want 
to express these as an equivalent index of refraction for matter waves. 

In optics, the refractive index p of a medium is defined (S76) in terms 
of the wave velocity as 

wave velocity in free space 



wave velocity in medium 



(5.2) 



Because the frequency remains constant, the refractive index is also 
given by 

wavelength in free space 



wavelength in medium 



(5.3) 



The wave velocity w (or "phase velocity") is a concept which applies 
strictly only to periodic fields which represent wave trains of infinite 
duration. For such fields the wave velocity is the product of the fre- 
quency v and the wavelength A, or 

w = \P (5.4) 

However, a wave train of finite extent, such as that representing a moving 
particle, cannot be represented in simple harmonic form by a single fre- 
quency. It must contain a mixture of frequencies in order that the wave 
train, under Fourier analysis, may have a beginning and an end. When 
two wave trains, having slightly different frequencies, are combined, 
their net amplitude as a function of both time and distance contains 
"beats/ 1 or "groups." These beats are propagated at a different veloc- 
ity, known as the group velocity g, which can be shown (p. 331 of S76) to 
be given quite generally by 

% dw dv ., .. 

( } 



Turning to the wave-mechanical description of a moving particle, 
we write 

X = - de Broglie (5.6) 

P 

W 

v = Schrodinger (5.7) 

h 

in which the momentum p and total energy W have their usual classical 
values 

W = iJlf r + U __ _ (5.8) 

p = A1V = \ / 2M(\V - U] (5.9) 



52 The, Atomic Nucleus [CH. 2 

where M = mass of particle 

V = velocity of particle (nonrelativistic) 
U = potential energy of particle 
Then the phase, or wave, velocity of the particle is 

hW W V , U W 

w = \ v =__=_=_+ 



. --- . 

p h p 2 MV V2M(W - U) 

On the other hand, the group velocity g, which can be obtained with the 
help of Eq. (5.5), is given by 



d /V2 


\M(hv - U)' 


) 

- < 5 ' n ) 
(5.12) 


dv\ 


h 

_ M 

P 



m _ m ^ d_ jV2M(W - U) 
g dv 



or g 

Thus we have the important result that the group velocity g does in fact 
correspond, under Eqs. (5.6) and (5.7), to the classical velocity V of the 
moving particle. The phase, or wave, velocity w corresponds only to the 
velocity of propagation of the individual waves comprising the wave 
train, whereas the group velocity g is the velocity V at which the energy 
or the particle actually travels. 

We may note in passing that the product of the wave velocity and the 
group velocity is a constant of the motion of the particle. Thus from 
Eqs. (5.10) and (5.11) we obtain the relationship 

,,,_EJ> E wg^W ,__ 



We can now utilize some of these relationships to express in a variety 
of ways the effective refractive index /* for matter waves passing from 
one region (analogous to free space in the optical case and denoted by 
subscripts zero) to a second region (denoted without subscripts). From 
optics we have 

M = ^ (5.14) 

W 

and because wg = constant (c* for electromagnetic waves and W/M for 
nonrelativistic matter waves) 

M = ^ (5.15) 

00 

For matter waves, utilizing Eqs. (5.0) and (5.12), we have 

V p \ 

M = pr = - = 15.16; 

V po X 

Then particles which are represented as matter waves, upon passing 
from a region in which U = UQ into a region in which the potential 



5] Radius of Nuclei 53 

energy is U, experience a change of wavelength and momentum which 
corresponds to their having entered a medium whose refractive index is 



IW - U 
W- Uo 



(5.17) 



If U varies with position, as it docs in the potential field of a nucleus, 
we see that refraction phenomena analogous to those encountered in 
classical physical optics are to be expected. Indeed, if U > W, the 
equivalent index of refraction becomes an imaginary number, and we 
may expect phenomena which are analogous to the interaction of electro- 
magnetic waves with conducting media, such as the reflection of light 
from metallic, surfaces. Also, the wave interaction of a charged particle 
incident on a pure coulomb potential barrier U(r) = Zze z /r does yield 
refraction which is equal to the Rutherford scattering, and when the 
wave is incident on a barrier U > W, for whirh the kinetic energy is 
negative and the refractive index is imaginary, the incident wave does 
penetrate exponentially into the barrier and has a finite probability of 
penetrating through the barrier. The quantitative evaluations of these 
interactions are carried through as special solutions of the Schrodinger 
wave equation. 

c. The Nonrelativistic Schrodinger Equation. The simplest differ- 
ential equatioa which represents a traveling wave in a homogeneous 
medium is 



(5.18) 

dz 2 w 2 W ^ ' 

where ^ = amplitude of wave motion 
w = wave velocity 

z = distance in direction of propagation 
i = time 

It is well known that this wave equation gives & correcv description of 
elastic waves in a string or in a membrane, of sound waves, and of 
electromagnetic waves in nonconductors. This equation has a large 
number of solutions, which are applicable to a variety of particular 
physical situations. 

For a plane wave in an isotropic, homogeneous medium we could use 
as solutions of this wave equation 



z/X) (5.19) 

or * = - 4 c i<- /w (5.20) 

where i = V 1 and A is the amplitude of the wave. In all these solu- 
tions, (vt z/X) represents travel in the +z direction and (vt + z/\) 
represents travel in the z direction. This follows at once from the 
fact that 



(-0 



const (5-21) 



54 The Atomic Nucleus [CH. 2 

represents a surface of constant phase. Differentiation gives 

^ = i>\ = w (5.22) 

at 

so that vX = w is the velocity of propagation of any particular feature 
of the wave, i.e., the "phase velocity," or "wave velocity." 

The periodic character of exponential solutions such as Eq. (5.20) 
is best seen from the conventional complex-plane presentation of complex 
numbers (p. 255 of S45) with the real parts plotted as abscissas and the 
imaginary parts plotted as or din at es. Then each complex quantity, such 
as e~ itf> } is represented by a point in the complex plane. Expansion in 
power series shows directly that 

c* = cos ? + i sin <p (5.23) 

cr** = cos v? i sin <p (5.24) 

Hence e i<p returns to the same value when the argument # changes by 
27T, 47P, . . . , etc. Therefore e lZvvt is periodic in time, with frequency v. 
In the general wave equation of Eq. (5.18) we can separate the 
variables if we elect to use only solutions of the form 

^ = f (z)0(f) (5.25) 

in which $(z) is some function of position only and $(/) is some function 
of time only. Then 

-' 

uZ " OS" 



and Eq. (5.18) becomes 



Separating the variables, we have two differential functions which must 
be equal to each other for all values of z and t and which therefore must 
be equal to some constant. We will call this separation constant fc 2 . 
Then Eq. (5.18) can be written 



Sc ions of these two separated differential equations include 

* = ^ (fcz) and ^ (2) = ea " (5 ' 27) 

L>Uo 

(wfcO and (0 = c-fc-*' (5.28) 

C Oo 



5] Radius of Nuclei 55 

In any of these forms, there is spatial periodicity when kz changes by 2v. 
Hence the corresponding motion has the wavelength X, where 

k = (5.29) 

The reciprocal length k, defined by Eq. (5.29), is of broad general use- 
fulness and is called the wave, number, or the propagation number. 

Among the periodic solutions of the wave equation, we then arbi- 
trarily choose the particular time-dependent function $(t) = e~ lwkt , and 
we write the wave function ty from Eqs. (5.25) and (5.28) as 

* = iKsJc- 2 "" (5.30) 

in which \l'(z) is any function of position only. When Eq. (5.30) is sub- 
stituted into the general wave equation, Eq. (5.18), we obtain at once 

A = (5.31) 

In a conservative system, the total energy IT of i\ particle remains 
constant and equal to the sum of the kinetic energy p-/2M and the poten- 
tial energy C7. Then 

W = ~M + r (5-32) 

or 7> 2 = 2Af(\V - U) 

and substitution of the do Broglic wavelength X = hip gives 



Then Eq. (5.31) can be written 

av + srw^.-n 

dz 2 /r 

which is known as "Schrodinger's amplitude equation," or simply as 
tichrodingcr 1 s equation. Tn three dimensions, Schroclinger's etjiiation 
becomes 

W + ^/c^_rJO , = o (5 . 35J 

where V 2 is the Laplaciaii operator aiid, in the cartesian coordina 
x, y, z, has the value 

v^- 32 + 

dx' 2 dy z ' ' dz*~ 

Equation (5.31) is a completely classical wave equation and is ^ 
whenever the spatial wave function \l/ oscillates with a constant & 



56 The Atomic Nucleus [CH. 2 

periodicity X. The transition to wave mechanics begins with the identi- 
fication of X as the de Broglie wavelength of matter waves, Eq. (5.33). 
The de Broglie relationship X = h/p can be regarded as an empirical 
relationship given by the experiments of Davisson and Germer and of 
others. Schrodinger's amplitude equation, Eq. (5.34) or (5.35), is there- 
fore semiclassical, provided that X, and consequently ( r , is constant. 
The transition to wave mechanics is completed when we postulate that 
Schrodinger's amplitude equation may be valid even when X, and there- 
fore U, is not a constant but varies from point to point. The validity 
and usefulness of Schrodinger's amplitude equation, when X and there- 
fore U and p are functions of the spatial coordinates, rest solely on the 
considerable success which this equation has experienced in matching 
experimental results, 

The Schrodingcr Equation Containing Time. Schriklinger's more gen- 
eral wave equation containing time makes use of the total wave function 
^ of Eq. (5.25) rather than just the spatial portion ^. 

Using Eqs. (5.6), (5.7), and (5.19), we could represent a plane wave 
moving in the +z direction by 

* = A sin -" (\Vt - pz) +B cos 2 7 - (\Vt - pz) (5.36) 

h li 

A differential equation which satisfies Eq. (5.32) could then be con- 
structed (R18) by utilizing: (1) the time derivative d^f'dt in order to 
obtain a term proportional to IT, (2) the derivative rT^'dc 2 to obtain a 
term proportional to p 2 , and (3) the product UV to obtain a term pro- 
portional to U. Such a differential equation could have the form 

(5.37) 

If we substitute Eq. (5.36) into Eq. (5.37) and equate coefficients of sine 
terms, and separately of cosine terms, so that Eq. (5.37) is valid for all 
t and z, then Eq. (5.32) is satisfied only if A 2 = -7? 2 , or .4 = iB. 
The choice of sign here is arbitrary, and most commonly the minus sign 
is chosen, so that A = iB. Then the wave function of Eq. (5.36) 
becomes 



where ^ contains the spatial parameters and the amplitude but is inde- 
pendent of time. With this choice of sign, the conscrvation-of-encrgy 
law, Eq. (5.32), is satisfied by Eq. (5.37) if its coefficients arc chosen as 

* (539) 

v 



ih dt dz 2 h z 

which is Schrodinger 1 s wave equation containing time. If the opposite 
choice of sign were made, that is, A = iB, then the signs of the time- 



5] Radius of Nuclei 57 

dependent factor in Eq. (5.38) and of the left side of Eq. (5.39) would 
both change. 

Equation (5.38) is equivalent to Eq. (5.30) and, when substituted 
into Eq. (5.39), leads at once to Eq. (5.31) and hence to Schrodinger's 
amplitude equation, Eq. (5.34), with which the great majority of our 
considerations will be concerned. 

d. Physical Significance of the Wave Function. Equations (5.34) 
and (5.35) have the form of "amplitude equations" representing the 
maximum value of & at x, y, z, as t takes on all possible values. In 
Eq. (5.25) we defined SF as a wave function in which time can be expressed 
as a separate factor. Therefore the phase of ^ at any instant is the same 
throughout the entire wave. Such waves are called standing waves, in 
contrast with traveling waves in which there is at any instant a progres- 
sion of phase along the wave train. For bound states, the solutions ^ 
of Schrodinger's equation therefore represent the maximum values, or 
amplitudes of the standing wave Sk as functions of position. 

The amplitude ^ is generally complex for the de Broglie waves which 
describe unbound material particles. This is in contrast to the analogous 
amplitude equations of acoustics and of electromagnetic theory, where 
the amplitudes are real quantities. However, in those theories, the 
state of the wave is described by two quantities, for example, and H 
in the electromagnetic wave. As N. F. Mott (M(38) has clearly pointed 
out, the de Broglie wave can also be thought of as defined by two quan- 
tities, say, / and g, but for convenience these are combined to form a 
complex wave function, ^ = / + ig. 

It is necessary that, at each point in space, the de Broglie wave 
associated with a particle, must be described by some parameter which 
does not oscillate with time. The absolute value |*| of the wave func- 
tion is such a quantity, if we regard the real and imaginary parts, / and g, 
of the wave function as 90 out of phase. For example, if A is some 
slowly varying real function of 2, we could regard a particular wave as 
made up of a real component 

/ = A cos 27r (vt - - J 
and an imaginary component, 90 ouc of phase, given by 

g = A sin 2w I vt -- J 
Then if * = / + ig 

we form the complex conjugate of SF, represented by the symbol **, by 
changing the sign of i wherever i occurs in ^, obtaining 



Then the product of * and its complex conjugate ** is 

f* + g 2 = A* (5.40) 



58 The Atomic Nucleus [CH. 2 

which is a real quantity equal to the square of the absolute value of 
and written |^| 2 . 

It will be noted that, when time is expressed as a separate factor, as 
in Eqs. (5.25) and (5.38), the absolute values of the total wave function 
^ and of the spatial wave function ^ are equal. Thus 

|^|2 = ^,* = ^,* e -2,ri,< e +2iri,* _ ^* = |^|2 (5.41) 

The solutions ^ of the wave equation which can correspond to physical 
reality must be everywhere single-valued, noninfinite, and continuous, and 
they must vanish at infinity. 

In optics, the intensity of light is proportional to the square of the 
amplitude of the electromagnetic- wave. In wave mechanics, the square 
of the amplitude is analogously related to the density of particles at a 
given position in space. When \f/ is normalized so that 



z = 1 (5.42) 

then \t\*dxdydz (5.43) 

corresponds physically to the probability of finding the particle described 
by ^ in the volume element, dx dy dz, if an experiment could be performed 
to look for it. Thus the physical interpretation of \\f/\ z is that it is a 
probability density, with dimensions of cm~ 3 . Therefore |^| 2 is large in 
those regions of space where the particle is likely to be and is small 
elsewhere. 

In the physical interpretation of solutions of the wave equation, the 
wave function ^ is taken as describing the behavior of a single particle 
and not merely the statistical distribution of the behavior of a large 
group of particles. This means that the wave can interfere with itself, 
in order that ^ may describe the motion of a single particle as a diffrac- 
tion phenomenon. 

Because of its close parallelism with other wave problems in physics, 
Schrodinger's equation is bound to work in those cases where A does not 
change much in a distance of one wavelength. But in the region of 
strong fields around nuclei and in atoms, the de Broglie wavelength can 
change a, great deal in a distance of one wavelength; consequently it had 
to be shown that Schrodinger's equation would describe the experimental 
findings in such cases. It is found that Eq. (5.35) does successfully 
describe many atomic and nuclear phenomena. In a number of impor- 
tant cases, however, the wave functions are still inaccurately known. 
In all but the simplest physical cases, an assortment of special mathe- 
matical methods may be needed in order to obtain the actual solutions 
of the Schrodinger equation for any particular problem. 

e. The Uncertainty Principle and the Complementarity Principle. 
Heisenberg (1927) has shown quite generally that the order of magnitude 
of the product of the uncertainties in the values of pairs of certain 
canonically conjugate variables is always at least as large as h/2w. Thus 
the uncertainty in momentum Ap, and the uncertainty in position Ax 



5] Radius of Nuclei 59 

in the direction of Ap, of a particle are related by 

Ap Arc > A s A (5.44) 

^7T 

Equation (5.44) can be expressed in 1111 equivalent form which is con- 
venient for numerical applications. For a particle having mass M, 
velocity V = fie, momentum p = pMCj and rest energy Me 1 



A(pc) AJT ~ he 
A(pMc 2 ) Ax ~ he = J.97 X 10~ n Mev-rm (5.44a) 

The uncertainties in angular momentum A7, and in angular position 
A<p, of a system are related by 

A J A^ > - h T= h (5.45) 

2?r 

If J is expressed in natural units, J = Ih, then Eq. (5.45) becomes 

Al A^ ~ = I radian (5.45a) 

The uncertainty in kinetic energy AT 1 , and in the time A/ during which 
the energy is measured, are related by 

AT' AZ > '- SE A = O.C6 X 10 ~ Mev-soc (5.4G) 

2?r 

The Heisenberg uncertainty principle, or the "principle of indeterminacy/ 1 
is expressed by these three quantitative relationships. Because of the 
smallness of fe, these uncertainties arc significant primarily in atomic or 
nuclear systems. 

Bohr has made the physical implications of the uncertainty principle 
especially clear and useful through his complementarity principle (1928). 
It can be shown, quite generally, that measuring instruments always 
interfere with and modify the system which they are intended to measure. 
In the domain of classical physics, it is usually possible to calculate the 
disturbance produced by the instrument and to correct for it exactly. 
But in the domain of small quantum numbers, as in observations on a 
single elementary particle, the exact magnitude of the influence of the 
measuring instruments cannot be determined precisely. The magnitudes 
of the minimum attainable uncertainties are just those specified b}' the 
uncertainty principle. This is often illustrated by the hypothetical 
observation of an electron with a light microscope, in which the scatter- 
ing of the light quantum into the microscope's optical system by the 
electron introduces just these same minimum uncertainties in the attain- 
able simultaneous knowledge of position and momentum (see, for exam- 
ple, p. 11 of SI 1 or p. 169 of S29). These effects arc produced by even 
perfectly ideal instruments, and they preclude our observation of too small 
momentum changes in small regions of space. For example, in an elastic 



60 



The Atomic Nucleus 



[CH. 2 



collision between two particles we cannot actually hope to observe (and 
therefore verify as true) a small momentum change Ap at an impact 
parameter x, if Ap is only of the order of h/x. 

Within this domain we must therefore forgo the possibility of experi- 
mental knowledge of the intimate details of the interaction. If two theories 
of the interaction specify two different models or mechanisms for the 
interaction within this domain, we have no way of experimentally deter- 
mining which, if either, actually occurs. This is a domain of "blackout 1 ' 
which prevents our observing the mechanism of the collision too inti- 
mately. Within this domain, whose boundaries are set quantitatively 
by the uncertainty principle, we cannot reject a particular model merely 

because it differs from the only 
model which we can set up on a 
basis of classical mechanics. The 
test of validity of the new theory 
cannot be at the level of the details 
of the interaction but is rather in 
the over-all success which the model 
may have in describing the things 
which can be observed, such as the 



Incident 
particles 

Reflected 
particles 



C7, = 0- 







Transmitted 
particles 



CD 



Fig. 6.1 A one-dimensional rectangular 
potential barrier of height V l'i = I* 
and thickness a. Particles incident from 
the left (region 1) ; whoso kinetic energy 
is less than the barrier height, have a 
finite probability of being transmitted 
through the barrier and into region 3. 



final angular distribution of .scat- 
tered particles. The wave me- 
chanics, or any other subsequent- 
theory, is therefore permitted to 
differ from the classical within just 
the domains specified by the un- 
certainty principle. 
f. Transmission of Particles through a Rectangular Barrier. One of 
the fruitful general results of the wave mechanics is its -quantitative 
description of the probability that a charged particle can pass through a 
potential barrier, even if the particle has insufficient energy to surmount 
the barrier. 

In Fig. 5.1, a particle which has mass M, velocity V, and kinetic 
energy T = ?MV 2 is moving from left to right in a region of space 
where the potential t/i is taken as zero. At z = 0, we imagine that an 
abrupt increase of the potential energy to the value U z = U occurs and 
that this continues for a distance z = a, where the potential again drops 
to zero. In classical mechanics, all particles whose incident kinetic 
energy is smaller than U would be thrown back by the barrier, while all 
particles of greater energy would pass the barrier. In the wave mechan- 
ics neither of these statements is exactly true. A fraction of the incident 
particles, when represented as waves, will be reflected, and the remainder 
will pass the barrier when T = U. The fraction transmitted will 
increase when T > T7, and it will decrease when T < U. The classical 
values will be approached most closely when the thickness of the bar- 
rier is large compared with the de Broglie wavelength of the incident 
particles. 

Localization of a Particle. Let us first apply the uncertainty prin- 



5] Radius of Nuclei 61 

ciple, in order to develop a plausibility argument concerning the trans- 
parency of this barrier. If we are seeking only to locate the particle, we 
can accept an uncertainty Ap in its momentum which equals the full 
value p of the momentum. To this maximum possible uncertainty in 
momentum there corresponds a minimum possible uncertainty of posi- 
tion, which is 

(Az) m ,,,~^ = ^=* (5.47) 

This is a very general result: A par Hdr cannot be localized more closely 
than its dc Broglie wavelength divided by 2ir. In the present case, if the 
barrier width a is comparable with or less than X/27T, we cannot say 
whether a particle whose momentum is p = h/\ will be found on the left 
side of the barrier or on the right side. But if the particle is found on the 
right, side of the burrier we should have to regard it as having successfully 
passed through the barrier. 

We can make this qualitative argument semiquaiititative. In ask- 
ing whether the particle is on the left side or the right side of the barrier, 
we accept an uncertainty of Az = a in the position of the particle. To 
this uncertainty in position, there corresponds an uncertainty in momen- 
tum, which is 

Ap ~ - (5.48) 

Instead of />, we now represent the momentum by (p Ap). We will 
first examine the case of (p + Ap) which is of special interest when 
T < U. Then the energy of the particle will not be represented by 
T = p*/2M but may be as much as 



T , = 



2M 



" "*" 



2M "" M "" 2M 

T + -vy Ap whenever Ap p 



" * "*" M a 

= T + * -; (5.49) 

This can be written as 

AT = 7" - T = -^7 (5.50) 

a/ V 

Here (a/F) is the time At required for the particle to travel a distance 
equal to the thickness of the barrier. Equation (5.50) is seen to be 
equivalent to AT At ~ A, that is, to Eq. (5.46). Suppose that the barrier 



62 The Atomic Nucleus [CH. 2 

is a = 10~ 12 cm thick (about the radius of a heavy nucleus) and that the 
particle is traveling at one-tenth the velocity of light (about the velocity 
of a 4-Mev proton) . Then 



hV 1 (fc/W)/F\ 

AT = - - = --- I I (m c 2 ) 

2w a 2ir a \c / 



~2Mev (5.51) 

This particle might therefore succeed in passing a barrier which is of the 
order of 2 Mev higher than its own kinetic energy. 

Conversely, we may consider the case of (p Ap). Here we have 
only to change the sign of the second term in Eq. (5.49). We obtain 
(T T r } = h/(a/V). The same particle therefore might fail to pass 
the barrier even if its own kinetic energy were of the order of 2 Mev 
greater than the barrier energy U. 

With the help of the Schrodinger equation, we can treat the problem 
quantitatively and can determine the actual reflection coefficient and 
transmission coefficient of the barrier. In the remainder of this section 
we emphasise the physical principles and the physical interpretation of 
the mathematical results. The corresponding algebraic details are 
carried out fully in a parallel treatment given in Sec. 1 of Appendix C. 

Wave Representation. The incident particles in region 1 are repre- 
sented by a plane wave moving in the direction of increasing z. The time- 
dependent factor in Eqs. (5.20) and (5.30) can be omitted, because 
v = W/h is a constant of the motion and therefore has the same value 
in regions 1, 2, and 3 of Fig. 5.1. Accordingly, the wave function for the 
incident particles can be written as 

tfw,,ii = A,c*** (5.52) 

where the subscripts 1 refer to region 1. The propagation number for 
the incident wave is 



, . 

The amplitude Ai of the incident wave could be taken as unity without 
loss of generality. However, AL will be retained in order to facilitate 
identification of the incident amplitude in subsequent equations. Equa- 
tion (5.52) is a solution of the wave equation, Eq. (5.31), in region ] , when 
ki has the value given by Eq. (5.53). 

The incident flux of particles is the probability density |^, nci de n t| 2 
multiplied by the group velocity Fi of the particles in region 1 ; thus 

Incident flux = ^ M \ t V l = I^ITi (5.54) 

Some particles will be turned back by the potential barrier. These 
reflected particles move toward the left in region 1 and can be represented 



5] Radius of Nuclei 63 

by a wave of amplitude Bi, propagated in the z direction, or 

" (5-55) 



The total disturbance in region 1 is then represented by the wave func- 
tion ^i, which has the value 

li = .4 !<"*" + Bic-*i* (5.50) 

This total wave function also is a solution of Schrodinger's equation for 
region 1, where U = 0. 

In region 2, under the barrier, we can expect a disturbance moving 
toward +z and also one reflected from the potential discontinuity at 
z = a and therefore moving toward z. The total disturbance could 
be written 

k *' z (5.57) 



where *', = JL^f-IL^) (5.58) 

n 

In region 2, the potential energy exceeds the incident energy T. Hence 
the kinetic energy (T U) is negative in region 2, and the propagation 
number kf is imaginary. It is mathematically convenient, but not 
mathematically necessary, to use in region 2 a real propagation number 
A- 2 defined as 

^/2M (U T) . f 

K/Z ^ = T " =: ifcz ^o.ijyj 

n 

Then kz is the wave number which would be associated with a hypo- 
thetical particle whose kinetic energy is positive and equal to the energy 
difference between the top of the barrier and the incident kinetic energy 
T. 

The disturbance under the barrier is then represented by 

1^2 = A z c~ kzZ + B 2 r k * s (5.60) 

Because kz is a real number, Eq. (5.60) shows that the disturbance under 
the barrier is n on oscillatory. 

In region 3, we can have a transmitted wave moving toward +z. 
There is no wave moving toward z because there is no potential change 
beyond z = a from which a reflected component could be produced. In 
other words, region 3 is a domain of constant refractive index from z = a 
to z = oo. The total wave function in region 3 therefore consists of a 
plane wave moving toward +z, with a propagation number ft 3 = fci, 
because U is zero in both regions. This gives for region 3 

^3 = Atf*!* (5.61) 

Boundary Conditions. The wave functions fa, $ 2 , and ^ 3 are the 
solutions of Schrodinger's equation in the three regions. Across the 
boundaries between these regions \j/ and d\fr/dz must be continuous. Only 
in this way can dfy/dz* remain finite across the boundaries and hence 
conform with noninfinite values of the total and potential energy W and 



64 The Atomic Nucleus [CH. 2 

U in Schrodinger's equation. Therefore the boundary conditions are 

fc.fc -*'-*l' atz = 

(5 ' 62) 



These boundary conditions give us four linear equations, from which 
the amplitudes A 2 , A 3 , BI, B 2 can be obtained in terms of the incident 
amplitude .-li. 

Transmission. The flux of transmitted particles, by analogy with 
Eq. (5.54), is for region 3 

\Ai\*V* (5.IB) 

where Fs is the group or particle velocity in region 3. 

The fractional transmission, or the probability for the transmission of 
a single particle through the barrier, is given by 



"" 



4 '' I r 

* i 3 " r 



3 



where we will call T ? the transmission coefficient, or, synonymously, the 
"transparency," of the barrier (BG8, F41). This is to be distinguished 
physically from a closely related quantity, the so-called penetration factor 
P, which is merely the ratio of the probability densities on the two sides 
of the barrier, i.e., 

P = {!-! (5-65) 

This systematic distinction between "transmission" and "penetration" 
follows the nomenclature adopted by Blatt and Weisskopf (B68) and is 
seldom found in the earlier literature, where "transmission" was often 
synonymous with "penetration," and usually (but not always) denoted 
T*. 

The penetration factor will arise later in connection with our dis- 
cussions of nuclear barriers and nuclear reactions [e.g., Eq. (8.fm)]. In 
general, T t = P(V^/Vi). It happens in the present problem that 
I's = Vi, because U = f or regions 1 and 3. In such special cases the 
transmission coefficient and the penetration factor are equal. 

In order to evaluate the transmission coefficient, we must determine 
A 3/Ai, the ratio of the transmitted amplitude to the incident amplitude. 
In general, A 3 will be complex, as will all other amplitudes except that 
of the incident plane wave AI. 

A more detailed discussion of the wave-mechanical treatment of this 
and other related barrier problems is given in Appendix C. It is shown 
that, when T < U, the exact solution of Eqs. (5.02) gives for the trans- 
mission coefficient of the rectangular barrier of Fig. 5.1 



5] 



Radius of Nuclei 



65 



The transmission coefficient Tj for the case in which the incident 
kinetic energy is T = 0.8 J7 is illustrated in Fig. 5.2. The barrier thick-- 
ness a is plotted in terms of the de 
Broglie wavelength Xi of the incident 
particles. Then 



while 




V2M(U - T) 



Hence k z a = ir(a/\i). It should he 
noted that the transmission coeffi- 
cient decreases slowly for barriers up 
to the order of a ~ Xi/4 (or l: 2 a ^ 1 ) 
in thickness. For barriers which 
are thicker than about a ~ Ai/2 (or 
Ar 2 a^2), log T z is seen to decrease 
substantially linearly as a increases. 
In this region, therefore, the trans- 
mission coefficient decreases ex- 
ponentially with increasing barrier 
thickness. 

Reflection. There are no sinks or 
sources of particles under the barrier 
of Fig. 5.1. Consequently those in- 
cident particles whirh are not trans- 
mitted must be reflected by the barrier, 
therefore, is 




1 2 3 4 5 6 
Barrier thickness aA, 

Fig. 5.2 The probability of transmis- 
sion TI df a rectangular barrier by parti- 
cles whoso kinetic energy is 0.8 of the 
barrier height. The thickness a of thp 
barrier is ^jven in terms of the do 
HrughV uuvolciifrth \i of the incident 
particles. The solid curve represents 
the exact expression, Eq. (5.66). The 
dotted line represents the approxima- 
tion lor thick barriers, as given by Eq. 
(5.70). 



The probability of reflection, 
(5.67) 



Reflectance = 1 L 

This relationship can be verified in detail by computing the amplitude 
BI of the reflected wave. The result of the computation turns out to be, 
as expected, 



or 



(5 . 68 ) 



We see that the probability of reflection is generally loss than unity and 
increases toward unity as the barrier becomes thicker. This is in sharp 
contrast with the classical model. In classical mechanics, the incident 
particles would all be turned back, or reflected, if T < T. Moreover, 
this reflection would occur sharply at the incident surface, z = 0, of the 
barrier. In contrast, the wave mechanics predicts a reflectance which 
depends on the barrier thickness. This means that the reflection process 
occurs not just at the incident surface, but within the barrier as well, and 



66 



The Atomic Nucleus 



[CH. 2 




also from the back, or emergent, surface, An analogous conclusion 
regarding the reflection and transmission of visible light by a metallic 
film or mirror is a well-known result in physical optics. In the limiting 
case of a very thick barrier, the classical and wave theories both predict 
100 per cent reflection. But the reflection is from the front surface 
alone in classical theory, whereas it occurs throughout a finite depth of 
the barrier in the wave-mechanical theory. 

Graphical Representation. It is helpful to visualize the boundary 

conditions and the general char- 
acter of the wave functions ^j, \[/ 2 , 
and ^ 3 in the three regions of Fig. 
5.1. 

To do so, we first note some 
general characteristics of any solu- 
tion ^ of Schrodinger's equation 



Fig. 6.3 Graphical representation of the 
total wave functions ^\ t ^ 2 , and ^3 in the 
three regions of Fig. 5.1. There are two 
vertical .scales. One is an energy scale, 
with respect to whieh the horizontal lines 
show the total energy W (equal in this 
case to the initial kinetic energy T) and 
the potential energies. The second verti- 
cal scale is the real part (or, alternatively, 
the imaginary part ) of the wave functions, 
which are plotted with respect to the en- 
ergy line W as a horizontal axis. This 
schematic representation of ^ can be re- 
garded as applying to some arbitrary value 
of time t, because the time factor e 2ir " f is 
common to all three regions. The bound- 
ary conditions are satisfied by making ^ 
go straight across the discontinuities of 
potential at z = and z = a. 



(5.09) 

In any region of positive kinetic 
energy (H' T), it is necessary 
that d z \l//dz* be of opposite sign to^. 
This- condition requires that, if ^ is 
positive, its slope must decrease us 
z increases. Similarly, if ^ is nega- 
tive, d\///dz must increase as z in- 
creases. This requires at once that 
^ be an oscillatory function of z. 
In Fig. 5.3, we therefore portray ^i 
to the left of the barrier as an os- 
cillatory function. Because the 
reflected component has a complex 
amplitude Si, we can regard Fig. 



5.3 as a representation of the real 
part (or, alternatively, the imaginary part) of \f/i. 

Inspection of the Schrodinger equation also shows that, in any region 
of negative kinetic energy U > W, it is necessary that 6>V/dz 2 and ^ be 
of the same sign. Then, if ^ is positive, its slope will increase as z 
increases; that is, ^ must always be convex toward the origin of coordi- 
nates. Then ^ is not oscillatory but will have general features similar 
to those of an exponentially decreasing function. In Fig. 5.3, the wave 
function ^ 2 in the region under the barrier is shown as such a function. 

At z = 0, the boundary conditions require that ^ and its slope d\l//dz 
have the same values in regions 1 and 2. Therefore the curves represent- 
ing ^ in the two regions must be joined at z = in such a way that they 
pass straight across the potential boundary. Thus the boundary conditions 
are easily visualized graphically, as shown in Fig. 5.3. 



5] 



Radius of Nuclei 



67 



At 2 = a, ^ must again pass straight across the potential boundary. 
In region 3, the kinetic energy again becomes positive. Therefore ^ 3 
is an oscillatory function but of smaller amplitude than ^i, as shown in 
Fig. 5.3. 

It is to be emphasized that ^i is not the incident wave but is the sum 
of the incident and reflected waves. Hence the real part of ^i is not 
necessarily a pure sinusoidal curve, but it does have to be oscillatory. 

Thick Barriers. In many of the cases which are of practical interest 
in nuclear physics, the barrier thickness a is large compared with the 
de Broglie wavelength X 2 = 27r/fc 2 of a particle whose energy is (C7 T). 
This condition corresponds to fc 2 a ^> 1 in Eq. (5.66). Such barriers 
could be described as either "thick" (large a) or as "high" (large /c 2 ). 
For k^a ^> 1, the exact relationship 
Eq. (5.66) can be represented with 
good approximation by 



(5.70) 



T, = 16^(1 - 




Fig. 6.4 Generalized potential barrier 
V(z) for a particle whose total energy is 
W. 



For T/U not too close to or 1 , the 
coefficient of the exponential term 
is of the order of unity. The domi- 
nant term it- the exponential. A 
plot of Eq. (5.70) for T = 0.817 is 
shown as the dotted line in Fig. 5.2, where the domain of validity of 
the approximation can be seen clearly. 

The exponential term in Eq. (5.70) can be derived by an entirely 
different method. Approximate solutions of the wave equation can 
be determined by the so-called Wentzel-Kramers-Brillouin (W.K.B.) 
method, if the potential U(z) does not vary too rapidly with z. Then an 
approximate solution of the wave equation can be obtained for barriers 
-jf arbitrary shape, such as the barrier shown in Fig. 5.4. The trans- 
mission coefficient for such barriers can be written as 

I,-*- (5.71) 

where the dimensionless exponent 7 Is known as the barrier transmission 
exponent. Then the W.K.B. method leads to the following approximate 
general solution for 7 



7 = V2M 
h 



[U(z) - 



(5.72) 



dz 



Here Zi and z 2 are the distances between which the barrier height U(z) 
is greater than the total energy W of the incident particle. _ 

For the special case of the rectangular barrier, VC7(z) W is con- 
stant and equal to VC7 T. Then Eq, (5.72) integrates directly to give 



68 The Atomic Nucleus [CH. 2 



y = 2a =- V2M(U - T) = 2Jfc 2 o (5.73) 

Ft 

wliich is in agreement with Eqs. (5.70) and (5.71). 

The integration of Eq. (5.72) can be carried out analytically for 
certain simple potentials, such as the coulomb barrier combined with a 
rectangular well. For more complicated potential barriers, Eq. (5.72) 
is evaluated by numerical integration. 

g. Transmission of Particles through a Nuclear Coulomb Barrier. 
The approximate expression for barrier penetration, as given by Eq. 
(5.72), is in one dimension. We must now go over to three dimensions, 
in order to evaluate the radial transparency of a nuclear coulomb barrier 
for a charged particle. We shall find that for certain restricted but very 
important cases the radial transmission coefficient can also be obtained 
from Eq. (5.72). 

It can be seen from Eq. (5.72) that the transmission exponent 7 for 
the radially symmetric barrier is the same for incoming and for outgoing 
particles. Thus nuclear disintegrations by charged particles and a-ray 
radioactive decay are based on the same general theory concerning the 
transmission of nuclear potential barriers. 

Wave Equation in Spherical Polar Coordinates. For the three-dimen- 
sional coordinate .system, it is most convenient to choose spherical polar 
coordinates, r, tf, <?. If the potential U depends only on r, and not on 
tf and v?, then it is possible to find wave functions ^ in which the variables 
r, tf, if> appear only in separate functions. Thus 

* = /2(r)e(0)*(*0 (5.74) 

where the " radial wave function " /?(r) depends only on r, the polar func- 
tion (#) depends only on tf, and the azimuthal function $(p) depends 
only on ^. For such a wave function, Schrodinger's equation can then 
be separated into three differential equations, one in r and /?(r), one in 
# and ft(tf), and one in ^ and #(^)- 

Modified Radial Wave Equation. Of these three differential equations, 
the radial equation is of direct interest here. The separation of the three- 
dimensional wave equation is carried out in Appendix C, Sec. 2, where 
it is shown that the radial wave equation is 



(5.75) 

This equation does not involve tf or <p, and the two companion differential 
equations^ one in tf and one in ^, do not involve r. 

It is mathematically convenient to use a modified radial wave Junction 
X defined as r times the radial wave function, or 

X = r R(r) (5.76) 

Upon substituting R(r) = \/r into Eq. (5.75), algebraic simplifications 
occur, and we obtain the simpler and more useful modified radial wave 



5] Radios of Nuclei 69 

equation, which is 



In Eqs. (5.75) and (5.77), r is the radial distance measured from the 
origin of the potential l'(r) } such as the center of a nucleus, and M is the 
reduced mass of the colliding particles, or of the disintegration products, 
whose separation is r. The quantity 1(1 + 1) in Eqs. (5.75) and (5.77) 
arises purely from the mathematical operation of separating the wave 
equation into radial and angular equations. In this operation it is found 
that the separation constant can have only the values 0, 2, 0, 12, 20, 
'JO, ... in order that the solution 0(0) of the polar equation (Legendre's 
equation) be finite. These numbers are conveniently represented by the 
quantity 

/(/ + j) 

where the index / is zero or a positive integer, I = 0, 1, 2, 3, . . . . The 
modified radial wave equation has a separate solution xi for each value 
of the index /, and there are corresponding solutions Si for the polar 
equation. The mathematical details are given in Appendix C, Sec. 2. 

Centrifugal Barrier. Comparison of Eq. (5.77) with Eq. (5.69) shows 
that the modified radial wave equation is markedly similar to the one- 
dimensional wave equation. However, in the three-dimensional case 
the potential is replaced )>y the quantity 

' 



The second term, therefore, has the dimensions of energy. In its denom- 
inator, the quantity Mr" will be recognized as the classical moment of 
inertia for two particles whose reduced mass is M and whose separation 
is r. In classical mechanics, rotational kinetic, energy can be written as 
J 2 /2/, where / is angular momentum and / is moment of inertia. Then 
by dimensional reasoning we can identify the second term in Eq. (5.78) 
as associated with the rotational kinetic energy of the two particles about 
their center of mass. Because this term has the same sign as the poten- 
tial energy I T (r) and thus physically augments the potential barrier, the 
quantity 

1 



is known as the centrifugal barrier in collision problems and in disintegra- 
tion problems. A schematic diagram of the centrifugal barrier will be 
found in Fig. 10 of Appendix C. 

Angular Momentum. Comparison of Eq. (5,79) with the classical 
expression for rotational kinetic energy J 2 /2I shows that a portion of 
Eq. (5.79) can be identified as the angular momentum J. Thus the 
magnitude of the angular momentum of the wave-mechanical system of 



70 The Atomic Nucleus [CH. 2 

two particles is taken as _ 

Jt = Vl(l + 1) h (5.80) 

In contrast with the classical angular momenta MVx as used in 
collision theory, the angular momentum of the quantum-mechanical 
system can have only the discrete quantized values given by Eq. (5.80) 
with Z = 0, 1, 2, 3, . . . . In both theories, of course, the angular 
momentum is a constant of the motion. 

Because the index I physically determines the angular momentum of 
the system, we call I hereafter the angular-momentum quantum number. 
Note that the magnitude of the angular momentum is not /A, as in the 
older quantum theory, but is Vl(l + 1) h. 

Plane Wave in Polar Coordinates. In collision problems, we express 
a collimated beam of monoenergetic particles as the usual plane wave 
e lk *. Such a plane wave represents a mixture of particles which have 
all possible values of angular momentum with respect to any scattering 
center being traversed by the wave. We need to locate the origin of 
spherical polar coordinates at some position to be occupied later by a 
scattering center arid then find an expression for the plane wave in 
those coordinates. In this way we express the plane wave as the sum 
of a number of " partial waves. 77 Each partial wave must be a solution 
of the wave equation when the scattering potential V(r) is zero, i.e., for 
uniform motion. Each partial wave will be characterized by a particular 
value of I and will therefore correspond physically to those particles in 
the incident beam for which the angular momentum about the scattering 
center is ^/l(l + 1) h. The sum of all the partial waves, for / = to 
I = , must equal the plane wave e ikz = e l * rcoB *. 

It is shown in Appendix C, Eq. (75), that the representation of the 
plane wave which satisfies these conditions is 

(21 + \)i l ji(kr) P,(cos tf) (5.81) 



j-o 



Here ji(kr) are the spherical Bessel functions of order I, and P z (cos tf) 
are the Legendre polynomials of order L 

A correlation between the I values of the partial waves and the 
angular momentum associated with classical impact parameters x can 
be made with the help of the uncertainty principle. It is shown in 
Appendix C, Sec. 4, that in a classical coulomb collision it would be 
impossible to obtain a precise experimental verification of the relation- 
ship between the classical impact parameter x and the deflection in an 
individual coulomb collision. The minimum uncertainty in the impact 
parameter (Ax) min for a particular observed deflection would be 



Then central collisions could be regarded as extending from x = to at 
least x ~ \ and thus including classical angular momenta between 



5] 



Radius of Nuclei 



71 



and at least J = MVx ~ ft. These correspond to the ''central collisions" 
I = of the wave theory. Similarly, collisions whose classical angular 
momentum lies between MVx ~ Ih and MVx ~ (/ + 1)A correspond to 
the Z-wave collisions of the wave theory, for which the quantized value 
of the angular momentum is the geometric mean between Ih and (I + l)ft, 

that is, VT(7 + 1) h. These values can be visualized from Fig. 8 of 
Appendix C. 

The individual spherical partial waves are designated by their numer- 
ical / values, or more commonly by borrowing the Rydberg letter notation 
from atomic spectroscopy. This is 



I 





1 


2 


3 


4 


5 




















Letter designation 


s 


V 


d 


f 





h 





The s Wave. Transmission through a barrier is, of course, most 
probable for those particles which have central collisions. In these cases 
no energy is "wasted" as rotational energy, and all the initial kinetic 
energy is available for attacking the potential barrier. The collisions 
which have no angular momentum are the I = or s-wave collisions. 
The s wave from Eq. (5.81) has the simple form 



sin AT 


kr 



(5.82) 



The s wave is the only partial wave which has no dependence on # and 
it is therefore spherically symmetric. The modified radial wave func- 
tion, for the s wave, is then 



Xo = r # (r) = - 



sin kr 



k 



(5.83) 



Transmission through a Nuclear Coulomb Barrier by s Wave.. For the 
I = partial wave, the modified radial wave equation becomes 



[W - t^(r)]xo = (5.84) 

14' III 

This is identical with the one-dimensional wave equation. The prob- 
ability that a particle will be found in a volume element between r and 
r + dr is 



We can therefore use the one-dimensional integral in Eq. (5.72) to calcu- 
late the radial transparency for s waves. 

Let the nuclear potential barrier be a coulomb barrier, cut off at the 
edge of an inner rectangular well, and given by 



t/W - o 



r < R 
r> R 



(5.85) 



72 



The Atomic Nucleus 



[CH. 2 



where R is the nuclear radius. The integration of Eq. (5.72) can be 
carried out explicitly for this potential. The general result is developed 
in Appendix C, Eq. (95), and is 

SrZze 2 



T " hV 

where B = Zze*/R = coulomb barrier height 

T = ?MV 2 = total kinetic energy of particles in C coordinates 

when widely separated 
M = reduced mass 

V = mutual velocity of approach or recession 

Figure 5.5 shows the behavior of the I transmission coefficient 
To = e~ 7 as given by Eq. (5.86) for three representative elements of low, 

medium, and large nuclear charge 
(Al, Sn, and U), using protons and 
a rays as the incident particles. In 
each case the effective radius has 
been taken as 



i^ 
. 





1.0 



O 

bo 



si - 6 

V 

| 04 

S 

.5 0.2 

I 




0.2 0.4 0.6 0.8 1.0 
(Kinetic energy )/( barrier height ) = T/B 

Fig. 6.6 Approximate barrier transpar- 
ency TO for s waves as given by Eq. 
(5.86). Curves are for protons and a 
rays passing through the coulomb bar- 
riers of aluminum (isAl 27 ), tin LoSn 118 ), 
and uranium (gall 288 ), on the assumption 
that these nuclei have radii R = 1.5 X 
10-" A* cm. 



R = 1.5 X 10-A cm 

The transmission exponent 7 
takes on a simpler form for the 
physically important case of a 
1 ' high " barrier. When the kinetic 
energy T is small compared with 
the barrier height B, the transmis- 
sion coefficient T = e~ 7 is given to 
a good approximation by 



7 = 



Sir , ,^ X1 
- (2Zze*MR)* 



(5.87) 



This expression can be used as a 
reasonable approximation for the 
treatment of a decay in the heavy elements, where B ~ 25 Mev and 
T ~ 5 Mev ~ B/5. The physics of Eq. (5.87) is portrayed more clearly 
by rearranging the variables, so that the barrier transmission exponent is 
given by 



7 = v\ 






where 



= 2ire 2 /hc = fine-structure constant 

=r V/c = relative velocity of particles Ze and ze } in terms of 

velocity of light 

= reduced mass in terms of rest mass ra of electron 
/j/ ro = nuclear radius in terms of classical electron radius r = 
= 2.818 X 10~ 13 cm 



5] 



Radius of Nuclei 



73 



In many practical cases, the first term in Eq. (5.88) predominates. 
We see that when the charge parameter 2Zz/137/9 is large, the trans- 
mission is small, and the classical limit of no barrier transmission is 
approached. 

When the first term is used alone, the approximate barrier trans- 
parency, for s waves through a very high or thick barrier, is called the 
Gamow factor G which is 



0) 'S G ~ 



(5.89) 



The Gamow factor corresponds to the transparency of a coulomb 
barrier which has zero internal radius. This can be seen by setting 




Fig. 5.6 Approximate physical interpretation of the coulomb-barrier transmission 
exponent -y. The Gamow factor corresponds to the integration of Eq. (5.72) ail the 
way to the origin r = 0. This is the entire shaded region, above T, and extending 
from r = to 6. The region shown with dotted shading corresponds to the deduction 
to be made because of the presence of the inner potential well between r and R, 
This dotted part corresponds to the second term in Eq. (5.88) and portrays the 
reduction of y as produced by R. The difference between the two terms corresponds 
to the integral through the region shown by the unbroken shading from r *= R to b. 
Note that the integral in Eq. (5.72) involves the square root of elements of area in 
this figure, rather than simply the area itself. 

R = 0, hence B = in Eq. (5.86), or by setting R = in Eq. (5.87). 
With this in mind, the two terms in Eqs. (5.87) and (5.88) for the barrier- 
penetration exponent can be given an approximate physical interpreta- 
tion which can be visualized as in Fig. 5.6. The first term corresponds 
to the integration of Eq. (5.72) from r = 0tor = b = Zze^/T, that is, 
over the entire shaded region in Fig. 5.6. The second term corresponds 
to the integration from r = to , which is shown as dotted shading 
in Fig. 5.6. This term is of opposite sign to the first term and corre- 



74 



The Atomic Nucleus 



[CH. 2 



spends to the reduction in 7 on account of the presence of the inner 
potential well. All the dependence of 7 on the well radius 7? is seen to be 
contained both mathematically and physically in this second term. 

Figure 5.7 makes use of the net shaded area of Fig. 5.6, in order to 
provide a visualization of the physical effect of changes in the param- 
eters Z, z, R, V, M, and T. 

Penetration When I j 0. When the angular momentum is not zero, 
the penetrability of a nuclear coulomb barrier is much more complicated 
than it is for s waves. If Z is large, the effects due to I are small in 



-Increase in Zor z (less transmission) 
^Increase in R (more transmission) 




.Increase in V, M or T 
_ ( more transmission ) 



Fig. 6.7 Pictorial representation of the effects of Z, z, R, V, M, and T on the net 
shaded area of Fig. 5.(i and hence on the barrier transmission. 

comparison with the profound dependence of 7 on V and on R. In 
general, the penetrability decreases as I increases. Some approximate 
analytical expressions have been deduced, and tables which arc more 
accurate than these have been prepared by numerical integration. These 
expressions and comments on the tables are given in Appendix C, Sec. 5. 

Problem 

Compute from the approximate formula Eq. (5.87) the transmission coeffi- 
cient for at least one of the barriers in Fig. 5.5 and compare with the values 
plotted. For about what values of T/B is the approximate formula acceptable? 

6. Lifetime of a-Ray Emitters 

We return now to reconsider the impasse which classical nuclear 
theory faced in 1928. It was shown experimentally that 8-Mev a rays 
are elastically scattered by a pure coulomb field surrounding the uranium 
nucleus but that the same nucleus emits only 4-Mev a rays in radioactive 
decay. The experimental situation was summarized in Fig. 4.2. 

In 1928 Gamow and, independently, Gurney and Condon first 
applied the wave mechanics to the problem of a decay (G2, G50). It 
was shown that the wave model does not oblige the a ray to go over 
the top of the barrier but allows the a ray to pass through the barrier 
instead. 



6] Radius of Nuclei 75 

As we have just seen, the qualitative model of wave penetration can 
be quantified if certain simple potential shapes are assumed. Equation 
(5. 80) gives us the basis for a quantitative theory of some aspects of a 
decay, if we are willing to represent the potential energy between a 
nucleus and an a ray by a coulomb potential which is cut off at a radius 7? 
by a rectangular potential well of poorly defined depth. Actually, we 
shall find that the experimental data agree so well with this simple model 
that we must accept it as a reasonable approximation to the ultimate 
truth. 

We have noted that Eq. (5.80) is valid only for the transmission of & 
waves through the barrier. Happily, the largest and most important 
class of rt-ray emitters does correspond to I = 0. These are the nuclei 
which have even atomic number Z and even mass number A. There 
are an even number of protons and an even number of neutrons in such 
nuclei and also in their decay products after the emission of an ray 
(helium nucleus, 2 IIe 4 ). Such "even-Z even- -4" nuclei are found quite 
universally to have zero total nuclear angular momentum (Chap. 4). 
Because angular momentum is conserved in all types of nuclear reactions, 
the a rays emitted in transitions between the ground levels of even-Z 
even--! nuclei must be emitted with / = 0, that is, as s waves. 

When the experimentally known values of a decay energy, T ~ 5 to 
8 Mev, are substituted in Kq. (f).S(>), with R 10" 1 - cm, the transmis- 
sion coefficient T = r" * is found to extend over a domain of about 
10"- to I0~ 4n . This range is just what is needed to relate T and R to 
the broad domain of known a decay half-periods. These extend from 
1.39 X 10 l(l yr for thorium, down to 0.8 sw for ThC', or a spread of 
about 10--\ 

a. Radioactive Decay Constant. The half-period for radioactive 
decay T is given by 



where the so-called radioactive decay constant X is the probability of decay 
per unit time for one nucleus. It is known experimentally (Chap. 15) that 
X is a constant for any particular type, of nucleus and especially that X 
is independent of the age of the iiu.-Jeus. 

For a particular type of radioactive atom, say radium, we can express 
X as 

X = X,,r-T (6.2) 

where e~' Y is the barrier transmission coefficient and Xn is the decay 
constant without barrier. Both X and X have dimensions of sec~ J . 

Decay Constant without Barrier. Many investigators have developed 
theoretical estimates of Xn. These differ from one another because of the 
variety of nuclear models which have been assumed and because the 
calculations for any particular model have been done with varying 
degrees of approximation. In many models, X is related to the average, 
spacing between energy levels in the nucleus (p. 573 of BOS). The 



76 The Atomic Nucleus [CH. 2 

various models give different results for Xo which are usually in agreement 
with experiment only within a factor of about 10 s . At present an empir- 
ical value for X is usually the best choice for quantitative work. Then 
the variation of T with the kinetic energy T and with R is given very 
accurately by the e~ T term. 

A useful rough estimate of X can be obtained in the following way: 
The a particle within the parent nucleus is regarded as a standing wave 
corresponding to a particle having some velocity y iM i d e. This particle 
would then hit the barrier about F in-ld e/-K times per second, which is a 
crude estimate of X . 

In addition, we must include some factor which represents the prob- 
ability that the a particle exists as a preformed particle in the nucleus. 
This probability is currently estimated (C28) as lying in the domain 
between 0.1 and 1 for nuclides which have even-Z and even-^4. Then 
we can express Xo roughly as 

Xo (probability that preformed a particle exists in nucleus) 

X (rate of hitting barrier) 



=* (0.1 to 1) X ^-===J (6.3) 

The equivalent velocity inside the parent nucleus can be estimated 
in various ways. One simple method is to associate F inllldB with the 
velocity of a nucleon which is confined to a region of space whose dimen- 
sions are comparable with the average spacing between nucleons. Then 
the uncertainty principle gives 

Ap = ~ M Ftau. (6.4) 



On this basis V lomid9 is of the order of 3 X 10 9 cm/sec if Ax is of the order 
of 2 X 10~ 13 cm and M is a proton or neutron. Then, roughly, 



Xo^(O.ltol) X( p) (6.5) 

\K/ 

^ 10* cm/sec 

~ 10- 12 cm 

~ 10 21 sec- 1 (6.6) 

Figure 6.1 illustrates the strong dependence of the half -period for a 
decay on nuclear radius. It will be noted that a 10 per cent change in R 
produces about a 40-fold change in the decay constant and half-period. 
Because of this very strong exponential dependence of X on R, it is 
possible to obtain very close estimates of the effective nuclear radius R 
even though X is known only approximately. 

b. Empirical Evaluation of Nuclear Unit Radius. We may now repre- 
sent the decay constant of Eq. (6.2) in the form 

X = Xoe~* 



6] 



Radius of Nuclei 



77 



where a and b are known functions of Z, 2, 7, M, and A given by Eq. 
(5.86) or (5.88). Then each measured case of a decay provides experi- 
mental data which make Eq. (6.7) an equation in two unknowns, the 
empirical constant X and the nuclear unit radius RQ. Consequently, 
data on any two suitable cases of a decay permit an empirical evaluation 
of the nuclear unit radius. When such calculations are carried through, 
values in the neighborhood of RQ = 1.5 X 10~ 13 cm and X ^ 1 X 10 21 
sec" 1 are obtained. In Fig. 6.1, the 
empirical value X = 1.2 X lO^sec* 1 
was used. 

A systematic study of the data 
on all known cases of a. decay in 
even-Z even- A nuclides has been 
made by Perlman and Ypsilantis 
(PI 6) . They calculated the nuclear 
radius R which would correspond to 
the observed value of half-period and 
a decay energy for these nuclei. 
The exact integral for 7 as given by 
Eq. (5.86) was used, and X was taken 
as simply V/R, where V is the ve- 
locity of the emitted a ray, with re- 
spect to the residual nucleus. The 
calculated radii show close agreement 
with the constant-density model 
R = RoA*. For all parent nuclei 
with Z > 86, the unweighted aver- 
age value of R Q for the daughter 
nucleus is 1.49 X 10~ 13 cm. Con- 
sidering the direction and magnitude 
of the most common experimental 
errors in the measurement of a decay 
energies, Perlman and Ypsilantis 
gave greater weight to the relatively few energy determinations which have 
been made by magnetic analysis and thereby selected as the best value 



If-period of radium m years 

i 
p 

3855 s 


\ 










\ 


\ 










\ 


^ 










^ 


.162Ch 
asobj 


rears, 
erved 






Theoretical ha 








\ 












\ 



1.0 1.2 1.4 1.6 1.8 2.0 
Nuclear unit radius R Q in 10" !1 cm 

Fig. 6.1 Illustration of the strong 
dependence of the theoretical half- 
pcriod for a decay upon small changes 
in the nuclear radius. The curve shows 
how rapidly the predicted half-period 
for radium varies with one's choice of 
the nuclear unit radius R in the con- 
stant-density model R = R A*. Note 
that the half-period T\ varies by 10 12 
when RQ varies by a factor of 2. 



R = 1.48 X 10- 13 A 4 cm 



(6.8) 



The radius R and mass number A refer to the decay product. 

Figure 6.2 presents their results. The points are experimental; the 
curves are theoretical. General agreement is evident. Those points 
which are below the curves, especially U 238 and Th 232 , may be due to small 
negative experimental errors in the values of the a decay energy. If 
the measured values are correct, then these two nuclei have radii which 
are about 4 per cent greater than Eq. (6.8). 

In Fig. 6.2, only the a-ray transitions between nuclear ground levels 
are plotted. However, the transitions to excited levels in the product 
nucleus also show excellent agreement with Eq. (6.8) among the even-Z 
even- A a emitters. The systematics of a decay in odd- A nuclides shows 



78 



The Atomic Nucleus 



[CH. 2 



a marked dependence on several factors besides R and T and is discussed 
in connection with Fig. 4.3 of Chap. 16. 

Shell Structure. The polonium isotopes (Z = 84) and seRn 212 are not 
plotted in Fig. 6.2. These nuclei show abnormally small radii, with 
deviations from Eq. (6.8) amounting to between 1 and 9 per cent. These 
deviations are probably real. In Chap. 11 we will discuss the abundant 
evidence for the existence of a shell structure in nuclei and for closed 
shells containing 82 or 126 nurleons. Nuclei containing 82 protons are 




-12 



4.0 



4.5 



7.5 



5.0 55 6.0 6.5 
a -Disintegration energy in Mev 

Fig. 6.2 Consist c a iu-y of the radii of even-Z even- ,4 nuclei with the constant-densitx 
model R = R*A*. The experimental points show the observed values of the half 
period and total decay energy for each nuclide. The mass number of the radio- 
active parent nuclide is given for each point. The curves represent the theoretical 
values for radioactive parents having the same atomic number (isotopes) and are 
drawn (PIG) from a decay theory, using 7 as given by TCq. (5.80) when 7? = 1.-18 X 
10~ 13 A* cm, and with X taken as simply V/R. These effective radii 7? refer to the 
daughter nuclides. Note that the values of Z and A given in the figure are for the 
parent nuclides. 

found to be especially tightly bound, and they are expected to have 
radii which are slightly smaller than the "standard" radii. The polo- 
nium isotopes decay by a-ray emission to lead (Z = 82). Therefore the 
effective radii for these transitions are small. 

The nuclide 8 6 Kn 212 is an isotope of radon which is produced in the 
spallation reaction of 340-Mev protons on thorium. In 8 r.ltn 212 , the 
neutron number is 12G, which corresponds to a closed neutron shell in the 
parent nuclide. The calculated radius for a decay of this nuclide is 
about 5 per cent less than the "standard" radius. Thus closed shells 
in either the parent or daughter nucleus do affect R measurably, but 
only by a few per cent. 

c. Effect of Finite Radius of the a Particle. The effective radius R 
of the inner rectangular potential well is a fictitious radius which was 



6] 



Radius of Nuclei 



79 



introduced in order to make Eq. (5.72) integrable, so that Eq. (5.86) 
could be obtained. We have seen that, when X is taken as V/R, this 
"well " radius is about 72 = 1.48 X lO^Ml cm. Other values of R cor- 
respond to other models for X . Thus Preston (P31) finds J? ^1.52 
0.02 X 10~ 13 cm when R and the inner potential well depth ?7 are 
determined from the decay energy T and the half-period. A more elab- 
orate analysis by Devaney (D33, 
B68) leads to R Q = 1 .57 X 10~ 13 cm 
when Xo is determined by the level 
spacing in the parent nucleus. 

Blatt and Weisskopf (p. 574 of 
H68) have pointed out that the cor- 
rected radius R A of the daughter 
product will be a little smaller than 
R. This is because of the Unite 
effective radius R a of the a particle. 
Neutron scattering experiments 
(D14) have indicated that the 
radius of the helium nucleus is 
about R a ~ 2.5 X 10* 13 cm. The 
short-range attractive nuclear forces 
will be felt when the effective sepa- 
ration of centers between the 
daughter nucleus and the a particle 
is anything less than 




R' = R a + RA (0.9) 

This distance K is the radius at 
which the corrected potential bar- 
rier begins to be lower than the 
assumed barrier, as indicated by the 
dotted potential curve in Fig. 6.3. 
The effective well radius R lies 
somewhere between R f and R A . It 
can be expressed analytically as 



R = RA + 



(6.10) 



Fig. 6.3 Schematic representation of the 
relationship between 1 he corrected nuclear 
radius RA, the effective radius R of the 
rectangular potential well, arid the effec- 
tive radius R a of the a particle. The 
corrected potential (dotted) breaks away 
from the assumed potential at r = 
RA + R a = R f . The particle and the 
daughter nucleus are still partially im- 
mersed in each other when the separation 
of their centers is R. The potential well 
radius can be expressed as R RA + p, 
with Ptt ~ 1.2 X 10- 13 cm. 



where p a is a semiempirical radius correction for the particle. We can 
eliminate R A between the last two equations to obtain 

R a - p tf = R f - R (6.11) 

The choice of p a is somewhat arbitrary. The value 

Pa = 1.2 X 10~ l3 cm (6.12) 

has been picked by Weisskopf (W21, B68) as being reasonable. This 
quantity also enters the interpretation of nuclear barrier transparency 
to bombarding a particles, as in (a,ra) reactions, where p a = 1.2 X 10~ ia 
cm gives reasonable agreement with observations (Chap. 14, Fig. 2.9). 



80 The Atomic Nucleus [CH. 2 

We can relate the corrected radii R A to the effective well radius R 
by means of Eqs. (6.10) and (6.12). Then 

R = R A + Pa =- RuA* + Pa 

I (6.13) 

where R OA is the unit corrected radius. For the heavy nuclei, .4* ~ 6; 
hence 

R -- (ff M + 0.2 X 10-' 3 ),4i cm (6.14) 

If we take the -decay data as corresponding to 7? ~ (1.5 0.1) X 
10 I3 .4i cm, then the unit corrected radius R (}A of the daughter product 
whose mass number is A becomes 

R l}A ~ (1.3 0.1) X 10- 13 cm (6.15) 

which is probably the best value now available from the appraisal of 
a-decay data. 

d. Electromagnetic Radius in Decay. W note from Eq. (6.15) 
that the effective radii R A for decay have a somewhat larger unit 
radius than would be expected from the electromagnetic unit radii deter- 
mined, for example, from /i-mesonic atoms, Eq. (3.1 1 ). A possible recon- 
ciliation of the two results has been discussed by Hill and Wheeler (1153) 
in terms of the so-called collective model of nuclei. The finite electric 
quadrupole moment found for many nuclidcs (Chap. 4) suggests that 
truly spherical nuclei are rare, and that most nuclei are slightly ellipsoidal, 
including many which have eveii-Z and cveii-A. From an ellipsoidal 
nucleus the a decay probability would be slightly greater than from a 
spherical nucleus of equal volume, because the a decay rate is really a 
function of both the average nuclear radius and the ellipticity. 

Hill and Wheeler have estimated that an ellipsoidal deformation which 
stretches the nuclear axis by 10 per cent would result in about a 16-fold 
increase in the a decay rate, as compared with a spherical nucleus of the 
same volume. Thus if the electromagnetic radius means the radius of a 
sphere whose volume equals that of the ellipsoidal nucleus, the ellipsoidal 
shape enhancement of the a decay rate would emerge from conventional 
a decay theory in the form of an apparent nuclear radius R A which is 
larger than the corresponding electromagnetic radius. Measurements 
of the quadrupole moments, and of the /i-mesonic "X rays/' for nuclides 
heavier than Bi 209 are clearly needed for the future quantitative clarifica- 
tion of this topic. 

Problems 

1. Use the uncertainty principle to show that a nudeon which is confined to 
a region of the order of Ax ~ 1 X 10~ 13 cm must have a velocity of the order of 
6X10* cm/sec and a kinetic energy of the order of 20 Mev. 

2. Calculate empirical values of X and R , as indicated by Eq. (6.7), using 
the data for any pair of the following carefully measured cases of a decay. 



7] 



Radius of Nuclei 



81 



Parent 


Daughter 


T a , Mev 


T, Mev 


Half-period 


X f sec- 1 


, B Ra 
,,Rn" 

, 4 RaA B 


,.Rn'" 

H4 RaA" 
B2 RaB" 4 


4.777 
5.486 
5.998 


4.863 
5.587 
6.110 


1620 yr 
3.825d 
3.05min 


1.27 X 10-" 
2.097 X 10- f 
3.78 X 10-" 



NOTE: The tabulated X for Ra corresponds to the approximately 94 per cent of the 
disintegrations which go to the ground level of Rn. These correspond to a, "partial 
half-period" of about 1,740 yr, if the tot&l half-period is 1,620 yr. 

3. Radium has been .found to emit a small percentage of 4.593-Mev a rays 
in addition to the main group whose energy is 4.777 Mev. This weak group of 
a rays corresponds to transitions to a 0.1 87- Mev excited level in radon. Esti- 
mate from a decay theory what fraction of the transitions should go to the excited 
level, if I = 2 for this transition and I for transitions to the ground level. 

4. Justify the extremely strong dependence of X on T and R by showing that 
the fractional change in X equals the absolute change in % that is, 

AX 



if Ay is small. 

7. Anomalous Scattering of a Particles 

We have seen that the elastic scattering of 4- to 8-Mev a rays by 
heavy nuclei follows the Rutherford scattering law (Chap. 1, Fig. 3.3). 
This scattering is therefore due to a simple inverse-square coulomb force 
between the scattering particles. In these classical experiments the 
numerical value of the parameter 2Zz/1370 was of the order of 50. 
Therefore the transparency of the nuclear barrier was negligible (Eq. 
5.88). 

In experiments conducted at small values of 2 Zz/ 137/3 some barrier 
transmission will occur. Then mechanisms other than the coulomb 
interaction will also become effective, and the observed scattering will 
become more complicated. The name anomalous scattering applies to 
all instances of elastic nuclear scattering in which any type of deviations 
from pure coulomb scattering is observable. Several separate nuclear 
mechanisms often conspire to produce these deviations. 

a. Classical Model of Anomalous Scattering. Rutherford was among 
the first to recognize that an estimate of the size of nuclei could be 
obtained by determining the smallest a-ray energy which would produce 
anomalous scattering. Classically, anomalous scattering should set in 
at an a-ray energy which is just large enough to bring the incident a ray 
to the edge of the scattering nucleus. In a strictly head-on collision, 
this would occur when the initial kinetic energy ?MV* in the center-of- 
mass system is just equal to the barrier height B = Zze*/R. Then the 
classical collision diameter b = 2Zze*/MV* would be a rough measure of 
the nuclear radius R. 

For scattering angles which are leas than 180, the minimum separa- 



82 The Atomic Nucleus [CH. 2 

tion p min , or closest distance of approach, between the particles is greater 
than the collision diameter b and is given classically by [see Appendix B, 
Eq. (82)] 



where is the deflection in the center-of-mass coordinates. Then for 
@ = 180, p min = 6; but p min = 26- when 6 is about 39, and p niin = 46 
when @ is about 16.5. 

Classically, anomalous scattering should set in at an or-ray energy 
which depends on the angle of scattering; 8 and which has a minimum 
value just equal to the barrier height if = 180. We shall ,see that 
the experiments contradict both these classical predictions. As the 
energy of the incident particles is increased, anomalous scattering actu- 
ally makes its appearance almost simultaneously at all angles of scatter- 
ing. Moreover, the onset occurs at a-ray energies which are well below 
the barrier height. 

In his observations of the scattering of a rays by hydrogen, Ruther- 
ford (R45) first studied anomalous scattering as early as 1919. Through- 
out the subsequent decade Rutherford inspired a number of workers who 
collected fundamental but puzzling data on the anomalous scattering of 
a rays by a number of low-Z elements. These data were admirabty 
compiled by Rutherford, Chadwick, and Ellis (R50) in 1930. Subse- 
quently, Mott and others applied the wave mechanics to the interpreta- 
tion of scattering in terms of a phase-shift analysis (Appendix C) and 
thus resolved many of the accumulated conflicts between theory and 
experiment. 

b. Wave Model of Anomalous Scattering. We can expect a measur- 
able amount of barrier transmission for bombarding energies which are 
distinctly below the barrier height (Fig. 5.5). Those particles which 
penetrate the coulomb barrier and reach the nuclear surface will experi- 
ence nuclear forces in addition to the coulomb forces and will be anoma- 
lously scattered. Anomalous scattering is thus intimately related to 
barrier transparency. 

Some of the particles which reach the nuclear surface may be able to 
penetrate it and thereby form a compound nucleus. For example, an 
a ray transmitted through the surface of an B 16 nucleus forms the com- 
pound nucleus i Ne 20 . Such a compound nucleus will not be in its 
ground level but will be highly excited because it comprises the rest 
masses of the interacting particles plus their mutual kinetic energy. 
Each such compound nucleus possesses a number of quantized excited 
levels, which are the quasi-stationary energy levels, or "virtual levels," 
of the nucleus. If the kinetic energy of the bombarding particle happens 
to be near or equal to that required for the formation of one of these 
excited levels, then penetration of the nuclear surface is facilitated. This 
resonance formation of the compound nucleus occurs when the magnitude 
and slope of the wave functions inside and outside the nuclear surface 



7] Radius of Nuclei 83 

can be matched at substantially full amplitude (compare Appendix C. 
Fig. 11; also Chap. 14, Figs. 1.1 and 1.2). 

Once the excited compound nucleus has been formed, it will experience 
one of a number of possible competing transitions (Chap. 14). One 
possibility is the reemission of the incident particle, or one identical 
with it, without loss of total kinetic energy in the system. The direc- 
tional distribution of reemission will depend on many factors, especially 
those involving angular momenta (Chap. 6). The total process of 
resonance formation of the compound nucleus and its subsequent dis- 
sociation by emission of a similar particle, without loss of kinetic energy 
in the system, is called elastic resonance scattering. 

In principle, the total elastic scattering can now be divided, somewhat 
arbitrarily, into three cooperating phenomena. 

1. Coulomb Potential Scattering. This is the classical Rutherford 
scattering. In the wave model its predominantly forward distribution 
is due to interference between partial waves of all angular momenta I. 
Large values of I arc effective here because of the long-range character 
of the coulomb force. 

2. Nuclear Potential Scattering. This is the reflection from the 
abrupt, change in potential at the surface of the nucleus, combined with 
"shadow" diffraction around the nucleus. Its existence presupposes the 
penetration of the coulomb barrier, which is greatest for partial waves 
of small angular momentum. For this reason, and because it depends 
on short-range forces, its principal contribution at low bombarding 
energies is to s-wavc scattering. The s wave is spherically symmetric. 
Therefore the nuclear potential scattering can give rise to anomalous 
scattering at all angles at the smallest bombarding energies which permit 
significant barrier penetration. As the Rutherford scattering is smallest 
in the backward direction (@ = 180), the ratio of anomalous to classical 
scattering is generally greatest in the backward direction. Theoretical 
estimates of the pure nuclear potential scattering are obtained by pre- 
suming that penetration of the nuclear surface is negligible if the energy 
is not near a resonance. The corresponding scattering can then be 
evaluated as that due to an impenetrable sphere. 

3. Nuclear Resonance Scattering. This occurs only at energies near 
a resonance level, where the incident particle can easily form a compound 
nucleus and a similar particle may be emitted before any other competing 
emission or radiative process takes place in the compound nucleus. In 
the incident plane wave, only that partial wave will be involved whose 
angular momentum I will permit formation of the particular excited level 
involved. 

Coherence. These three processes must be regarded in the wave 
theory as taking place simultaneously. Each can, in principle, be 
described by an appropriate phase shift for each partial wave. The net 
result is very complicated analytically (B41, F45, M69) because all 
three effects add coherently. There are, therefore, opportunities for the 
occurrence of many interference minima and maxima in the net angular 
distribution. In a few cases involving resonance scattering it is now 



84 



The Atomic Nucleus 



[CH. 2 



possible to deduce the angular momentum of the excited compound 
level from the angular distribution of the scattering intensity (K29, 
L13). 

The three processes may be better visualized by a grouping into the 
"external scattering" or "total potential scattering" of a perfectly reflect- 
ing sphere surrounded by a coulomb field and the "internal scattering" 
due to resonance formation of a compound nucleus. These processes are 
indicated schematically in Fig. 7.1 

c. General Characteristics of the Experimental Results on Anomalous 
Scattering. The simplest cases of elastic scattering oi charged particles 
are those in which the incident and target particles both have zero spin. 



, f Nuclear! 

Potential ] < potential f 

scattering I [ scattering J 

(External f f Coulomb] 
scattering)! < potential > ( 
LscattermgJ 




f ^ f Resonance 

f Reemission of scattering 
)< captured type f< (|nterna , 
particle J [scattering) 




Excited levels of the 1 
compound nucleus which I . 
are responsible for f 

resonance scattering J 

Fig. 7.1 Schematic representation of the three coherent rlajslir- scattering processes 
whose cooperative effect is the total anomalous scattering. 

Then an analytical representation which is not too formidable can be 
obtained (p. 319 of M69) for the scattering amplitude f(fl) and for the 
differentia] cross section !/(tf)| 2 dfl, as defined by Eq. (107) of Appendix 
C. Among the light particles which are available as projectiles, only 
the a particle has zero spin. These particular theoretical restrictions 
are satisfied, then, in collisions between a rays arid target nuclei which 
have even-Z and even-^4 and, hence, zero spin. The corresponding 
experimental work is meager, and most of it has been done only with 
sources of natural a rays, such as RaO'. There results an inevitable 
lark of high resolution in energy and angle, which usually makes precise 
analysis difficult. Nevertheless, the over-all features of elastic, scatter- 
ing of charged particles are well portrayed in these experiments. 

Figures 7.2 and 7.3 show the experimental results for the scattering 
of a rays by oxygen nuclei. Bearing in mind the experimental uncertain- 
ties, which are suggested in part by the vertical and horizontal lines 
through the points, the following general characteristics may be seen. 

1. Onset below Barrier Energy. The coulomb barrier height, between 
B O 18 and 2 He 4 , is about 6.1 Mev if fl~ 1.5 X 10- I3 A* cm. An a-ray 



7] 



Radius of Nuclei 



85 



kinetic energy T a = 7.6 Mev (laboratory system) would be required to 
produce T = 6.1 Mev in the center-of-mass coordinates. Deviations 
from classical scattering are clearly evident at T a < 5 Mev, which is 
only about two-thirds of the barrier height. From Fig. 5.5, we could 




4567 
a- Ray energy 7^ in Mev 

Fig. 7.2 Elastic scattering of rays by O 16 . The ratio of the observed scattering 
to that expected classically is plotted against the energy in the laboratory coordinates 
of the incident a rays. The mean sc: 11- ring angle in laboratory coordinates is shown 
for each of the five curves, but the angular widths were actually quite large and 
somewhat overlapping. The rays were from RaC' (7.08 Mev) and, in some cases, 
ThC' (8.77 Mev) and were slowed down to the smaller energies by absorbing foils. 
This absorption introduces an inhomogeiieity of incident energy, because of straggling. 
Below about T a ^ 4.5 Mev, the observed scattering was indistinguishable from classi- 
cal. At the higher energies, all the principal features of nuclear potential scattering 
and resonance scattering are seen. [Brubakcr (B136).| 

estimate that the barrier penetration may be of the order of 10 to 20 per 
cent for this a-ray energy. 

2. Simultaneous Onset at All Angles. At the lowest energies, the 
anomalous scattering is seen to vary smoothly with energy. This mono- 
tonic deviation, at any particular scattering angle, is characteristic of the 
potential scattering. Its shape is due to the coherent combination of the 
potential scattering amplitudes from the nuclear surface and from the 
coulomb barrier. From classical theory, Eq. (7.1), one would expect 



86 



Tlie Atomic Nucleus 



[CH. 2 



the anomalies to appear at the smallest energies for the largest scattering 
angles. This is clearly not the case. Within the accuracy of measure- 
ment, Fig. 7.2 shows that at all directions the anomalies begin at the 
same energy. This occurs at the smallest energy for which s-wavc pene- 
tration of the coulomb barrier is sufficient to produce a detectable ampli- 
tude of nuclear potential scattering. 




a-Ray energy 7^ in Mev 



Fig. 7.3 Elastic scattering of a rays by O 16 at a mean angle of 157 n in the laboratory 
coordinates. Sources and techniques are somewhat similar to Kip;. 7.2, but the angu- 
lar spread is confined to about 15. The curve drawn represents the theoretical vari- 
ation of p-wave resonance scattering, from two levels at about 5.5 Mev and 6.5 Mev, 
superimposed on the monotonic deviation due to nuclear potential scattering. AH in 
Fig. 7.2, detectable deviations from classical scattering occur at all energies above 
about 4.5 Mev, which is only about two-thirds the barrier height. [Ferguson and 
Walker (F30).] 



3. Resonances. At T a ~ 5.5 Mev the two upper curves of Fig. 7.2 
are clearly not monotonic. The irregularity is more apparent in Fig. 
7.3, where the effects of two disc-re tc resonance levels can be seen. In 
Fig. 7.3 the curve has been drawn with a general shape and amplitude 
which correspond to the theoretical resonance scattering for p waves, 
superimposed on a monotonic increase in potential scattering. The fit 
is distinctly better than could be obtained by assuming s-wave or d-wave 
resonance scattering (F30, R28). This implies that both the correspond- 
ing excited levels in the compound nucleus Ne 2u have an angular momen- 
tum of unity, because they are formed from spinless particles by capture 



7] 



Badius of Nuclei 



87 



in the I = 1 wave. These two isolated resonance levels occur at bom- 
barding energies of T a ~ 5.5 and 6.5 Mev. It can be shown from mass- 
energy relationships that they therefore correspond to excited levels at 
about 10.1 and 9.0 Mev about the ground level of ioNe 20 . Five addi- 
tional scattering anomalies, corresponding to excited levels between 6.738 
Mev and 7.854 Mev in Ne 20 , have been found by Cameron (C5) T who 
made precision measurements of the () 16 (a,a) elastic scattering, using 
electrostatically accelerated helium nuclei over the energy range from 
0.94 Mev to 4.0 Mev. 

4. Angular Distribution. The angular distribution is noiumiform. 
Note from Fig. 7.2 that the sign of the initial potential scattering anomaly 
can be either positive or negative, depending on the angle of observation. 
The potential scattering anomaly is most apparent in the backward 
direction, where the classical scattering is smallest. The over-all angular 
distribution contains maxima and minima which are due to constructive 
and destructive interference between the three components of scattering 
intensity. In any particular direction, such as the 157 observations of 
Fig. 7.3, the peak and valley due to resonance .scattering may dominate 
the mcmotonir background of potential scattering. The angular distri- 
bution of resonance scattering is determined primarily by the Legendre 
polynomials P/(eos ti) which characterize the partial waves of angular- 
momentum quantum number / [see, for example, Appendix C, Eq. (118)]. 
In particular, nodes can orr-ur at angles which are determined by the 
condition P,(ros tf) = 0. For example, /M90) = 0, P 2 (125) = 0. 
Thus the angular distribution of maxima and minima of resonance scatter- 
ing can be used to determine the angular momentum of the resonance level 
in the case of a collision between two spinless particles (R28, C5). 

d. Nuclear Radii. The smallest bombarding energy at which nuclear 
potential scattering is detectable depends upon the experimental resolu- 
tion and upon the shape and position of peaks due to resonance scatter- 
ing, which may mask the onset of potential scattering. As a means of 
measuring nuclear radii, anomalous scattering therefore is usually less 
accurate than several other contemporary methods. Such results as 

TABLE 7.1. NUCLEAR BARRIER HEICJHT It, EFFECTIVE RADII'S A, ATIW 

EFFECTIVE UNIT RAI-IUS R = 7?/.4* 

Based on Pollard's (P25) summary of the minimum energy T (center of mass) for 
anomalous scattering of a rays. 



Element 


Z 


T, Mev 


B, Mev 


/?, 1(T 13 cm 


o, 10~ 13 cm 


Ho 


2 


1.4 


2.4 


2.4 


1.5 


Li 


3 


2.0 


3.3 


2.6 


1.4 


Be 


4 


2 4 


4 


2.9 


1.4 


B 


5 


2 8 


4.5 


3.2 


1.5 


C 


li 


3.1 


5.1 


3.4 


1.5 


N 


7 


3 5 


5.6 


3.6 


1.5 


Mg 


12 


5.4 


8.5 


4.0 


1.4 


Al 


13 


5.8 


9.0 


4.1 


1.4 



88 



The Atomic Nucleus 



[CH. 2 



have been obtained are in acceptable agreement with the values obtained 

by all other methods. 

Pollard (P25), in 1935, correlated the data then available on the 

energy of onset of anomalies in the scattering of a rays by eight light 

elements. Pollard's interpretation of the data accumulated by various 

investigators was based on the rea- 
sonable assumption that nuclear 
potential scattering would become 
experimentally detectable when the 
barrier transmission is about 10 per 
cent for $ waves. With this as- 
sumption, a recalculation of all the 
data led to the values of nuclear 
radii which are summarized in 
Table 7.1 and in Fig. 7.4. It will 
be seen that these data are in accord 
with the constant-density nuclear 
model 

R = R Al (7.2) 

with an effective nuclear unit 
radius in the domain of RQ = 1.4 to 
1.5 X 10~ 13 cm. When the effec- 

(mass number)3 ^ radij ^ CQm ^ d f or the finite 

Fig 7.4 Effective nuclear radius /?, from ^ Qf lhe fl ^ iu ^ manmi| . of 
Table i.l, vs. A . E ^ (&.15), the unit corrected radii 

for the target nuclei are again in the domain of R$ A ^ (1.3 0.1) X 10~ 13 
cm. 




Problems 

1. The scattering of a rays by hydrogen was found by Chadwick and Bieler 
(CIS) to be in accord with the Rutherford scattering law for 1.9-, 2.8-, and 3.3- 
Mev a rays but to be markedly anomalous for 4.4-, 5.7-, 7.5-, and 8.6-Mev a rays 
(laboratory coordinates). 

(a) Show that the numerical value of the parameter 2Zz/ 137/3 is about 0.6 
for the 4.4-Mev a rays. 

(&) Comment on the degree of validity which you would expect for classical 
theory in such a collision. 

(c) Calculate 2Zz/137/3 for some of the other a-ray energies used. How can 
such small variations in 2Zz/137/3 be expected to spell the difference between 
classical and anomalous scattering? 

2. (a) When a rays are scattered by hydrogen, show that the kinetic energy 
in the center-of-mass coordinates is only one-fifth of the laboratory kinetic energy 
of the incident a ray. 

(6) Could the same nuclear interaction be studied by bombarding helium 
with accelerated protons? 

(c) Determine what fraction of the proton energy would then be available in 
the C coordinates. 

(.d) If anomalous scattering is observed with 4.4-Mev a rays on hydrogen, 



8] Radius of Nuclei 89 

what energy protons should be used to produce the same nuclear effects when 
helium is bombarded by protons? 

() Does 2Z2/1370 have a different value for the 4.4-Mev a rays and for the 
proton energy determined in (d)? 

(/) Show that, in general, the parameter 2Z/ 137/3 is not dependent upon 
which of the interacting particles is the target in the laboratory, so long as the 
kinetic energy in the center-of-mass coordinates is kept constant. 

3. The elastic scattering observed when helium is bombarded by 1-Mev to 
4-Mev protons has been shown (C55) to be largely due to resonance scattering, 
involving the formation and prompt dissociation of excited levels of the compound 
nucleus Li 5 , which is formed by the coalescence of He 4 and H 1 . The mirror 
nucleus of Li 5 is He 5 , which should have an analogous internal structure and could 
be studied by scattering fast neutrons in helium (A3). Would you anticipate 
that the neutron energy required to excite these levels in He 5 would be about the 
same as the proton energy required to excite the analogous levels in Li 5 , or would 
you think that the coulomb barrier in the (p-a) interaction would require that 
greater energies be used in this case? Why? 

8. Cross Sections for Nuclear Reactions Produced 
by Charged Particles 

Classically, nuclear reactions initiated by charged particles should 
begin to take place when the bombarding energy T in center-of-mass 
coordinates is just equal to the coulomb barrier height B = Zze*/R, 
because then the classical closest possible distance of approach is just 
equal to the nuclear radius R. Actually, of course, these reactions take 
place abundantly at T considerably less than B. This fundamental fact 
was dramatically proved in the pioneer experiments by Cockcroft and 
Walton (C27) in 1932. 

Using high-voltage transformers and rectifiers in voltage doubling 
circuits, and a suitable ion source and discharge tube, Cockcroft and 
Walton produced a beam of protons which had been accelerated to about 
0.6 Mev. This energy is far below the barrier height, even for the light- 
est elements (B ~ 1.5 Mev for Li + p). But the success of Gamow's 
barrier-transmission concepts as applied to a decay stimulated Cockcroft 
and Walton to believe that these low-energy protons could penetrate 
into nuclei and possibly produce observable disintegrations. A proton 
beam current of up to 5 ^a provided enough incident particles to compen- 
sate for their very small individual probability of penetrating the nuclear 
barrier. At proton energies as small as 0.12 Mev, the (p,a) reaction was 
observed in lithium targets, Fig. 8.1, and the (p,a) reaction was reported 
for a number of heavier target elements as well (B, C, F, Al, . . .)- The 
daring and brilliant success of these experiments should be kept in view 
today, when the work has long since taken its place in the archives of 
physics. 

a. Bohr's Compound-nucleus Model of Nuclear Reactions. The 
experimental and theoretical aspects of nuclear reactions are discussed 
in more detail in several later chapters. Here we shall note some of the 
fundamental physical concepts which underlie our current views about 



90 



The Atomic Nucleus 



[CH. 2 



nuclear reactions, especially with regard to the influence of nuclear 
radius. 

Bohr, in 1936, first clearly emphasized the so-called compound- 
nucleus model of nuclear reactions, which has since been well verified 

in a large class of nuclear reactions. 
It is assumed that when some target 
nucleus A is bombarded by an 
incident nuclear particle a, the two 
may coalesce to form a compound 
nucleus (A + a), where the paren- 
theses denote (F62) that, the com- 
pound nucleus is produced at an 
excitation energy which is dictated 
by the bombarding energy. In the 
compound nucleus (A + a), there 
are assumed to be strong interac- 
tions between all the nuclcons. 
The incident nuclear-particle a loses 
its independent identity, and the 
total energy of the excited com- 
pound nucleus is shared in a com- 
plicated manner by all the nucleons 
present. The compound nucleus 
(A + a) is thought of as being in 
a quasi-stationary quantum state, 
whose mean life is long (~ 10" 188 
sec) compared with the time for a 
proton to cross the nucleus (~ 10~" 
sec). Identically the same com- 
pound nucleus, and in the same 
level of excitation, can be produced 
by the collision (usually at a dif- 
ferent bombarding energy) of other 
nuclei, say, B and b, so that it is 
possible to have 

(A + a) = (B + 6) 




200 300 
Kilovolts 

Fig. 8.1 The first "excitation function" 
for artificially accelerated particles, as 
obtained by Cockcroft and Walton (C27). 
The yield of a particles, from a thick 
target of lithium, is plotted as a function 
of the energy of the incident protons. 
The absolute yield was estimated to be 
about one a particle per 10 8 protons at 
0.5 Mev. Note that the (p,a) reaction 
is detectable here at an incident energy 
which is only about one-tenth the esti- 
mated coulomb-barrier height (5 ~ 1.5 
Mev), and note that the yield increases 
rapidly (roughly exponentially) with 
bombarding energy. The reaction has 
since been observed at proton energies 
as small as 10 kev. This excitation func- 
tion for Li(p,a) was promptly confirmed, 
and was extended to 700-kev protons 
from a small cyclotron, by Lawrence, 
Livingston, and White (L14). [From 
Cockcroft and Walton (C27).] 



(See, for example, Figs. 1.4 and 2.8 
of Chap. 14.) In the Bohr postu- 
lates, the properties of the com- 
pound nucleus (A + a) are inde- 
pendent of its mode of formation, i.e., the compound nucleus "forgets 11 
how it was formed. 

The second step in the nuclear reaction is the dissociation of the 
compound nucleus. This dissociation can generally take place in a 
large number of ways, sometimes called "exit channels," subject to the 
conservation laws for mass-energy, charge, angular momentum, etc. 
The competition among various alternative modes of dissociation does 



8] Radius of Nuclei 91 

not depend on the manner in which the compound nucleus was formed, 
i.e., on the "entrance channel." Schematically, any nuclear reaction in 
which a compound nucleus is formed can be represented as 

Entrance Compound Exit 
Channel nucleus channels 

A + a (elastic scattering) 

A* -I- a (inelastic scattering) 

% r B + b (nuclear transformation) 

A + fl * (A + a ) (g i } 

xT^^ C+6 + c (nuclear transformation) v ' 



D + d (nuclear transformation) 

etc. (nuclear transformation) 

The asterisk, as in -4*, denotes an excited level of a nucleus. As an 
explicit example, the reaction on F 19 in which a proton is captured and 
an a particle is emitted would be written 

9 F 14 + iH' -* GoNe 20 ) - ,He 4 + H 16 (8.2) 

The same compound nucleus might instead emit a neutron, leaving 
ioNe 19 as the residual nucleus, according to 

9 F 19 + ,IP --> (, Nc w ) -> oH 1 + 10 Xe (8.3) 

In the more compact notation which is usually used for nuclear reac- 
tions, these two competing reactions would be written F 19 (p,a)O 16 and 
F ll *(?vONe 19 , without explicit designation of the compound nucleus. 

The cross section for any nuclear reaction which involves a compound 
nucleus can then be written as 

all b * a (8.4) 



where ff coir .(a) is the cross section for formation of the compound nucleus. 
The partial level width T 6 is proportional to the probability that the 
compound nucleus will dissociate by the emission of the particle 6, and 
the total level width F = 21", is the sum of the partial widths for all 
possible modes of dissociation. Thus F&/T is simply the fraction of the 
compound nuclei which dissociates by emission of />. Equation (8.4) is 
applicable to all oases of reactions (b y^ a) and to inelastic scattering. 
It is not applicable to elastic scattering, <r(a,a) = <T SCJ because of inter- 
ference effects [Chap. 14, Eq. (1.16)]. 

b. Cross Section for Formation of the Compound Nucleus. One 
major objective of the theoretical treatment of nuclear reactions (Chap. 
14) is the prediction of <7 com as a function of the incident energy and other 
parameters of the colliding particles. The incident particles are repre- 
sented as a plane wave, whose rationalized de Broglie wavelength of 
relative motion is 

X = J * (8.5) 

MV V2MT 



92 The Atomic Nucleus [CH. 2 

where M = reduced mass 

V = velocity of relative motion 

T ?M V z = kinetic energy in center-of-mass coordinates 
This plane wave is the sum of partial waves, corresponding to particles 
whose angular-momentum quantum numbers are I. For each partial 
wave, the maximum possible reaction cross section is (21 + IV* 2 [Appen- 
dix C, Eq. (85)]. Then the actual cross section, for each partial wave L 
can be represented as 

.i = (2? + l)*X*Ti (8.6) 



in which T ( is the over-all harrier-transmission coefficient. For purposes 
of visualization, we may write 

T z = P,(D&(D (8.6a) 

in which Pi(T) is a Gamow-type coulomb-penetration factor representing 
the chance that the incident particle, with kinetic energy T, can penetrate 
the coulomb and centrifugal barrier and thus reach the nuclear surface, 
while /(?') is called the "sticking probability" and represents the chance 
that the particle can pass through the potential discontinuity at the 
nuclear surface and bo absorbed to form the compound nucleus. When 
all values of / are considered, wi can write the cross section for formation 
of the compound nucleus as 



T ) (8.7) 

1=0 

In the actual calculation of a c , jni along the lines represented sche- 
matically by Eq. (8.7 J, the over-Mil transmission probabilu-y P/f; cannot 
in fact be treated purely as two sequential probabilities, but this simplified 
viewpoint is convenient for giving a physical picture of the absorption 
process. The theoretical values of <r mm for charged particles depend in a 
complicated way on Z, z, T, R, and on assumed properties of the interior 
of the nucleus, and they cannot be expressed in any simple analytical 
form. In the continuum theory of nuclear reactions, which is most 
applicable for reasonably large T and Z, <r c(liu is averaged over any indi- 
vidual resonances which may be present. The theoretical results for 
ow are given in the form of tables and graphs, such as Fig. 2.10 of Chap. 
14. Some simple and important qualitative generalizations should be 
noted at this time: 

1. <r com is small but finite for bombarding energies T which are far 
below the coulomb barrier height B = Zze z /R. 

2. (Toom increases very rapidly with !T, when T < B, behaving roughly 
like a Gamow-type barrier penetration. 

3. ffoom definitely does not reach its maximum value at T = B. The 
barrier height is B only for s waves. For I > the centrifugal barrier 
must be added. 

4. (Toom approaches asymptotically a maximum value which is simply 
the geometrical area of the target nucleus, vR z , when T y> B. 



8] Radius of Nuclei 93 

Semiclassical Approximation for o^. An instructive approximation 
for the cross section cr coni for large bombarding energies can be obtained 
easily and is found to be in surprisingly good agreement with the results 
obtained by the detailed wave-mechanical calculation. 

In the classical collision between two charged particles Ze and ze, the 
closest distance of approach p min is given by Eq. (7.1), which can be 
written 

< 8 - 81 



where the incident energy T and the scattering angle are both in 
center-of-mass coordinates, and where the impact parameter is 

Zze* . 8 



The classical cross section for forming the compound nucleus, by col- 
lision of the charged particles, is simply o- fnm = wx z , where x is the largest 
value of the impact parameter for which the charged particles come in 
contact, that is, p min < R', if R' is the effective nuclear radius. If we 
set R' = (R + X), where R is the radius of the target nucleus and X 
represents the "size" of the projectile, or the lack of definition of its 
impact parameter, as in Eq. (5.81a), then we find at once that 

(8.9) 



, TT Zzc* D R 

where U = ^ . = B 



+ X R + X 

is the coulomb potential at a separation R + X, and T = h 2 /2M\ 2 is the 
incident- kinetic energy in C coordinates. Equation (8.9) gives meaningful 
results obviously only if T > B > J7, that is, for incident energies which 
are well above the barrier height for s waves. The detailed wave- 
mechanical calculations give values of tT mm which are only 15 per cent 
lower than Eq. (8.9) when T/B = 1.2 and which are essentially equiv- 
alent when T/B > 2. As T increases, X decreases and 0-^ > wR 2 in the 
limit of T B. 

Effects of Nuclear Radius on <r com . In the higher-energy domain, the 
effects of the nuclear radius R on the cross section <r wm for formation of the 
compound nucleus are evident from Eq. (8.9). For fixed values of Z, z, 
Af, and 7 T , an increase in R increases o- com , both by increasing the asymp- 
totic limit irR z and b}' lowering the coulomb barrier height B. Also, at 
small bombarding energies, an increase in R increases <r e( ,, u . 

When T < B, an increase in R makes the coulomb barrier lower and 
thinner (Fig. 5.7) and also reduces the centrifugal barrier, all these effects 
tending to increase a com . The magnitude of these changes is shown in 
Figs. 2.9 and 2.10 of Chap. 14, for two representative values of the 
nuclear unit radius R = 1.3 X 10~ 13 cm and R Q = 1.5 X 10~ 13 cm. 



94 The Atomic Nucleus [CH. 2 

The presently available experimental data on the cross section for 
formation of the compound nucleus, when interpreted in terms of the 
theory (S30) of charged-particle reactions as available at the beginning 
of 1954, correspond to nuclear unit radii near the domain of 

#o = (1.4 0.1) X 10- 13 cm (8.10) 

on the constant-density model R = 7? Al. This radius is a "nuclear- 
force radius," and because of the assumptions which enter the theory of 
nuclear reactions, R may be expected to differ by the order of 1 X 10~ 13 
cm from the radii deduced from other typos of experiments. 

9. Awf/eor Cross Sections for the Attenuation of Fast Neutrons 

The attenuation of a collimatod m on oen ergot ic beam of fast neutrons, 
by a wide selection of absorbing materials, has been measured in a num- 
ber of experiments. In this way the total nuclear cross section, for 
absorption plus scattering, can be determined as a function of mass 
number .1. Some of the conclusions, which bear on the question of 
nuclear radii, will be summarized here, but we defer the details for dis- 
cussion in Chap. 14. 

Because it possesses no charge, a fast neutron passes easily through 
bulk matter. In a close collision with a nucleus, there is no coulomb 
deflection, and so the target cross section might be expected to be simply 
the geometrical cross section irR z . Ascribing, as in Kq. (8.9), a "size" \ 
to the neutron, a better estimate of the cross section for a direct encounter 

^ ba ~T(/t: + x) 2 (9.1) 

where 0- ab * = when there is no elastic reemission of neutrons after 
formation of the compound nucleus. This simple relationship corre- 
sponds to z = in Eq. (8.9), and it is found to be a good representation 
of the detailed wave-mechanical theory (F49) whenever R ^> X. Equa- 
tion (9.1) has been thought of as valid for neutrons of greater than about 
10 Mev, for which X = 1.44 X 10~ 13 cm. Equation (9.1) represents the 
actual absorption cross section only to the extent that all neutrons which 
strike the target sphere are actually absorbed by the nucleus. Recent 
evidence has shown clearly that swift incident neutrons have a small but 
finite probability of passing through nuclear matter without being 
absorbed (see Fig. 2.3 of Chap. 14). Thus nuclei can be described as 
slightly "translucent," rather than completely "opaque," to incident 
fast neutrons. 

Considering the neutron beam as a plane wave, we note that each 
nucleus should cast a shadow, just as would be the case for an opaque 
disk intercepting a beam of light. This shadow, in the wave-optical 
model, is the result of interference from waves scattered from near the 
edge of the opaque sphere. It can be shown easily that exactly the same 
amount of incident energy is diffracted as is absorbed by the opaque 
sphere (see Fig. 2.5 of Chap. 14). In the case of fast neutrons, with 



9] Radius of Nuclei 95 

X <$C /?, this diffraction, or "shadow scattering/' corresponds to a small- 
angle elastic scattering, for which the cross section a ac is the same as 
<Tb B , or 

<r, r ~?r(/e + X) 2 (9.2) 

Then the total nuclear cross section tr t is 

fft = <r.u + IT,, ^ 2T(/Z + X) 2 (9.3) 

or just twice the effective geometrical area of the nucleus. 

When the measured values of the total attenuation cross section <r t 
are interpreted in terms of Eq. (9.3), the data are fairly consistent (see, 
Fig. 2.2 of Chap. 14) with nuclear unit radii in the domain of 

flo^ (1.4 0.1) X K)- 13 cm (9.4) 

on the constant-density model R = /?/!*. Again, as in the case of 
charged-particle interactions, this is a "nuclear-force radius. 7 ' Refine- 
ments of the theory, and some rcinterpretation of the experimental data, 
can be expected when the degree of "transparency" of nuclei to fast 
neutrons becomes more precisely evaluated. 



CHAPTER 3 

Mass of Nuclei and of Neutral Atoms 



Much of our present knowledge about the structure of nuclei and the 
forces between nucleons is derived from carefully measured values of the 
mass of nuclei. These measurements present a variety of special experi- 
mental problems. Most of the accurate mass values now available have 
been obtained either by mass spectroscopy or by measurements of the 
energy released or absorbed in various nuclear reactions. We shall sur- 
vey these two principal methods, and some fundamental results obtained 
from them, in this chapter. The measurement of atomic mass by micro- 
wave spectroscopy is discussed in Chap. 5, Sec. 3. 

1. The Discovery of Isotopes and Isobars 

In Daltoii's atomic theory (1808) each chemical element consisted 
of an assembly of identical atoms. Front's hypothesis (1815) visualized 
each such atom as a close aggregate of hydrogen atoms. These two 
concepts maintained their simple attractiveness until the middle of the 
nineteenth century. 

a. The Discovery of Nonintegral Values of Chemical Atomic Weight. 
By the latter half of the nineteenth century, chemical-atomic-weight 
determinations had disclosed several elements which definitely do not 
have integral whole-number atomic weights on the oxygen-equals-16 
(chemicai) scale. Although the atomic weight of carbon is 12.00; 
fluorine, 19.00; and sodium, 23.00, that of neon is 20.2, chlorine is 35.46, 
and magnesium is 24.32. These fractional atomic weights were incom- 
patible with continued acceptance of both Dal ton's and Prout's hypothe- 
ses, and in due course Prout's theory was discarded, only to be reestab- 
lished in a modified form after the discovery of isotopes and of the 
neutron. 

With the atomic-weight data before him, Sir William Crookes com- 
bined clear thinking and happy guesswork, reminiscent of the Greek 
atomists, to prophesy correctly the now basic concept of isotopes when 
he said in his 1886 address before the British Association (C56): "I 
conceive, therefore, that when we say the atomic weight of, for instance, 
calcium is 40, we really express the fact that, while the majority of 
calcium atoms have an actual atomic weight of 40, there are not a few 
represented by 39 or 41, a less number of 38 or 42, and so on." Except 

96 



1] Mass of Nuclei and of Neutral Atoms 97 

that calcium later turned out to be a mixture of isotopes having mass 
numbers of 40, 42, 43, 44, 46, and 48, Crookes's hypothesis states our 
contemporary beliefs quite perfectly. Before these ideas could become 
tenable, it was necessary to show experimentally that atoms of different 
weights can have the same chemical properties. This was accomplished by 
two widely different techniques just before the outbreak of World War I. 

b. Radiochemical Discovery of Isotopes, Isobars, and Isomers. 
Experimental proof of the chemical identity of atoms of different weight 
was first definitely established in 1911 by Soddy, who proposed the name 
isotopes for such atomic species (S58, S57). In the first 15 years follow- 
ing the discovery of natural radioactivity in 1896, chemists and physicists 
had separated many of the 40 radioactive species found in uranium and 
thorium minerals. Although the available quantities of many of these 
species are very small, the chemical behavior of any of them can be 
observed accurately by detecting the presence of the element in chemical 
precipitates, filtrates, etc., through its a or ft radiations. Each of the 
natural radioactive species has its own characteristic radiations and 
decay constant which identify it uniquely. Thus it was possible for 
Soddy to establish the chemical identity of two new trios of radioactive 
substances: thorium ( 90 Th 232 ), radiothorium ( 90 RdTh 22fi , or, in the 
newer simplified radiochemical notation, 90 Th 22B ), and ionium ( 90 Io 230 , 
or .oTh" ); also mesothorium-] ( 88 MsThi 88 , or 88 Ra 228 ), thorium X 
( 88 ThX 224 , or 8B Ra 224 ), and radium ( 88 Ra 226 ). 

At this time the mass and charge of the a and ft rays had been deter- 
mined, and ihe existence of atomic nuclei was just being established by 
Rutherford's interpretation of the a-ray-scattering experiments of Geiger 
and Marsden. Kadiothorium is a decay product of thorium, correspond- 
ing to the loss of one a. particle and two ft particles from the parent Th 232 
nucleus. According to the displacement law, RdTh 228 and Th 232 should 
have the same atomic number. Therefore the two species have the same 
nuclear charge but differ in mass by four units, corresponding to the 
emitted a particle. Somewhat similar considerations established the 
masses of the four other species involved. 

While thus demonstrating the existence of isotopes among the radio- 
active substances, Soddy correctly inferred that some of the common 
elements are also mixtures of chemically nonseparable species which 
differ by whole units in atomic weight. Consequently, the average 
atomic weight might be nonintegral, as in neon, chlorine, and others. 

In addition to pointing out the existence of isotopes among the radio- 
active Clements, Soddy called attention to the substances 90 RdTh 228 (or 
90 Th 22H ) and M MsThJ" (or 88 Ra 228 ). These two have the same mass 
number but different chemical properties, due to the difference in their 
nuclear charges. Such species are called isobars. 

Later Soddy suggested (S59) a further basis of classification of nuclei 
to meet the possibility that nuclei which have the same mass number and 
atomic number might still exhibit distinct radioactive properties, or they 
might differ in "any new property concerned with the nucleus of the 
atom." Such isobaric isotopes with distinguishable nuclear properties 



98 The Atomic Nucleus [en. 3 

are called isomers. The first case of nuclear isomerism was observed in 
1921 by Hahn for the isomeric pair uranium X 2 (giUX^ 34 , or 9 iPa 234 ) and 
uranium Z (gjlIZ 234 , or 91 Pa M4 ), both of which have atomic number 91 
a,nd mass number 234 but whose radioactive properties are widely differ- 
ent. Feather and Bretscher first showed in 1938 that UX 2 is only a 
long-lived excited nuclear level of UZ. Numerous other isomeric pairs 
have been discovered among the artificially radioactive substances. 

c. Discovery of Stable Isotopes by Positive-ray Analysis. While 
using a new parabola positive-ray apparatus (II 19) in the fall of 1012, 
Sir J. ,T. Thomson discovered in neon samples a faint line of unknown 
origin having a probable mass value of 22 in addition to a strong lino at 
about mass 20. Neon is the lightest clement having a definitely non- 
integral atomic weight (20.20). F. W. Aston immediately undertook a 
new precision measurement of the density and atomic weight of noon 
and endeavored unsuccessfully to concentrate appreciably the mass 22 
material by some 3,000 fractionations of noon over charcoal cooled by 
liquid air, and by repeated diffusion experiments (A37). The mass 22 
persisted with the mass-20 neon parabola, and while it seemed that the 
mass-20 parabola corresponded to slightly loss than an atomic weight of 
20.2, the experimental accuracy of 10 per cent was inadequate to clinch 
the matter. The existence of two isotopes of noon was strongly indi- 
cated by all these experiments, but none of them was absolutely convinc- 
ing when World War I interrupted the work. 

By the close of the war in 1919 the existence of stable, isotopes had 
been put beyond doubt by further work on the radioactive elements and 
by the accurate measurement of the atomic weight of ordinary lead 
(207.22), thorium lead (207.77), and uranium lead (200. Or>), and by the 
observations of Paneth and Ilevcsy (P4) on the inseparability of lead 
from radium D ( H RaD , or B ,Pb). 

2. Nomenclature of Nuclei 

a. Nomenclature of Individual Nuclei 

Atomic Number. The atomic number, or "proton number," equals 
the number of protons in a nucleus (Chap. 1) and is always denoted by 
the symbol Z. Atoms having the same Z are isotopes and in chemistry 
are usually studied as an unseparated group because they have similar 
configurations of valence electrons and substantially identical chemical 
properties. 

In sharp contrast, the nuclear properties of isotopes arc generally 
highly dissimilar. For example, nNa 22 is a positron /3-ray emitter, 
iiNa" is the stable sodium isotope, and n Na 24 is a negatron /3-ray emitter. 
In nuclear physics each isotope of an element needs to be studied as an 
individual nuclear species. The successful separation of stable isotopes, 
for independent study, is therefore of special importance to progress in 
nuclear physics (Chap. 7). 

Mass Number. The mass number A is the integer nearest to the 
exact atomic isotopic weight. In all cases, the mass number is equal to 



2] Mass of Nuclei and of Neutral Atoms 99 

the total number of protons and neutrons in a nucleus. It is therefore 
also called the "nucleon number." Atoms having the same mass num- 
ber arc isobars. The chemical properties of isobars are generally dis- 
similar, but their nuclear properties tend to present many parallel 
features, especially with regard to radius, binding energy, and spacing of 
excited levels. 

Neutron Number. With good experimental justification (Chap. 8), 
the number of neutrons in any nucleus is taken as (A Z). The 
neutron number (^l Z) can be represented by the single symbol N 
whenever confusion with other definitions of this much-overworked 
lettpr can be excluded. Nuclei having the same neutron number are 
isoloncs. Because of the symmetry of nuclear forces in protons and 
neutrons, isotones and isotopes play very similar roles in nuclear physics. 

Unpaired Neutron Number. The number of neutrons which are in 
excess of the number of protons in a nucleus is (A 2Z) = (N Z). 
This quantity reflects the asymmetry of a nucleus in neutrons and pro- 
tons. It is of importance in considerations of the binding energy of 
nuclei, especially in the liquid-drop model (Chap. 11). The name "iso- 
topic number" for the quantity (A 2Z) was used in 1921 by Harkins 
in his discussions of the relative natural abundance of isotopes, light 
elements having A 2Z = being by far the most abundant. The dis- 
covery of the neutron in 1932 endowed the quantity (A 2Z) = (N Z) 
with basic physical significance as an index of asj^mmetry. The name 
"isotopic number" is seldom used because it docs not connote this 
asymmetry. Instead, (N Z) is called the number of a unpaired neu- 
trons," or the number of "binding neutrons/ 7 or the "neutron excess." 
Nuclei having the same value of N Z are called isodiaphcres. 

Atomic Mass. The exact value of the mass of a neutral atom, rela- 
tive to the mass of a neutral atom of the oxygen isotope of mass number 
16, is the "atomic, mass," or "isotopic mass," M. Note that even for a 
single isotope this is not equivalent to the chemical atomic weight, which 
is based on a different mass scale (Chap. 7). 

Nuclear ^fass. In nuclear physics it is the mass of the nucleus itself 
which is usually of primary interest. This value is seldom, if ever, tabu- 
lated because all the necessary calculations usually can be carried out 
using only the neutral atomic masses. Moreover, it is the neutral atomic 
masses which are usually measured. When needed, the nuclear mass M f 
is given by 

M' = M - [Zm - B,(Z)] (2.1) 

where m n is the rest mass of one electron and the total binding energy 
B e (Z) of all the electrons in an atom is given approximately by tho Thomas- 
Fermi model, with zin empirical proportionality constant, as (F57) 

B r (Z) = 15.73Z 3 ev (2.2) 

Illustrative values of B e (Z) are given in Table 2.1. For the heaviest 
elements, the total electron binding energy approaches 1 Mev. But 
even this value is negligible compared with the binding energy of the 



100 



The Atomic Nucleus 



[CH. 3 



nucleus (Chap. 9), which is of the order of 8 Mev per nucleon. It is 
customary to neglect the electron binding energy in all but the most 
precise nuclear calculations. 

TABLE 2.1. APPROXIMATE TOTAL BINDING ENERGY OP ALL THE ATOMIC 

ELECTRONS IN ATOMS OF ATOMIC NUMBER Z 
[According to the Thomas-Fermi model, Eq. (2.2)] 



Element 


Ne 


Ca 


Zn 


Sn 


Yb 


Tli 
















z 


10 


20 


30 


50 


70 


90 
















B.(Z). kev 


3.4 


17 


44 


145 


318 


570 



b. Nomenclature of Nuclear Species. Kohman (K32) first proposed 
that any individual atomic species be called a nuchdc, rather than an 
"isotope" as had been conventional for some time, because the word 
"isotopes" literally connotes different nuclear species which have the 
same chemical properties. Kohman suggested a set of self-consistent 
definitions, which have been generally adopted. 

These definitions may be visualized and compared in Table 2.2. All 
have proved useful, especially in systematic presentations of the known 
nuclei, such as Sullivan's "Trilinear Chart of Nuclear Species" (S80). 

TABLE 2.2. SUMMARY OF CURRENT NOMENCLATURE OF NUCLEAR SPECIES 
(When the neutron number N = A Z is pertinent, it is written as a subscript 
beneath the mass number.) 



Term 


Characterized by 


Examples 


Remarks 


Nuclides 
Isotopes 


Z,A 
Constant Z 


fi 1 , LoSn' M , M U 
N, 7 N 14 , 7 N" 


>700 known 
3 to 19 known per ele- 


Isotones 
Isobars . . . 


Constant A - Z - N 
Constant A 


6 cj 4 , iN 8 o;- 

C 14 , T N" 


ment 
The neutron analogue of 
isotopes 
In ft decay the parent 


Isodiapheres . . 
Tsomers 


Constant 
A - 2Z = .V - Z 

Constant Z and A 


[ 6 c; 2 , 7 N} 4 , .en 

f -Q n 22B -p.,222] 

LHBitauB, heltiinel 

aijBrB Om (4.4 nr)j 


and product are isobars 
In decay the parent 
and product are iso- 
diuphrres 
Metastablc excited level, 






36 Br BO (18 min) 


~100 known 



Problems 

1. Explain qualitatively why the total binding energy of atomic electrons 
should vary faster than Z 2 . 

2. Derive an approximate expression for the change in the total binding 
energy of atomic electrons (a) following a decay and (b) following ft decay. 
Evaluate in kev for Z ~ 50. 

Ans.: (a) -13.5 kev; (&) 6.7 kev. 



3] Mass of Nuclei and oj Neulral Atoms 101 

3. Mass Spedroscopy 

F. W. Aston returned to the new but broad problem of isotopes as 
soon as World War I was concluded. He developed the first mass 
spectrograph for the accurate determination of atomic weights by the 
analysis of positive rays. Aston's brilliant work soon established the 
existence of many stable isotopes among the nonradioactive elements and 
brought him the Nobel Prize (for chemistry) in 1922. The data and 
techniques which had been developed up to 1941 are ably drawn together 
in Professor Aston's book "Mass Spectra and Isotopes" (A3 7). 

Mass spectroscopy has matured into a basic field in nuclear physics. 
Various workers have developed mass spectroscopes of many varieties, 
each designed for a particular type of duty. These include the accurate 
determination of atomic masses, the measurement of isotopic abundance 
ratios, the identification of the stable and radioactive nuclides found in 
nature, and the identification of products of nuclear reactions. Mass 
spectroscopes have also been developed for a variety of service uses. 
These include the chemical analysis of complicated vapor mixtures such 
as the products of human respiration; the measurement of isotopic dilu- 
tion in tracer studies with stable isotopes; the bulk separation of isotopes, 
as in the "calutron"; and the routine testing for gas leaks of a variety of 
enclosures ranging from basketball bladders to naval gun turrets. 

Although very different in detail, all these instruments embody 
merely a varying combination of a few fundamental components. We 
shall examine these basic principles in the following subsection. 

a. Basic Components of Mass Spectroscopes. In an ion source the 
substance to be examined is obtained in the form of free atomic or molecu- 
lar ions carrying single or multiple positive charges. 

Regardless of the type of source used, its exit slits provide the remain- 
der of the apparatus with a diverging bundle of positive ions containing a 
continuous distribution of velocities V and a discrete set of ne/M values, 
corresponding to the charge ne and mass M of the various atomic and 
molecular ions emitted by the source. Because the ionic charge ne can 
only be an integral multiple of the electronic charge e y and is usually e or 
2e, it is always possible to determine ne by inspection of the final record. 
Aside from the directional distribution of the ion beam which then enters 
the mass spectroscope's focusing and analyzing portions, there are two 
parameters, M and V, in the properties of the individual ions. 

It is therefore evident from the simple theory of equations that two 
independent operations must be performed on the beam in order to 
eliminate the parameter V and obtain the desired mass M . Fortunately, 
the properties of kinetic energy ?M V 2 and momentum M V of each ion 
are combinations of these independent parameters which may be inde- 
pendently and consecutively determined by the action on the ion of 
known electrostatic and magnetic fields. Thus, if both the energy and 
momentum of an ion are determined, its mass is uniquely specified. 

Most mass spectroscopes therefore consist essentially of a combination 
of an energy (electrostatic) and a momentum (magnetic) filter. Alterna- 



102 The Atomic Nucleus [CH. 3 

lively, a velocity filter (crossed electrostatic and magnetic fields) may 
be combined with either a momentum or an energy filter. Different 
types of mass spectroscopes result from combinations and permutations 
of the options available among these filter systems. 

Positive-ion Source. The selection of a source depends somewhat 
on the element or compounds whose mass spectrum is sought. 

Positive ions of the alkali metals (Li, Na, K, Cs) may be obtained 
readily by heating certain of their minerals (K48). Particularly with 
this type of source, and to a lesser degree with other types, the true 
isotopic abundance ratios of the solid material may not be faithfully 
reproduced in the positive ions resulting from evaporation. Where 
accurate isotopic abundance ratios are to be determined, great care is 
necessary in the selection and operation of the ion source. 

Ions of very refractory materials, such as gold, uranium, and others, 
may often be obtained from an oscillating spark discharge (D21) between 
electrodes containing the materials to be examined. 

Positive ions of many elements and molecules may be obtained from 
near the cathode of a high-voltage discharge tube containing the appropri- 
ate gas at a low pressure. A small amount of neon in the tube will 
stabilize the high-voltage discharge and produce controlled evaporation 
of metals, halides, and some other compounds which may be packed into 
small depressions prepared in the cathode (B5). The dense stream of 
high-velocity electrons along the axis of the discharge tube serves to 
dissociate and ionize the heterogeneous vapors of the substances evapo- 
rated. The positive ions are drawn out of this versatile source through 
an axial opening in the cathode. 

When the material to be examined can be put in gaseous or vapor 
form, it can be allowed to leak into a low-pressure ion source in which the 
ions are produced by bombardment with low-voltage electrons (< 100 ev) 
from an electron gun (N16). The ionized atoms and molecular frag- 
ments so formed are then drawn off through accelerator slits into the 
analyzing portion of the apparatus. 

Energy Filters. Since the force on a charge ne (esu) due to an electro- 
static field is simply neB, electrodes can be arranged in an arc (Fig. 
3. la) so that ions originally directed along their central tangent may be 
deflected to follow a circular path with a radius of curvature p. The 
centripetal force M V 2 /p is then provided by the electrostatic force ne& 
due to the radial field , which is of constant magnitude along the arc of 
radius p. Since KfV 2 /p = nc, it immediately follows that ions having 
any M and V, but possessing a nonrelativistic kinetic energy -^MV 2 , and 
entering the field while directed exactly along this arc, will follow the arc 
of radius p, where 

(3.1) 



The ions are not changed in energy by passing through this cylindrical- 
condenser filter, because they enter and leave at the same potential, 
For the central ray, whoso radius is exactly p, the electrostatic force is 
always normal to the motion. 



3] 



Mass of Nuclei and of Neutral Atoms 



103 



The ion optics of cylindrical-condenser filters (H45, W8) and of sector 
magnets has been reviewed and extended by Bainb ridge (B4), by John- 
son and Nier (JIG), and others. Cylindrical-condenser sectors possess 
optimum direction-focusing properties, which are analogous to the well- 
known 180 magnetic focusing, when the sector angle is ir/\/2 radians, or 
about 127. Under these conditions all ions having the same kinetic energy 
but diverging in direction by the order of 1 as they enter the filter are 
brought to a focus at the exit slit. This focusing action greatly increases 
the intensity of the beam, without impairing the energy resolution of the 
filter, by permitting the use of wider entrance and exit slits. Its use is 
illustrated in the mass spectrograph 
shown in Fig. 3.3, where the ions 
enter the energy filter with a large 
kinetic energy and a moderate uni- 
formity of direction as they emerge 
from the axis of the 20,000-volt, low- 
pressure discharge-tube ion source. 

An alternative, but less exact, 
energy filter is obtained by accel- 
erating low- velocity ions obtained 
from a low-voltage ion source. If 
the original velocity of the positive 
ions is low enough to be neglected, 
they may be accelerated by passing 
from entrance to exit slits between 
which is a potential difference of U 
electrostatic volts (Fig. 3.16). (c) (d > 

They then emerge with a uni- Fi &- " < a) Cylindrical-condenser en- 
form kinetic energy of P filter ' . (6) accelerator energy filter, 

(c) magnetic momentum filter, and (d) 

TsMV 2 = neU (3 2) vc l c ity filter. Mass spectroscopes are 

combinations of fundamental elements, 

on which is superposed the origi- arranged to focus ions having identical 
nal low-voltage energy distribution values of nc/M : J or ^^mple the mass 
with which they entered the accel- ^w of Fig. 3.3 combines the 

, ,, J elements (a) and (c). 

erator filter. 

Momentum Filters. The force on a particle of charge ne (esu, or 
ne/c emu) moving with velocity V in a magnetic field of B gauss is 
simply BncV/c and is directed at right angles to the field. Therefore, a 
moving ion is deflected in a circular path by a uniform magnetic field 
directed normal to its plane of motion. Equating the centrifugal force 
to the magnetic centripetal force, we have MV 2 /p = BneV/c, from which 
it immediately follows that the momentum M V of the ion determines its 
radius of curvature p, as 

"hflT D fQ Q^ 

M V = Bp (6.6 1 

C 

The uniform magnetic field therefore acts as a momentum filter (Fig. 




104 



The Atomic Nucleus 



[CH. 3 



3.1c). The deflecting force on the moving ion is always normal to the 
path; hence the initial momentum is unaltered by the filter. 

The direction-focusing properties of a uniform magnetic field are of 
special importance in mass spectroscopy [as they are also in 0-ray spectros- 
copy (PI 8)] because they permit greatly increased beam intensities 
without serious loss of resolution. The well-known 180 focusing prop- 
erty of a uniform magnetic field is a special case of a more general theo- 
rem. In Fig. 3.2 a slightly diverging beam of charged particles is 
deflected by a wedge-shaped sector magnet and is "focused" at B. The 
nature of this focusing action is such that the entrance slit A, the apex 
of the magnet 0, and the focal position B all lie on a straight line. 



E 




Fig. 8.2 Focusing action of a sector magnet OEF, whose field is normal to the plane 
of the paper and is uniform within OEF and zero outside OEF. [From Stephens (S70).] 
(1) Crossing for rays whose initial direction is a but which have the same momen- 
tum as those which cross at B. (2) Crossing for the "central" ray which enters and 
emerges from the magnet faces at 90. (3) Crossing for central rays whose momen- 
tum is larger than those crossing at B. 

Actually, only the central ray from A, which enters the edge OE of 
the magnetic field at normal incidence, passes exactly through the 
"focus" at B. Those initial rays which diverge from the central ray 
by a small angle a (such that a 2 1) cross the base line AOB inside 
the focus at B. The so-called lateral spread, or aberration, marked S in 
Fig. 3.2, can be shown to be (M28, S70) 

(34) 



sm 7 



sin 



for rays having the same momentum. Note that the spread is pro- 
portional to the line width at the focus B and increases as the square of 
the acceptance angle a. For a symmetrical sector-type magnet, with 
the half angle of the wedge equal to tf = 7, the lateral spread becomes 
simply 



S = (a sin 



= pa 



(3.5) 



where p is the radius of curvature of the central ion path in the magnetic 
field. Note that this result is independent of the angle of the magnetic 
sector. The classical "semicircular focusing" is simply the special case 
2* = 180. 



3] Mass of Nuclei and of Neutral Atoms 105 

It is helpful to develop from Fig. 3.2 a qualitative explanation of the 
focusing action of sector magnets. The initial ray which diverges by 
+a must traverse a longer path than the central ray in the magnetic 
field. It is therefore acted on longer and is deflected more than the 
central ray. Conversely, an initial ray which diverges by a traverses a 
shorter path in the magnetic field and is deflected less than the central 
ray. The three rays therefore have the opportunity of approaching or 
crossing each other and thus of forming a focus. The mathematical 
analysis shows that, in fact, the outer ray (+a) is always overcorrected 
and crosses the base line and the inner ray ( a) slightly on the near 
side, or magnet side, of B, as shown in Fig. 3.2. The "line shape" at B 
is therefore asymmetric, with the central ray at its outer edge. 

So far, we have considered only rays which all have the same momen- 
tum. Now let the initial central ray be made up of one group whose 
momentum is MV and another whose momentum is MV + AAfF. 
Where will the outer edge of the higher-momentum group fall? It can 
be shown that the lateral velocity dispersion, as measured by the separation 
D in Fig. 3.2, is given by (S70) 

n /&MV\ /sin &\ . . . . . , , Q ft . 

D = a( - ) ( - - ) (sin + sin 7) (3-6) 

\ M I / \sm 7/ 

In a symmetrical sector, of half angle tf = 7, the velocity dispersion is 
therefore given by 



n , . .. AMV n*\ 

D = 2(a sm ) ~ = 2 P (3.7) 



which is independent of magnet angle and proportional to the path 
radius p. 

Velocity Filter. The electrostatic and magnetic forces on a charged 
particle, moving through electric and magnetic fields which are directed 
at right angles to one another, may be balanced so that particles of 
velocity V and any charge experience no sideward force. The electro- 
static force is &ne, while the magnetic force is BVne/c. Equating these, 
and eliminating ne, we have 

F = c| (3.8) 

Figure 3. Id illustrates the arrangement of electrostatic and magnetic 
fields, at right angles to each other, which deflects all particles except 
those having the velocity V. 

Trochoidal Filter. Crossed electric and magnetic fields deflect any 
moving charged particle whose velocity differs from Eq. (3.8) into a 
path which is trorhoidal when projected onto a plane normal to the 
magnetic field. The trochoids for particles having the same specific 
charge ne/M have a common crossing, or focus, whose position is inde- 
pendent of the velocity of the particles. This property has been utilized 
in a mass spectrometer which involves only a single filter (B69, M51). 



106 The Atomic Nucleus [CH. 3 

The mass scale in the trochoidal mass spectrometer is rigorously linear, 
since the focal distance from the source is proportional to (/B 2 )/(ne/M). 
Angular Velocity Filters. The angular velocity w of a charged par- 
ticle, about an axis parallel to a uniform magnetic field B, is given at 
once by rearrangement of Eq. (3.3) and is 

V D ne/c f . 

u = = B (3.9) 

p M 

Therefore particles which have the same specific charge ne/M will traverse 
circular paths in the same period of time, regardless of their speed V and 
their path radius p. This is the well-known cyclotron resonance frequency 
''condition. The conventional cyclotron has, in fact, been used as a mass 
spectrometer in first proving the existence of the stable isotope He 3 in 
atmospheric and commercial helium (A25). 

A magnetic time-of-flight mass spectrometer has been developed which 
utilizes Eq. (3.9). Low-velocity positive ions (~ 10 ev) from a pulsed 
ion source traverse about 10 revolutions of a helical path (p ~ 12 cm) 
about the axis of a uniform field of 450 gauss. Their travel time is then 
of the order of 10 ^sec/amu. With the aid of timing equipment based 
on the principle of Loran navigation receivers, the time of flight can be 
measured with an accuracy corresponding to about 0.001 amu (H26, 
R17). 

b. Mass Spectroscopes. Any source and filter combination which 
provides a means of forming and observing mass spectra is called a mass 
spectroscope. Although there are innumerable individual forms, most 
of them fall into two broad general classes: 

1. The mass spectrographs are, in Aston J s definition, " those forms of 
apparatus capable of producing a focused mass spectrum of lines on a 
photographic plate." 

2. The mass spectrometers are forms "in which the focused beam of 
rays is brought up to a fixed slit, and there detected and measured 
electrically" (A37). 

Aston built the first mass spectrograph ; Dempster devised the earliest 
mass spectrometer. The term mass spectroscopes connotes collectively 
the mass spectrographs and the mass spectrometers. 

Mass spectroscopes which achieve both direction focusing and velocity 
focusing, usually by employing two direction-focusing elements (Fig. 
3.1a and 3.1c), are known as double-focusing instruments. Those 
which have direction-focusing action in only one element (usually 
because the electrostatic filter is of the accelerator type, Fig. 3.16) are 
called single-focusing. 

Double-focusing Mass Spectrograph. When the 127 electrostatic 
analyzer is employed, the resulting velocity dispersion can be just 
annulled by a momentum filter if the total magnetic sector angle is 60. 
The stray field at the edge of the magnet can be corrected for by cutting 
back the edges of the poles by about 1.6 gap widths. This combination 
of energy and momentum filters, with good direction-focusing values, has 
been so arranged by Bainbridge that it possesses the further advantage 



3] 



Mass of Nuclei and of Neutral Atoms 



107 



of a linear mass scale. The resulting mass spectrograph of Bainbridge 
and Jordan, in which p is about 25 cm for both filters, will separate ions 
having a mass difference of 1 per cent by about 5 mm on the photo- 
graphic plate. This permits a resolving power M/AM of about 10,000 
and makes this instrument valuable particularly for the accurate deter- 
mination of nuclear masses. A schematic diagram of this mass spectro- 
graph is seen in Fig. 3.3, where the components described earlier can be 
identified. The photographic registration is well suited to the accurate 
measurement of mass differences, as 
in the doublet method. It is less de- 
sirable for the measurement of iso- 
topic abundance ratios, because of 
the limited and nonlinear contrast 
scale of a photographic emulsion. 



Ion source 
~ 20,000 volts 




Vacuum 
pump 



Vacuum 
pump 



60 Magnetic 
analyzer 




Electrostatic 
accelerator 
^4000 volts 



Ion source 



Output meter 



Fig. 3.4 Simplified schematic drawing of 
the wi<m spectrometer of Nier and Roberts 
(N21, JIG). The mean radius of the 
electrostatic analyzer is about 19 cm and 
of the magnetic analyzer, about 15 cm. 
The entrance slit, apex of the magnet, 
and the exit slit of the magnetic analyzer 
are collinear. 



Fig. 3.3 Simplified schematic 1 diagram of 
the fftajw spcctrograph of Itainbridge and 
Jordan (B5). The mean radius of the 
ion path in both the electrostatic analj-zer 
and the magnetic analyzer is about 25 cm. 
The mass scale is linear at the photo- 
graphic plate, where ions of discrete ne/M 
values are separately focused. Note that 
the exit slit of the energy analyzer, the 
apex of the magnet, and the photographic 
plate are collinear. 



Double-focusing Mass Spectrometer. In a mass spectrometer, final 
focusing need be achieved at only one point. The ion current which 
passes through this point can then emerge through a slit and can be 
measured with the high precision which is characteristic of electrical null 
methods. 

Let ions having a discrete set of values of the specific charge ne/M, 
and a continuous distribution of velocities V, pass successively through a 
cylindrical-condenser energy filter and a momentum filter. Let p e be 
the radius of the ion path in the energy filter. Arrange entrance and 
exit slits in the momentum filter such that only those ions whose radius 
of curvature is p m will be transmitted. Then elimination of V between 



108 



The Atomic Nucleus 



[CH. 3 



Eqs. (3.1) and (3.3) shows that the only ions which can be transmitted 
successively through both filters must have the specific charge 

*C* = K- (3.10) 



ne 



where K is a constant for any particular slit arrangement. 



Ion Source 



Electrostatic 
accelerator 




Vacuum pump 



60 magnetic 
analyzer 



Lighter 
ions 



Heavier ions 



Ion collectors 
"To electrometer circuits 
Fig. 3.5 Schematic diagram of a 
single-focusing mass spectrome- 
ter, developed by Nier (N17) 
especially for the routine meas- 
urement of relative isotopic abun- 
dances. This instrument incor- 
porates two fixed ion collectors. 
By comparing the simultaneous 
ion currents to the two collectors, 
the ratio of ions of two nearby 
mass numbers can be determined. 
For example, the relative abun- 
dance of C" to C 1 * in carbon 
dioxide is obtained from the rela- 
tive ion currents of masses 45 
and 44, (C ia OjV and (COJ*) + , 
with correction, if needed, for 
(C 12 16 O 17 )+. As a single-collec- 
tor instrument, the resolution can 
be pushed^to allow accurate mass 
measurements on atomic or mo- 
lecular ions. 



The mass spectrum of discrete ne/M 
values can therefore be scanned by chang- 
ing , or B, or both. In order to avoid 
uncertainties due to hysteresis in the mag- 
net, it is generally preferable to hold B 
constant and to sweep through the mass 
spectrum by varying the potential applied 
to the electrostatic filter. The ion path 
through the mass spectrometer can there- 
fore be thought of as a paved highway, 
along which ions of various mass can be 
sent by adjusting C to a value appropriate 
for the particular ion. 

The transmitted ion current will gen- 
erally be in the domain of 10~ 12 amp or less 
and can be measured accurately with a 
vibrating-reed electrometer, a vacuum- 
tube electrometer, or an electron multi- 
plier tube. 

Figure 3.4 shows schematically a dou- 
ble-focusing mass spectrometer developed 
by Nier and Roberts (N21, J16). The 
ions are produced by electron impact or 
thermionic emission and then accelerated 
through a potential difference of about 
4,000 volts as they leave the ion source. 
This acceleration is equivalent to an energy 
filter of the type shown in Fig. 3.1b, so thali 
most of the ion current leaving the ion 
source is reasonably homogeneous in en- 
ergy. The ions then pass through a 90 
cylindrical-condenser electrostatic ana- 
lyzer followed by a 60 magnetic analyzer. 
The mass spectrum is scanned by varying 
the field in the electrostatic analyzer. 
Simultaneously, the accelerating potential 
at the ion-source exit is varied proportion- 



ately, both potentials being obtained from 
a common potential divider. The resolving power M/&M is comparable 
with the mass spectrograph shown in Fig. 3.3. 

Single-focusing Mass Spectrometer. Many varieties of single-focusing 
mass spectrometers have been developed. Usually, these are similar to 



3] 



Mass of Nuclei and of Neutral Atoms 



109 



the double-focusing mass spectrometer, but the electrostatic analyzer is 
omitted. Where only moderate resolution is required, as in the measure- 
ment of isotopic abundance ratios and for gas analyses in general, high 
ion intensities can be obtained by the use of relatively wide slits. A 
typical instrument (N17) is illustrated in Fig. 3.5. 

Sufficiently high resolution can be obtained from such instruments 
for accurate mass measurements on atomic or molecular ions (N10). 
However, molecular ion fragments cannot be used because these frag- 
ments acquire a sufficient initial kinetic energ3^ during the dissociation 
process in the ion source to spoil the resolution (N21). Accurate mass 
measurements are therefore usually carried out on double-focusing 
instruments. 




* Increasing mass 

Fig. 3.6 Mass triplet for singly ionized molecular ions each having mass 28. The 
resolving power is such that the peaks have a width AAf at half height of AM ~ M / 
4,600. The doublet separation (NJV - (C 1Z O 16 )+ is about 0.01 amu. [From a 
record taken on the mass spectrometer shown in Fig. 3 A, by Collins, Nier, and Johnson 

(C35).| 

c. The Doublet Method in Mass Spectroscopy. The high dispersion 
available in mass spectroscopes of the type illustrated by Figs. 3.3 and 
3.4 permits the complete resolution of ions having the same integral 
value of ne/M but slightly different exact numerical values. Thus the 
high accuracy always associated with measurements based on null meth- 
ods becomes available in these determinations, because differences in 
mass can be determined with high precision. By adjustment of the ion 
source or the exposure time, mass doublets may be obtained in which both 
components have about the same intensity and the same integral value 
of nc/M. The doublet separation is then a measure of the fractional 
difference in mass. The doublet method is illustrated by Fig. 3.6, which 
shows a doublet (actually a triplet) of singly ionized molecular ions, each 
of which has a total mass number of 28. 

The Fundamental O, C, D, H Doublets. Following an original experi- 
ment by Aston, many mass spectroscopists have measured a group of 
so-called "fundamental doublets" from which the masses of C 12 , H 2 , and 
H 1 can be obtained relative to O 16 . These three doublets are: (C 12 Hi) 4 
- (O 16 )+ (H 2 .)* - (C 12 )++, and (H l z ) + - (H 2 )+. For simplicity in nota- 
tion, we shall write 



C 12 = C 



H 2 = D H 1 = H 



Then the three fundamental doublets can be written as three simultane- 
ous equations in O, C, D, and H, with the observed doublet separations 
of a, 6, and c. 



110 



The Atomic Nucleus 

M/ne ^ 2: (Hi)+ - (H 2 )+ = 2H - D = a 
M/ne ~ 6: (H?)+ - (C 12 )++ = 3D - JC = b 
'. ^ 16: (C 12 HJ)+ - (O le )+ = C + 4H - 



fen. 3 



(3.11) 



Simultaneous solution of these three relationships, using = 16, gives 



H 1 H = 1 + Tir(6a + 26 + c) 
H 2 = D = 2 + i(-2a + 2b + c) 
C 12 - C = 12 + T(-6o - 26 + 3c) 



(3.12) 



AH masses and mass differences are in atomic mass units (abbreviated 
amu) defined by O 16 equals exactly 16 amu. 

The mass-spectroscopic literature has been summarized from time to 
time, with a view to deducing by least-squares analysis the "best values" 
of the three fundamental doublets and the corresponding masses of ( 11 ~, 

TABLE 3.1. DOUBLET SEPARATION AND DERIVED MASSES 

Summary, for the three fundamental doublets, of illustrative pre-1950 "host 
values' 1 of doublet separation and derived masses, with probable errors, from least- 
squares analysis of available mass-spectroscopic results. The L-ist line gives the 
results for the same quantities as obtained purely from nuclear-reaction data. 





Doublet separation 
AM, 10~ 3 amu 


Mass excess 
(M - A), l()- 3 amu 


H 2 -D 


D 3 -iC 


CH 4 - 


II 1 


D 2 


C' 2 


Mattauch 
(1940) 








8 130 
003 


14.722 
0.006 


3 861 
024 


Cohen and 
Hornyak 
(1947) 


1.539 
0.002 


42.230 
0.019 


36 369 
0.021 


8.1284 
0.0027 


14.718 
005 


3 847 
().U16 


Bainbridge 
(1948) 


1.5380 
0.0021 


42.228 
0.019 


36.369 
0.021 


8.1283 
0.0028 


14.7186 
0.0055 


3.856 
0.019 


Li et al. (1951) 

(L27) 


1.5494 
0.0024 


42.302 
0.016 


36.372 
0.019 


8.142 
0.003 


14.735 
0.006 


3.804 
0.017 



J. Mattauch, Phys. Rev., B7: 1155 (1940). 

E. R. Cohen and W. F. Hornyak, Phys. Rev., 72: 1127L (1947). 
K. T. Bainbridge, Isotopic Weights of the Fundamental Isotopes, Nail. Research 
Council Nuclear Science Rept. I, 1948. 

H 2 , and H 1 . The final results of three illustrative compilations are com- 
pared in Table 3.1. There are also given in Table 3.1 the entirely inde- 
pendent numerical values for the same doublets and atoms, derived from 
observations of the energy evolved in nuclear reactions. The basis of 
these data will be discussed in Sec. 4 of this chapter. Here we must note 
that the "best data' 1 are often mutually inconsistent, and by many times 
the probable errors assigned. 



3] 



Mass of Nuclei and of Neutral Atoms 



111 



As orientation, we uee in Fig. 3.7 a graphical comparison of measure- 
ments on the important, and troublesome, methane-oxygen doublet. 
The direct mass-spectroscopic measurements of this doublet vary by 
about 10~ 4 amu. As can be seen from Eqs. (3.12), this uncertainty 
in the quantity c has only a small absolute effect on the derived mass of 
II 1 and D 2 but affects the derived mass of C 12 by nearly + 10~ 4 amu. 

Doublet separation 
C 1 X - O 16 



0036 30 amu 003640 



0.036 50 





Bambridge and 
















Jordan (1936) 










3649 


in 

"c 


Mattauch (1938) 















36406 


? 


Jordan (1941) 


h 


O 

3632 


-4 








O /> 

S 


Ewald(1951) 
Ogata and 






36371 


10 


j 






Matsuda (1951) 








36447 






Roberts (1951) 










O^ 
















36478 




s 1* 


Nier(1951) 










O 


H 


^ w .0 

Q o -o 


Collins, Nier, 
and Johnson ( 1951) 








CH 
36427 


36484 




E 


Li, Whaling, Fowler, 














o _ m 


and Launtsen ( 1951 ,i 






36372 








c c S 
















0) "" 
















O 

















Fig. 3.7 Graphical (omparison of sonu* evaluations of the mass doublet Clh O. 
The reported numerical values are given in 11) aniu. Each author's* derlared prob- 
able error is shown graphically. fc>ix observations which are represent ativn of the 
direct in ass-spec troscopic results are given at the top. Below these are shown two 
evaluations calculated from differences between doublets containing sulfur t-oinpounds 
and measured with the mass spectrometer shown in Fig. 3.4. The solid point at the 
bottom is the calculated value which would agree with the energy liberation in a 
cycle of nuclear reactions. * [Rrfcrrnrrs: K. 7\ liwubridife and E. B. Jordan, Phyts. 
tor., 60: 282 (1930); J. Mattaurh, Physik. Z. y 39: 892 (1938); E. B. Jordan, Phys. 
Rev., 60: 710 (HM1); //. Euwld, Z. Naturfortch., 6a: 293 (1951); K. Ogata and H. 
Malxtulu, Phy*. Rev,, 83: 180 (1951); T. K. Rolnrtx, Phys. /^'., 81: 024L (1951); 
A. O. Nier, Phys. Rcr., 81: 024L (1951); T. L. Collinx, A. O. Xicr, and \V. H. John- 
son, Jr., Phys. Rev., 84: 717 (1951); C. \V. Li, W. Whaling, W. A. Fowler, and C. C. 
Launtsen, Phys. Rw., 83: 512 (1951). J 

Even so, the doublet method forms the basis for tables in which the 
uncertainties in atomic maws are only a few parts per million. 

These uncertainties lie far outside the probable errors assigned on a 
basis of the statistical reproducibility of the results. They are therefore 
due to unknown systematic errors in the particular techniques used. 
These systematic errors can presumably be isolated and reduced by 
extending the work to a number of other doublets. 



112 The Atomic Nucleus [CH. 3 

The S, O, C, H Doublets. Nier has utilized sulfur in order to evaluate 
the C 12 mass through a shorter cycle of doublets (N19, C35), using the 
mass spectrometer shown in Fig. 3.4. By introducing a mixture of O z 
and H2S into the ion source, the doublet (0 2 ) + (S 32 )+ is obtained in the 
ion beam. A mixture of C4H 6 and SO 2 in the ion source gives the doublet 
(C 4 ) + - (S 82 16 )+. These doublet separations can be written 

20" - S = a 

4CJ12 - S a2 lfl = 6 (6A6) 

There are now only two unknowns, C 12 and S 32 . The simultaneous solu- 
tions of Eq. (3.13) are 

C 12 = 12+i(b - a) (3.14) 

S 32 = 32 - a 

The mass of other light atoms can be obtained from additional doublets. 
For example, the hydrogen mass is obtained in this work by including 
the propane-carbon dioxide doublet (C^HJ C 12 Oo 6 ). These doublets 
have given (C35) 

H 1 = 1.008 146 0.000 003 

C 12 = 12.003 842 0.000 004 (3.15) 

S 82 = 31.982 236 0.000 007 

which are markedly different from the pre-1050 mass-spectroscopic "best 
values" shown in Table 3.1. 

Secondary Standards of Mass. Other atoms of low atomic weight may 
be compared with H 1 , H 2 , or C 12 by further application of the doublet 
method. For example, N 14 is obtained from the (N1 4 )+ - (C 12 O 16 )+ 
doublet, then the rare isotopes N 16 from (N lft )+ - (N^H 1 )* and O 18 
from (0 18 ) + (O 16 HJ) + . In this way, accurate masses of many of the 
principal stable isotopes of the lighter elements have been determined. 

Agreement on a few secondary standards of mass is an important 
prerequisite to the gradual completion of self-consistent mass tables, 
from which accurate calculations of nuclear binding energy can be made. 
Representative pre-1950 and post-1950 mass values for the original 
secondary standards H 1 , H 2 , and C 12 are compared graphically in Figs. 
3.8, 3.9, and 3.10. Even the more recent mass-spectroscopic values 
(except those of Ewald) tend to exceed the values derived from nuclear 
reactions, and by several times the probable error of measurement. In 
1954, general agreement on mass substandards was still to be achieved. 
In the meantime, all new mass values must be given relative to an arbitrary 
choice of reference substandard. 

The masses of the secondary standards H 1 , H z , and C 12 appear to be 
known only to about 20 X 10~ B amu at present, although each indi- 
vidual evaluation is always reported to the nearest 10~ 6 amu. 

Masses of Heavy Nuclei. Knowledge of the masses of middleweight 
and heavy nuclei is particularly important for the evaluation of nuclear 
forces and nuclear shell structure. Many of the mass values now avail- 
able were obtained by using the mass spectroscope as an absolute instru- 



S3] 



Mass of Nuclei and of Nealral Atoms 



113 



Atomic mass of H 1 

1.008 120 140 160 1.008 180 amu 



5 


Mattauch(1940) 




-OH 

8130 








m fundamel 
OCDH 
doublets 


Cohen et al(l 947) 
Bambridge (1948) 
Roberts (1951) 




8*128 
8*128 




1 




o 










8.169 




**" 


Ewald(1951) 




8.141 






15 1 


Nier ( 1951 ) 






i-C 

8.1 


H 

59 




^-5) o 


Collins etal (1951) 






HD- 






" 








8.146 






EIS 


Lietal(1951) 




t- 
8 


H 

142 






3 ID 

c o> 















Fig. 3.8 Graphical comparison of values recommended for the mass of hydrogen. 
The first three entries are from the compilations reported in Table 3.1. Roberts's 
value uses the first two doublets of Eq. (3.11) combined with the doublet difference 
2(D 2 O 16 iA 40 ) (CaH 4 A 4 ' 1 ). The numerical value shown under each point is 
the mass excess (M A) in millimass units (10~ 3 amu). Space has been left for 
the reader to add later "best values" as they develop in the literature. 



Atomic mass of H 2 
2.014720 740 760 2.014 780 amu 



I 


Mattauch (1940) 


14 


ol 
722 








pS!|j 


Cohen et al ( 1947) 


t 0-H 

14.716 








isl 


Bainbridge (1948) 


I-OH 

14.719 








1 


Roberts (1951) 








H 
1 


1 
4.765 




Ewald(1951) 




I-O-H 

14732 








Eg| 


Nier (1951) 








i 


i 


iS g 










147 


78 


1. " 


Lietal(1951) 




14.735 









Fig. 8.0 Graphical comparison of values for the mass of deuterium, 
notation are similar to Fig. 3.8. 



Unite and 



114 



The Atomic Nucleus 



[CH. 3 



merit. This involves careful calibration of the mass scale and is anal- 
ogous to methods regularly employed in optical spectroscopy. For 
heavy nuclei a precision as high as 1 part in 10 B corresponds to an uncer- 
tainty of the order of 0.002 amu, which is undesirably large. 

The doublet method provides much greater accuracy. The masses 
of heavy nuclei may be measured by comparing multiply charged ions 
of the heavy elements with singly charged light ions, since both can then 
possess comparable ne/M values. 



Atomic mass of C 12 
12 003 800 amu 12003840 



12.003 880 













1 




_ 


Mattauch (1940) 








k 















3861 




Of 


Cohen et al (1947) 














E x 










3847 






^3 ^ A 


Bambridce ( 1948 ) 














O o 










3856 






O 


Roberts ( 1951 ) 


3803 












Ewald(1951) 


1 . O t 
















3807 










E xS 

ig 2 


Nier (1951) 








t OH 
3850 








Collins et al ( 1951 ) 






i-CH 






" 








3 


342 






o> v c 

5" e 


Lietal(1951) 














3804 


" = 
















= CD 
















e^ 

u. 

















Fig. 3.10 Graphical comparisoii of values for the mass of C 12 . 
are similar to Fig. 8.8. 



Units and not al ion 



Dempster (D21) began the invasion of the heavy nuclidc masses by 
the doublet method in 1938, using such pairs as 

(<) 16 )+ - (Ti 4h )+++ and (Ti 5 ")^ - (Au l ' J7 ) ++++ 

Doublet measurements among the middleweight and heavy nuclei 
will be one of the most important contributions to nuclear physics which 
mass spectroscopy will make in the next decade. These heavy masses 
cannot be determined on an absolute scale from the energetics of nuclear 
reactions involving only hydrogen and helium ions, because the heavy 
masses are so far removed from that of oxygen. Nuclear reactions 
induced by accelerated heavy ions (carbon, sulfur, etc.) will eventually 
permit extension of the nuclear-reaction mass scale to the heavy nuclides. 
Until this is accomplished, it is necessary to have mass-spec, troscopically 
determined secondary standards among the heavier nuclides. Then 
nuclear-reaction data can be used to complete the mass data over a wider 
domain of mass numbers. 

Examples of systematic studies of this type are beginning to appear. 
In the mass region from A = 31 to 93, Colling Nier, and Johnson (C35, 



3] 



Mass of Nuclei and of \eutral Aloms 



115 



C36, C34) have reported about 70 atomic masses, from S 32 to Nb 93 
(Table 5.2), relative to their own substandards of H 1 , C 12 , and S 32 . 
These new masses, when combined with data on nudear-reaction energies, 
give mass values for many of the unstable nuclides in this mass domain 
(C36, C34). 

Problems 

1. In a radial electrostatic filter, find the field strength in volts per centi- 
meter necessary to bend singly ionized atomic, oxygon ions in an arc having a 
25-cm radius, if the ions have fallen through a potential difference of 20,000 volts 
before entering the filter. Arts.: 1,0*00 volts, cm. 

2. For the same oxygen ions, find the magnetic field necessary to bend them 
in an arc having a 25-cm radius. A w*.: 3,260 gauss. 

3. What is the optimum relationship between the angles d and 7 of the magnet 
faces in Fig. 3.2 for minimum line width, or "spread," in a homogeneous mag- 
netic field, Kq. (3.4), if (d + 7) is held constant? 

4. Derive from first principles the expressions for the lateral spread and 
velocity dispersion in the focus produced by a homogeneous magnetic field, for the 
particular case of a symmetrical sector magnet, and (a z 1). 

5. How would you modify the sector magnet of Fig. 3.2 to obtain an instru- 
ment which would (a) accept a bundle of rays diverging by a and produce a 
substantially parallel, or "collimated," beam and (It] accept a parallel beam of 
rays and produce a beam converging to a focal region? 

6. In a crossed magnetic and electric field velocity filter, find the magnetic 
field necessary to filter 20,000-ev singly charged argon ions (A 40 ) if the electric 
field is 2,000 volts, cm. Ans.: ti,430 gauss. 

7. The two stable isotopes of silver Ag 107 and Ag 109 are to be separated, using 
electromagnetic means. The singly charged ions are first accelerated through 




an electrostatic potential of 10 kv and then deflected in a uniform magnetic field 
through a semicircular path of radius 1 in. 

(a) What magnetic field intensity is required? 

(6) Assuming that the entrance and exit slits are the same size, and that the 
entrance slit is imaged perfectly on the exit slit, calculate the maximum slit 
width for which the two isotopes will be completely separated. 

Ans.: (a) 1,490 gauss; (fc) 1.87 cm. 

8. A uniform magnetic field is to be used as a momentum filter for high-energy 
particles. A slit system is adjusted to allow the passage only of particles having 
a radius of curvature p. With a magnetic field of B gauss, it is found that the 



116 The Atomic Nucleus [CH. 3 

filter transmits polonium a rays, whose energy is 5.30 Mev. The magnetic field 
10 now raised to 2.3B and deuterons are passed into the filter. What is the energy 
of the deuterons transmitted by the filter? Ana.: 14.0 Mev. 

9. Protons are accelerated through a potential difference of 2,50 million volts 
in a certain discharge tube and are then deflected into a circular orbit of radius p 
in a uniform magnetic field of 8,000 gauss. Without altering the slit system, 
what magnetic field would be required to deflect through the same path in the 
magnetic field a beam of doubly ionized helium (artificial a rays) which had been 
accelerated through a potential difference of 2.00 million volts? Ans.: 10.1 
kilogauss. 

10. The following is a diagrammatic representation of a mass spectrograph 
used by A. J. Dempster. Ionized particles from the source are accelerated 
through the slits by the potential U into a region where they are deflected by a 
uniform magnetic field B and recorded on the photographic plate. 



Photo plate 




U 



(a) Derive an expression for y in terms of the charge on the particle ne, its 
M, and the fields of U and B. 

(b) The apparatus can be used to determine mass differences in the following 
way: With U and B held constant, a line is recorded at y } for singly charged light 
hydrogen molecules (HJ)+ and then a line at y z for singly charged atomic deu- 
terium (H 2 )+. Call the absolute separation between these lines \y\ y t \ Ay, 
and their average value (y\ + yJ/2 s y. Show that the absolute value of the 
mass difference is 



grams 



where H 1 and H 2 represent the neutral atomic masses of hydrogen and deuterium, 
and the binding energy of the atomic electrons has been neglected in comparison 
with their mass. 

(c) In order to eliminate errors in B, U, and # from the determinations, it is 
customary to hold B and U constant and then compare the separations for several 
mass doublets. Under these conditions, y remains essentially constant. In the 
apparatus described, the absolute values of two doublet separations are 

A|/H = 0.097 mm for (HJ) + and (H a )+ 
Ay H . = 0.897 mm for (HJ) + and (He<)++ 

In both these doublets, y is larger for (HJ) + than for the other member of the 
doublet. From these data, find the mass of the neutral H 2 atom if the neutral 
atomic masses for hydrogen and helium are H 1 = 1.008 14 amu and 

He 4 = 4.003 87 amu 

Include the effects, if any, of the electron masses, as in (He)++, but neglect the 
atomic binding energy of these electrons. Ans.: 2.014 73 amu. 

11. Show that the mass correction for electron deficiencies is the same in both 



4] Mass of Nucki and of Neutral Atoms 117 

ions of a doublet. Thus the mass difference for the corresponding neutral mole- 
cules is the same as the observed doublet separation for the ions, and no net 
correction is needed for electron masses. For example, 

(D 3 ) + - (C 12 ) + + = 3D - C 

Is this relationship also true when the ratios of charge to mass number are unequal, 
as in (Ti 60 )+ - (Au 197 )++++ = Ti BO - |Au 197 ? 

12. From the mass-spectroscopic doublets (N19) 

C0 2 - CS = b = 17.78 X 10~ 3 amu 
C 3 H 8 - C0 2 = c = 72.97 X 10~ 3 amu 
C 6 H 4 - CS 2 = d = 87.33 X 10~ 3 amu 

(where S ^ S 32 , e. O 16 , C = C 12 , and H ^ H 1 ), show that 
H = 1 + A(4& + 5c - 2d) 



and determine the mass of hydrogen which is obtained from this group of doublets. 
Ans.: 1.008 166 amu. 

13. If the nitrogen-carbon monoxide doublet separation 

(N I4 ) 2 - (C 12 16 ) = 11.28 millimass units 

what would you cite as the atomic mass of N 14 ? [Data from Nier, Phys. Rev., 
81:624L(1951).] 

14. From the doublets (C36) : 

(S^OJ 6 )* - (Zn 64 ) + = (326.82 0.20) X 10~ 4 amu 
(OJ 6 ) + - (Zn G4 )++ = (252.46 0.22) X 10 ~ 4 amu 

determine the atomic mass of S 32 and Zn 64 . Ans.: 31.982 190 0.000 049 amu; 
63.949 508 0.000 044 amu. 

4. Atomic Mass from Nuclear Disintegration Energies 

Einstein's special theory of relativity requires that the inertial mass m 
of any body moving at velocity V with respect to the observer be given 
by 

w 

m = --- ===== (4 l) 

Vl - (F/c) 2 l } 

where c is the velocity of light and ra is the rest mass of the body. The 
relativistic momentum is mV. Direct experimental verification of this 
dependence of mass on velocity is obtained from observations of the 
magnetic deflection of the high-speed electrons emitted by radioactive 
substances (Zl). 

The kinetic energy T is given by 

T = moc 2 [ . 1 _ - ll 
LVl - (V/cY \ 

= me 2 WoC 2 = (m m )c 2 = Arac 2 (4.2) 

where moc 2 is the rest energy and me 2 is called the total energy. The 



118 The Atomic Nucleus [CH. 3 

kinetic energy therefore is equivalent to an increase of mass 

T 

(m 77Z ) = - 

The principle of conservation of energy includes mass as a form of 
energy, on a par with chemical, electrical, mechanical, and other common 
forms. The well-known chemical law relating to the conservation of 
mass in all chemical reactions is not rigorously true. The actual devia- 
tions from strict mass conservation arc simply too small to be detected 
in the case of chemical reactions. Nevertheless, the release of energy T 
in a chemical reaction must be accompanied by a slight reduction Aw 
in the total rest mass of the reacting components, in accord with 

- n - Aw (4.3) 

f- 

The energy per atom involved in nuclear processes is vastly greater 
than in chemical processes and thus constitutes the best test and an every- 
day application of the so-called Einstein law, Eq. (4.3). 

a. The Mass Equivalent of Energy. We need to evaluate the numer- 
ical factors connecting mass and energy. The unit of energy used most 
commonly in nuclear physics is 1 million electron-volts, abbreviated 
Mev. This is the kinetic energy acquired by an individual particle 
carrying a single electronic charge and falling through a potential differ- 
ence of 10 6 volts. By definition, 1 wt at volt = (c/ 10 s ) volts; hence 



-. TV/T ,/^c i, f I statvolt 1 

1 Mev = iO 6 volts I ---- 
I (c/10 s ) volts J 



c esu 






= - IO 11 ergs (4.4) 

c 

For a single particle, the numerical equivalence between atomic mass 
units and energy is 

1 c 2 

1 amu = g = ^ ergs (4.5) 

in which N is the number of atoms in 1 mole on the physical sra/r, e.g., in 
16 g of O 16 .f (For the unified scale, C 12 = 12 u, sec p. 264.) 

~ If F is the faraday constant, in coulombs per mole on the physical 
scale, then 

(4.6) 



1 faraday - N. - (F 5*SS*5\ \(*/M" \ =F 
\ mole / [1 coulomb J 10 



10 mole 



Eliminating Avogadro's number (N) between Eqs. (4.5) and (4.6), then 
using the conversion factor of Eq. (4.4), we have 

, lOce f 1 Mev "I c z in ., __ ,. _. 

1 amu = - ergs - - = 10~ 13 Mev (4.7) 

F | (IO 14 e/c) ergsj F 

t For the conversions to the unified mass scale (C 12 - 12) sec pp. 37-39 of the 
Instructor's Manual to accompany The Atomic Nucleus. 



4] Mass of Nuclei and of Neutral Atoms 119 

It should be noted that the relationship between Mev and ergs, Eq. (4.4), 
depends on the numerical value adopted for the charge on the electron, 
whereas the relationship between Mev and amu, Eq. (4.7), is independent 
of e. Using the 1952 values (D44) of 

e = (4.8029 0.0002) X 10~ 10 esu 
c = (2.997 929 0.000 008) X 10 10 cm/sec 
and F = 90,520.1 2.5 coulombs/mole physical scale 

gives the currently adopted relationships 

1 Mev = (1.602 07 + 0.000 07) X 10~ G erg (4.8) 

1 amu = 931.102 0.024 Mev (4.9) 

The reciprocal value is often useful and is 

1 Mev = 0.001 073 93 0.000 000 03 amu (4.10) 

or, in round numbers. 1 Mev ~ 1 millimasH unit. 

The rest mass of one electron w? is 1/1,830.1,3 times the proton mass, 
or 0.000 548 76 amu, and corresponds to an energy 

Wor 2 = 0.510 98 + 0.000 02 Mev (4.11) 

This "rest energy 1 ' m c 2 of an electron is used commonly as a natural 
basic unit of atomic and nuclear energy. 

b. Experimental Verification of the Equivalence of Mass and Energy. 
The first direct experimental comparison of the energy liberated in a 
nuclear reaction and the accompanying change in total rest mass was 
made in 1933 by Bainbridge (B3), using the reaction Li 7 + H 1 He 4 
+ He 4 . A more accurate comparison is obtained from nuclear reactions 
in which the masses are known from mass-spcctroscopic doublet data 
(J19). 

Q Values of Reactions Compared with Mass-spcctroscopic Doublets. 
When nitrogen is bombarded by high-speed deuterons a nuclear reaction 
takes place in which energetic protons are produced. The nuclear reac- 
tion is 

N 14 + H 2 -> N u + H l + Q (4.12) 

where Q is the kinetic energy released in the reaction when N 15 is formed 
in its ground level. A discussion of the methods of measuring Q values 
will be found in Chap. 12. Here it is sufficient to recognize Q as the 
difference between the kinetic energy of the products and the kinetic 
energy of the original particles in any nuclear reaction. When the 
products have greater kinetic energy than the original particles, Q is 
positive, and the reaction is said to be exocrgic since it releases kinetic 
energy by the conversion of a portion of the rest mass into kinetic energy. 
Conversely, when the products have less kinetic energy than the original 
particles, the reaction is called cndocrgic, and Q is negative and numer- 
ically equal to the kinetic energy converted into rest mass in the reaction. 
The reaction Eq. (4.12) is strongly exoergic; energy measurements 
on the ejected protons give Q = 8.615 0.009 Mev. Equation (4.12) 



120 



The Atomic Nucleus 



[CH. 3 



can be rewritten to represent the difference between two mass doublets. 
Solving for Q, after adding H 1 to both sides, we obtain 

Q = (N 14 H! - N 1B ) - (2H 1 - H 2 ) (4.13) 



Excited 

compound 

nucleus 




Substituting the mass differences observed for these doublets (J19), we 
find 

Q = (0.010 74 0.0002) - (0.001 53 0.000 04) amu 

= 0.009 21 0.0002 amu (4.14) 

= 8.57 0.2 Mev 

Thus the combined rest mass of N 14 and II 2 is 0.009 21 amu greater than 
the total rest mass of the products N 15 and H 1 , and the mass decrease 

in the reaction appears quantita- 
tively as kinetic energy of the reac- 
tion products. 

This mass-energy balance, and 
many others like it, shows also that 
the, stable atoms produced by nuclear 
disintegration are indistinguishable 
from their sister atoms found in na- 
ture. Regardless of their mode of 
origin, the ground levels of all nu- 
clei of any given nuclide appear to 
be completely identical. 

The exact equivalence of mass 
and energy is one of the most 
firmly established principles of mod- 
ern physics. By means of this 
principle and the large number of 
available nuclear reactions, relative 
masses of the atoms can be estab- 
lished independently of mass-spec- 
troscopic data. There are many 
cases, particularly the neutron and 
a number of rare or radioactive nu- 
clides, where the mass spectrograph cannot be used, and so disintegration 
masses are used to complement the mass-spectroscopic data in compiling 
our mass tables. 

Energy Diagram for Nuclear Reactions. It is useful to visualize the 
energetics of nuclear reactions by means of energy diagrams. Figure 4.1 
portrays the energetics of the reaction of Eq. (4.12). The total rest 
mass of the original particles is shown in the left-hand scale of the dia- 
gram and has the numerical value 



Ground level 

of compound 

nucleus 

"""&*'" 



16.03- 



I 16.02- 



16.0L- 



16.00- 



20- 



1 10- 



o- 



Fig. 4.1 Typical graphical representa- 
tion of the energetics of a nuclear re- 
action. The numerical equivalence of 
changes in rest mass and kinetic energy 
is shown on the matched energy and mass 
scales. 



N 14 = 14.007 515 

IP = 2.014 735 

N 14 + H 2 



16.022 250 = 16 amu + 20.71 Mev 



(4.15) 



The total rest mass of the reaction products, when N 1B is formed in its 



4] Mass of Nuclei and of Neutral Atoms 121 

ground level, is shown at the right and is 

N 16 = 15.004 863 
H 1 = 1.008 142 
N" + H 1 = 16.013 005 ~ 16 amu + 12.10 Mev (4.16) 

The reaction takes place when N 14 and H 2 approach each other with a 
kinetic energy in center-of-mass coordinates which is adequate for trans- 
mission through the potential barrier. The colliding particles form a 
compound nucleus, which in this case is N 14 + H 2 ^> (O 16 ). Thiscom- 
pound nucleus has an internal excitation energy which is generally so 
high that it lies in the region of broad overlapping excited levels which 
act like a continuum. Thus the compound nucleus caD be formed over a 
broad and continuous domain of incident kinetic energy TO (in). The 
mass of the excited compound nucleus is shown in the center of Fig. 4.1. 

Generally, the compound nucleus can disintegrate in any one of a 
number of competing modes, or "exit channels.' 1 If a proton is emitted, 
and the residual N 1B is left in its lowest possible level of internal energy, 
then the mutual kinetic energy of the reaction products will be T (out) 
in center-of-mass coordinates, as shown in Fig. 4.1. The Q value of the 
reaction is 

Q = To (out) - To (in) (4.17) 

and this is equal to the mass difference between the original particles 
and the reaction products, Eqs. (4.15) and (4.16), as can be seen graph- 
ically in Fig. 4.1. 

Atomic masses, rather than nuclear masses, usually are used through- 
out. At each stage of the reaction the atomic masses exceed the nuclear 
masses by the same number of atomic electrons. If the small differences 
in the binding energies of the atomic electrons are neglected, then all 
mass differences in Eqs. (4.12) to (4.16) and in Fig. 4.1 arc the same for 
atomic masses as they would be for nuclear masses. 

c. The Energetics of Radioactive Decay. Quite accurate determina- 
tions of atomic-mass differences can be obtained from measurements of 
the energy of a, ft, and y rays, provided that the radioactive-decay scheme 
of the parent nuclide is known. 

Negatron ft Decay. In that class of ft decay which involves the emis- 
sion of a negative electron, a nucleus of charge Z and nudcar mass Z M' 
transforms spontaneously into a nucleus of charge (Z + 1) and mass 
z+iM'. Mass-energetically, this can be written 

Z M' = z+iM' + m + v + T0- + T; + T u > + T y (4.18) 

where m = rest mass of ft ray 

v a rest mass of accompanying antineutrino 
TV = kinetic energy of ft ray 
TV = kinetic energy of antineutrino 
TV = kinetic energy of recoil nucleus z+iAf' 
T 7 = total T-ray energy emitted after ft ray 



122 The Atomic Nucleus [CH. 3 

Experimentally, v = 0; T0 + T v = T^* is the measured maximum 
kinetic energy or k 'end point'' of the continuous /9-ray spectrum, and 
T M ' 7'*- Then we can call the observed decay energy To, where to a 
good approximation 

To = r m + T T (4.19) 

Then the relationship between the nuclear masses, Eq. (4.18), becomes 
Z M' = z ^M r + m + To (4.20) 

If we now add the mass of Z atomic electrons to both sides, we have 

( Z M' + Zm ) = [,+iJlf + (Z + |)TO O ] + TO (4.21) 

which, if the small difference in the binding energy of the atomic electrons 
be neglected, is 

Z M = Z ^M + T for 0~ decay (4.21a) 

where 7 M and z+i3/ are 'he neutral atomic masses of the parent and 
product of negatron ft decay. Utilizing the letter Q, Avith appropriate 
subscripts, to denote the differences in atomic mass which occur in radio- 
active decay, we write for ft~ decay 

Qp -= Z M - z+iM = To = T mn + 7% (4.22) 

Thus, to a good approximation, the total kinetic energy of decay To 
equals the difference in neutral atomic mass Qp- between the isoharic 
parent and decay-product atoms. 

Energy-level Diagram* for Negatron 8 Decay. Many types of energy- 
level diagram have beer proposed for displaying the decay schemes of 
radioactive substances. The two most fundamental varieties are shown 
in Fig. 4.2; where the energetics of the negatron decay of Al 28 are 
plotted. This nurlide decays by emitting a simple continuous spectrum 
of negatron ft rays whose maximum kinetic energy is 2.865 0.010 Mey 
(M71). After the emission of the ft ray, each residual nucleus of Si ?H 
is left in an excite.d level at about 1.78 Mev above its ground level. The 
prompt emission by Si 2 * of a single 7 ray, whose measured quantum 
energy is 1.782 6.010 Mev (M71), puts this decay product into its 
ground level. 

The measured decay energy T of Al~ s is, from Eq. (4.19), 2.865 
+ 1.782^4.65 Mev, or 4.99 X 10~ 3 ainu. The change in nuclear 
mass, Eq. (4.20), equals this sum of the -ray and 7-ray energies plus the 
rest mass m Q of the emitted ft ray. 

The right-hand side of Fig. 4.2 is the Al 28 decay scheme on the nuclear 
mass-energy scrJe. The ground level of Al 28 is 

T Q + m = 4.G5 + 0.51 - 5.16 Mev = 5.54 X 10~ 3 amu 

above the ground level of Si 28 . The diagonal line represents the rest 
energy moc 2 = 0-51 Mev of the ft ray and also the increase of one unit in 
nuclear charge when the 0-ray negatron is emitted. The vertical line 
represents the kinetic energy 2.865 Mev of the electron-neutrino pair, as 



4] 



Mass of Nuclei and of Neulral Atoms 



123 



measured by the maximum energy of the /3-ray spectrum. The wavj r 
line shows the final 7-ray transition in Si' Jg . 

The left-hand side of Fig. 4.2 shows the same transitions but drawn 
on the more common atomic mass-energy scale. Tho atomic mass of 
Al 28 includes the mass of its 13 atomic electrons. One more? atomic 
electron is required in the product atom Si 28 than in the parent Al- s 
From the standpoint of over-all mass balance, we can imagine that the 
emitted /5-ray electron eventually joins the product atom as this addi- 
tional atomic electron. Hence no mass rorrectioji T/IU is needed, and thu 



27.991 




27.985 - 



Fig. 4.2 Negalron $ decay scheme for A1 2R . At loft, the* conventional diagram in 
which nuclear energy levels and transitions aro plotted against a scale of atomic mass. 
At right, the nuclear energy-level and mass diagram. An absolute mass scale for the 
nuclear mas^-encrgy diagram would equal the atomic-imisH seal* 1 diminished by about 
the mass of 14 atomic electrons, Eq. (2.1). 



difference in atomic mass Qp is given by T Q . 
we find from Eq. (4.22) that 



Thus, using atomic masses, 



where Qp- = 4.65 Mev = 0.004 99 amu 

Taking (L26) Si 28 as 27.985 77 amu, we find 

Al 28 = 27.990 76 amu 

These atomic-mass values, as well as the corresponding Qp- and ra values, 
are shown in the dual scales of mass and of energy in Fig. 4.2. 



124 The Atomic Nucleus [CH. 3 

Positron ft Decay. In the class of ft decay which involves the emission 
of a positive electron, a nucleus of charge Z and nuclear mass Z M ' trans- 
forms into a nucleus of charge (Z 1) and mass Z -\M'. In a manner 
which is completely analogous to that of Eqs. (4.18) to (4.20), we con- 
clude that for positron ft decay the change in nuclear mass is the same 
as for negation ft decay, that is, (T + m ), where m is the positron mass. 
Thus for positron decay, Eq. (4.20) becomes 

Z M' = z-iM' + m + To (4.23) 

When we now add the mass of Z atomic electrons to both sides and, as 
before, neglect the difference in the binding energy of the atomic electrons, 
we find 

Z M = z-iM + 2m + To for 0+ decay (4.23a) 

where Z M and Z -\M are the neutral atomic masses of the parent and 
product of positron decay. Writing Qp+ as the atomic mass difference in 
positron ft decay, we have 

Qp+ = Z M - z-iM = 2m c 2 + To = 2moc 2 + T m + T T (4.24) 

In contrast with all other typos of disintegration and decay reactions, 
the positron decay energy T is not directly equal to the change in neutral 
atomic mass. The positron decay energy must be increased by the rest 
energy 2m c 2 of two electrons, as seen in Eq. (4.24). 

The physical origin of this 2wi c 2 correction term should be understood 
clearly. When a radioactive nucleus emits a positron, the nuclear charge 
decreases by one unit, and the product nucleus requires one less atomic 
electron than its parent did in order to form a neutral atom. The decay 
therefore liberates an electron from the extrariuclear structure at the 
same time that it emits a positron from the nucleus. Because their 
masses are equal, we can account for the atomic electron and the positron 
by adding 2m to the products of the reaction, as in Eq. (4.23a). The 
subsequent events are worth following. The positron lives only about 
10~ 10 sec and then combines with some atomic electron. Both are anni- 
hilated, and the mass energy 2m c 2 appears ordinarily as two photons 
each having a quantum energy of ra c 2 = 0.511 Mev. Thus in Eq. 
(4.24) we may regard the annihilation quanta as an additional energy of 
2moc 2 = 1.022 Mev which is always emitted as an ultimate consequence 
of positron ft decay. 

Energy-level Diagrams for Positron ft Decay. Diagrams for positron 
decay schemes require a slightly special treatment in order to present 
the 2moc 2 term. In Fig. 4.3, the nuclear mass-energy diagram on the 
right is completely analogous to the ncgatron ft decay situation in Fig. 
4.2. This identity arises from Eqs. (4.20) and (4.23). From a nuclear 
standpoint, either type of ft decay requires an expenditure of w for the 
production of the rest mass of the negatron or the positron ft ray. 

The atomic-mass diagrams differ. One common method of repre- 
senting the 2moc 2 term is shown on the left of Fig. 4.3. Here the two 



4] 



Mass of Nuclei and of Neutral Atoms 



125 



electron masses are regarded as subtracted from the mass of the parent 
Na 22 ; then the diagonal line represents only the kinetic energy of the 
positron-neutrino pair, as evaluated by the maximum energy of the 
positron 0-ray spectrum. 

The atomic mass of Na 22 , from Eq. (4.24) or Fig. 4.3, is given by 



Na 22 = Ne 22 

= 21.998 36 amu + 2 X (0.000 549) amu + 0.542 Mev 

+ 1.277 Mev 
= 22.001 41 amu 

Electron-capture Transitions. Radioactive decay by electron capture 
competes with all cases of positron decay. The parent nucleus Z M' 
captures one of its own atomic electrons and emits a neutrino. The final 
decay product nucleus has charge (Z 1) and mass z ~iM' after the 



22.002 



22.001 



22.000 







"5 

*> 21.999 



21.998 



,Na 22 



2m c : 



^0.542 Mev 



1.277 Mev 




^0 

1 



Atomic mass-energy 
diagram 




Nuclear mass-energy 
diagram 



'I 



Fig. 4.3 Positron ft decay scheme of Na 22 (competing modes < 1 per cent not shown). 
Scales analogous to Fig. 4.2. Note in the atomic-mass diagram that among the prod- 
ucts of the decay process the total mass 2mo of the positron and one atomic electron 
r-an be regarded physically as cither a mass product (before annihilation) or an energy 
product (after annihilation). Compare Eq. (4.24). 

emission of any 7 rays which accompany the over-all transition. Mass- 
energetically, the transition corresponds to 

Z M' + m Q = z .,M f + v + T, + T M > + T y (4.25) 

Experimentally, the total 7-ray energy 7% is relatively easy to measure, 
the recoil energy T M has been measured in a few cases (Chap. 17, Sec. 2), 
the neutrino energy 7\ cannot be measured directly, and the rest mass v 
of the neutrino is zero. In general, T v T*: The decay energy 

To = T v + T M ' + T, ~ T, + T y (4.26) 

is therefore not directly measurable for electron-capture (EC) transitions. 



126 



The Atomic Nucleus 



[CH. 3 



Now we can add (Z 1) electrons to both sides of Eq. (4.25) and, 
if differences in atomic binding energy are neglected, we obtain 



or 



Z M = 
zM - z-i 



+ To 

T = T, + 



(4.27) 



Thus, to a good approximation, the electron-capture decay energy TQ 
is equal to the difference in atomic mass Q E c between the isobaric parent 
and product. 

Energy-level Diagrams for Electron Capture. Figure 4.4 illustrates a 
case of electron capture for which the competing positron decay is 
excluded energetically. The difference in atomic mass 4 Be 7 3 Li 7 is 
known from nuclear disintegration data to be only about 0.93 X 10"' 



E 7.020 

ro 

.5 

E 7.019 



"5 7.018 



0.478Mev(y 
""" 



Atomic mass-energy 
diagram 



r 

0.478 



Nuclear mass-energy 
diagram 



"i 



Fig. 4.4 Electron-capture decay scheme of Be 7 . Scales analogous to Fig. 4.2. The 
neutrino energies T v are shown dotted. About 1 1 per cent of the transitions involve 
a low-energy neutrino group followed by a 0.478-Mev y ray. The remaining tran- 
sitions go directly to the ground level of Li 7 and involve neutrinos whose kinetic 
energy equals, to a good approximation, the difference in atomic mass QEC between 
the isobaric parent and product atoms. 

amu, or 0.86 Mev. Because the Be atom contains one more atomic 
electron than the Li atom, the difference in nuclear mass is only 



(0.93 



X 10- 3 ) - (0.55 X 10- 3 ) = (0.38 X 10~ 3 ) amu = 0.35 Mev 



The minimum mass difference required for positron decay [Eqs. (4.23) 
and (4.23a)] is therefore not available. 

In the nuclear mass-energy diagram, on the right-hand side of Fig. 4.4, 
the diagonal line shows the Be 7 nucleus gaining the mass m of the 
captured electron and, at the same time, decreasing its charge to that of 
Li. Simultaneously, the neutrino is emitted. Its energy is represented 
by the vertical line, which is dotted as an indication that this kinetic 
energy cannot be measured directly. In about 11 per cent of the Be 7 
* Li 7 transitions a low-energy neutrino is emitted, and the Li 7 nucleus 
is left in an excited level at 0.478 Mev. A 7-ray transition to the ground 
level of Li 7 follows promptly. 

Note that in an electron-capture transition the neutrinos are emitted 
in one or more monoenergetic groups, in contrast with the continuous 



4] 



Mass of Nuclei and of Neutral Atoms 



127 



distributions of neutrino energy found in ft decay. Each neutrino in the 
transition to the ground level of Li 7 has a kinetic energy of 0.86 Mev. 
The neutrinos in the transition to the excited level of Li 7 each have 
T v = 0.86 - 0.478 = 0.38 Mev. 

Competition between Electron Capture and Positron ft-Ray Emission. 
Equation (4.27) shows that the electron-capture transition, which is 
permitted energetically whenever Q EC > 0, can occur whenever Z M 
> X -\M. Comparison with Eq. (4.24) shows that the competing posi- 
tron ft decay is possible energetically only if Z M > 2mo + Z -\M. Transi- 
tions between isobars whose mass is nearly the same may therefore take 



0.001 
8 0.0005 

ra 

u 



> 
3 
& 



r 



Atomic mass-energy 
diagram 




Nuclear mass-energy 
diagram 



0.5 .E 




Fig. 4.5 These schematic diagrams emphasize the difference between the energetics 
of electron capture and positron ft decay. Electron-capture transitions can take place 
to any level in the domain shown (2m c 2 wide) but positron transitions are excluded. 
Compare Eqs. (4.24) and (4.27). The nuclear diagram shows more clearly that the 
origin of this 2m r 2 band is the difference between the capture of m Q and the emission 
of m by a nucleus. Note from the nuclear diagram that the electron-capture tran- 
sitions can "climb," i.e., the final nucleus can be even heavier than the initial nucleus. 

place by electron capture even when positron ft decay is excluded ener- 
getically. In the domain of mass differences 



< ( Z M - Z -,M) < 2m 



(4.28) 



only electron- capture transitions can take place between isobars, as can 
be visualized from Fig. 4.5. 

Dual ft Decay. There is a large class of radioactive nuclei which 
have odd Z and even A. Many of these have stable neighboring isobars 
at both Z 1 and Z + 1. These radioactive nuclei may therefore 
transform by negatron ft decay, by electron capture, and by positron 
ft decay. A familiar example is Cu 64 , which is illustrated in the decay 
schemes and energy-level diagrams of Fig. 4.6. Such cases of dual ft 
decay provide a means of determining mass differences between pairs of 
stable isobars which differ in atomic number by two units. 

In the case of Cu 64 



29Cu 64 = BO Zn 64 + T 
2 gCu 64 = 2H Ni 64 + 2w c 2 



(4.29) 
(4.30) 



128 The Atomic Nucleus 

Eliminating the radioactive nuclide by subtraction, we obtain 

30 Zn 4 - 



= 2m c 2 + T 

= 1.02 + 0.66 - 0.57 Mev 

= 1.11 Mev = 0.001 19 amu 



[CH. 3 



(4.31) 



Note that, systematic errors in the measurement of the end points of the 
two /8 spectra tend to cancel in the determination of the mass difference. 



"0.002 
i/i 



| 0.001 
15 

.1 



5 




,Cu" 




Atomic mass-energy 
diagram 




Nuclear mass-energy 
diagram 



Fig, 4.6 Principal dual 0-decay and electron-capture transitions of Cu 84 . The 
approximate relative abundance of the competing transitions is shown on the nuclear 
mass-energy diagram. Note that the difference in nuclear mass between Ni 04 and 
Zn B4 is very small, and also that the excited level at 1.35 Mev in Ni 64 lies above the 
nuclear mass of the ground level of Cu 64 and can only be reached by the electron- 
capture transition. 

a. Decay. In a. decay, the parent nucleus zM r emits a helium nucleus 
Mass-energetically, the transition can be written 



a. 



M' = z- 



T T 



(4.32) 



The kinetic energy TV of the residual recoil nucleus is of the order of 
2 per cent of the laboratory kinetic energy T a of the a ray and cannot be 
neglected in a mass-energy balance. The total kinetic energy T of the 
heavy particles, which is often called the "disintegration energy," is 



= T a 



(4.33) 



where Mo is the reduced mass of the a ray M a and of the recoil nucleus. 
Equation (4.32) can be put in terms of neutral atomic masses by 
adding Zm to both sides and neglecting differences in the binding energy 
of the atomic electrons. This gives 



or 



He 4 + To 
.M - He 4 



(4.34) 



where He 4 is the neutral atomic mass of the helium atom. The decay 
scheme of Ra 226 is represented in the atomic mass-energy diagram of 
Fig. 4.7. 



4] 



Mass of Nuclei and of Neutral Aloms 



129 



Effects of Electron Binding Energy. Equation (4.34) represents the 
usual experimental situation. Measurements of T a = TQ(M Q /M a ) and 
Ty lead to experimental values of the mass difference (zM z-*M) 
between the parent and product neutral atoms. If Z ~ 90, Eq. (2.2) 
will show that the difference in electron binding energy between Z M and 
z-zM is ~ 0.03 Mev, which is in fact much greater than the error of 
measurement of T a . This just means that in Eq. (4.32), for bare nuclei, 



226.104 



226.103 



, 226.102 



g 226 101 

E 



226 100 



o 

V 



226 099 



226 098 - 



226097 - 



Ra 226 




3 E 



Fig. 4.7 The decay scheme of Ra" 6 , on a scale of neutral atomic mass. Note that 
the rest mass of a neutral He 4 atom is added in with the atomic mass of the decay 
product Rn 222 . This is done to hold the mass scale in the domain of 226 amu and 
still allow the energy scale to show the total "disintegration energies" of the a ray 
and recoil atom T in ccnter-of-mass coordinates. The lower-energy a transition and 
its 0.187-Mev 7 rny occur in only about 6 per cent of the transitions of Ra Mi . The 
atomic-mass scale is the tentative one proposed by Stern (871), based on the assump- 
tion that Pb 1M = 206.045 19 amu. 

the actual kinetic energy of the emitted a ray is, in fact, greater than the 
observed T a in the laboratory, by ~ 0.03 Mev for Z ~ 90. As the 
emitted a ray emerges through the negatively charged cloud of atomic 
electrons, it is decelerated to the energy T a as observed in the laboratory. 
Thus, if electron binding energy is not neglected, Eq. (4.34) remains 
valid, but T a in Eq. (4.32) would be changed to a larger value, say, T' a . 
This legitimate correction is usually ignored. 

Analogous considerations apply also to ft decay, except that the 
effects of electron binding are smaller and are usually comparable with 



130 The Atomic Nucleus [en. 3 

present experimental uncertainties. For ft decay, Eqs. (4.19), (4.22), 
(4.24), and (4.27) represent correctly the usual experimental situation. 

d. The Neutron -Hydrogen Mass Difference. Accurate knowledge 
of the mass of the neutron is of fundamental importance in evaluating the 
binding energy of nuclei and the nature of the forces between nucleons. 
The neutron, being uncharged, cannot be studied directly with a mass 
spectroscope. Its mass must be determined from the energetics of 
nuclear reactions. 

The quantity which is actually determined from the mi clear-reaction 
data is the neutron-hydrogen mass difference (7* H 1 ). C. W. Li et al. 
(L27) have evaluated (n H 1 ) from eight independent cycles of nuclear 
reactions. The weighted mean value is 

n - H 1 = 0.7823 0.001 Mev (4.35) 

Threshold for Reactions with Tritium. The most direct and accurate 
determination of (n H 1 ) is obtained by measuring the Q value of the 
reaction 

H 3 + H 1 -> n + He 3 + Q (4.36) 

In contrast with the reactions discussed in the previous section, this 
reaction has a negative Q value. Then Eq. (4.17) shows that the reaction 
cannot take place if the incident kinetic energy in the center-of-mass 
coordinates TQ (in) is less than a definite minimum value given by 

[Po (in)U = -Q (4.37) 

At this minimum incident energy the products of the reaction are sta- 
tionary in the ceriter-of-mass coordinates. But the reaction is detect- 
able at this incident energy because the velocity of the products in the 
laboratory is finite and equal to the velocity of center of mass. 

The minimum kinetic energy of the bombarding particle, in the 
laboratory coordinates, which is just sufficient to produce a reaction is 
called the threshold energy. If MI is the mass of the bombarding particle 
and MQ is the reduced mass of the system, then 

7\ = ^ To (in) (4.38) 

MQ 

where T l is the laboratory kinetic energy of Mi. The threshold energy 
(Ti)^ n is therefore given by 

(I-O. = - ; Q (4.39) 

for all reactions which have negative Q values. 

The threshold proton energy for the reaction H 3 (p,n)He B of Eq. 
(4.36) has been measured relative to an Al 27 (p,7)Si 2H resonance whose 
value is taken as a substandard of proton energy at 0.9933 Mev. When 
H 8 is bombarded, neutrons first appear at from the direction of the 
bombarding protons when the proton energy is 26 1 kev above the 
reference energy, or at an absolute energy of 1.019 0.001 Mev (T7). 



4] 



Moss of Nuclei and of Neutral Atoms 



131 



Then, from Eq. (4.39), Q = -0.764 0.001 Mev for the 
reaction. 

The negatron /8-ray spectrum of H' has an end point of only 0.0185 
0.0002 Mev. By combining the two reactions 



H* + H 1 = n + He' - 0.764 Mev 
H = He 1 + 0.018 Mev 



(4.40) 
(4.41) 



we obtain the present "best value" for the neutron-hydrogen mass 

4.027 r 

1.5 



.E 4.026 



4025 



4.004 



r o (in) 




He 4 



1.0 



0.5 1? 



-19.5 



-20 



Fig. 4.8 Graphical representation of the energetics of the reaction H 3 (pfW)He* and 
of the negatron decay H a (3~)He a ; from which (n H 1 ) is determined. This dia- 
gram combines the principles of Fig. 4.1 for plotting reaction energetics with the 
methods of Fig. 4.2 for plotting radioactive decay, by adding in the mass of one 
neutron on the decay scheme. In this way, all masses are in the domain of 4 amu 
and can be plotted together. Then both the Q value of the reaction H a (p,n)He a and 
the (n II 1 ) mass difference appear as energy-level separations on the diagram. 
Note that the ground level of the compound nucleus He 4 is really far off scale, at 
about 20 Mev. The compound nucleus in the H 3 (p,n)He 3 reaction is a highly 
excited level in the continuum of overlapping excited levels of He 4 . 



difference. 



n - H 1 = 0.782 0.001 Mev 

= (0.840 0.001) X 10- 3 amu 



(4.42) 



The energetics of the nuclear reactions of Eqs. (4.40) and (4.41) can be 
visualized in Fig. 4.8. 

Radioactive Decay of the Free Neutron. The neutron is about 2.5 
electron masses heavier than a proton. A neutron which evades capture 
by some nucleus should therefore undergo ordinary negatron ft decay. 



n 



H' + Q f - 



(4.43) 



132 The Atomic Nucleus [CH. 3 

The radioactive decay of free neutrons has been observed directly by 
passing a collimated beam of slow neutrons between two magnetic-lens 
spectrometers arranged end to end. The /3-ray spectrum has been 
measured, using coincidences between protons focused in one spectrom- 
eter and ft rays focused in the other (R22, S54). In this way, the end 
point of the 0-ray spectrum of the free neutron is found to be 0.782 
0.013 Mev. This result confirms the more accurate but less direct 
value of (n H 1 ) obtained in Eq. (4.42). 

e. Binding Energy of the Deuteron. In the deuteron, the nature of 
the fundamental nuclear forces between nucleons is approachable as a 
two-body problem. The binding energy of the deuteron is therefore an 
experimental quantity of special importance. The binding energy of any 
system of particles is the difference between the mass of the free con- 
stituents and the mass of the bound system. Then the binding energy 
5(H 2 ) of the deuteron is 

B(H 2 ) = n + H 1 - H 2 (4.44) 

The most direct determination of 5(H 2 ) is the measurement, of the 
quantum energy of the photons which are emitted when slow neutrons 
are captured by hydrogen. These "capture 7 rays" from the reaction 

H 1 + n = H 2 + T y 

have an energy of 2.229 + 0.005 Mev, as measured relative to the 
standard ThC" y ray taken as 2.615 Mev. Adding 1.3 kev for the recoil 
energy of the deuterium nucleus, and including a probable error of 0.004 
Mev in the ThC" standard y ray, Bell and Elliott (B29) obtain for the 
Q value of the reaction 

H 1 + n = H 2 + Q 

Q = (H 2 ) = 2.230 0.007 Mev (4.45) 

The binding energy of the deuteron has been computed by Li et al. 
(L27) from the energetics of six independent cycles of nuclear reactions. 
Their weighted mean value is 

B(H 2 ) = n + H 1 - H 2 = 2.225 0.002 Mev (4.46) 

f. Mass of Rare Nuclides. Mass-spcctroscopic determinations of 
the mass of stable nuclides whose natural abundance is small, such as 
O 17 , are often difficult or impossible. Many of these masses can be 
obtained from nuclear-reaction data, for example, O 17 from the reaction 
18 (d,p)0 17 . In this way complete mass tables will become available 
eventually. 

g. Reaction Cycles. The atomic masses of all known nuclides, up to 
at least A = 33, have now been obtained relative to O 16 entirely from 
nuclear-reaction data. The work of C. W. Li et al. (L27, L26) is the first 
example of such a compilation. 

The O 18 mass can be related to that of lighter nuclides in a reaction 



4] Mass of Nuclei and of Neutral Atoms 

such as O 16 (d,)N". Then 

+ He 4 - H 2 + Q 



133 



Other reactions can be obtained relating these products to the other light 
nuclei. Simultaneous equations can then be set up relating a number of 
independent reactions, called a nuclear cycle, to obtain the mass of any 
required nuclide. 

As an illustration, the mass of H 1 is given in terms of O 18 by (L27) 



H' = 
where 



- (Oi - Q 2 - 



+ 5Q C 
+ Q 4 + 



+ Qe + QT - Q.)] (4.47) 



Q a = n - H 1 

QX = 16 (d,a)N 14 

Q 4 = C ll (d l a)B 
Q 7 = Li(p,*)He 3 



Q b = n + H' - H 2 
Q 2 = C"G9-)N" 
Q 6 = B(d,a)Be B 
Q 8 = H 2 (d,n)He 3 



Q c = 2H 2 - He 4 
Q = Be*(p,a)Li ( 



It is evident that many more terms are involved here than in the anal- 
ogous derivation of H 1 from a small number of mass-spectroscopic 
doublets, as in Eq. (3.12). However, the accuracy of the Q-value 
determinations is now sufficiently high so that good atomic-mass values 
can be obtained for the light nuclides. In general, the mass-spectro- 
scopic masses are consistently larger than the nuclear-reaction cycle 
masses and by more than the probable error of measurement. 

Problems 

1. The kinetic energy of the two nuclei produced in the fission of U 235 is about 
170 Mev. Approximately what fraction of the original mass of (U 285 + n) 
appears as kinetic energy? 

2. In any nuclear reaction, develop a simple argument based on the mass 
definition of the Q value which will show that the difference in kinetic energy of 
the incident and residual particles is the same in laboratory coordinates as it is in 
center-of-mass coordinates. 

3. From the following nuclear-reaction data (L26), calculate the atomic-mass 
difference (Al 28 Si 28 ). Compare with the ft decay energetics of Fig. 4.2. 



Reaction 


Q value, 
Mev 


Doublet 


Mass difference, 
amu 


3i 29 (d,tt)Al 27 


5 494 
6.246 
5.994 


(2H 1 - H 2 ) 
(2H 2 - He<) 


1.549 X H)- 3 
25.596 X 10- 



4. Show that in decay the maximum value of the ratio of the recoil energy 
T M to the maximum 0-ray energy T m ^ depends on the mass of the nuclide and is 
give i by 



M 



(4.48) 



134 



The Atomic Nucleus 



[CH. 3 



Derive analogous expressions for the recoil energy resulting from the emission of 
a neutrino whose energy is T v and from the emission of a 7 ray whose energy is T y . 
Evaluate the recoil energy for M ~ 50 amu if T mK ~ 1 Mev; T 7 ~ 1 Mev; 
T 9 ~ 1 Mev. 

6. Estimate the difference between the binding energy of the atomic electrons 
in Al" arid Si 28 . Compare with the combined experimental uncertainty of the 
0-ray and y-ray energy in the Al"(/J~)Si" transition. 

6. In the decay of I 1 ", as given in Fig. 4.9, 

(a) Show that the total disintegration energy Q is independent of which com- 
peting mode of decay a particular nucleus follows. 

(6) Compute the atomic-mass difference in amu between the ground levels of 
I 131 and Xe 13 '. 



8.0 day 53!' 




- 12dayXe 13 ,l 



Stable 54 Xe 131 



10 



o.a 



o.s 



0.4 



0.2 



Fig. 4.9 The decay scheme proposed by Bell and Graham (B30) for the principal 
transitions in the complicated decay of I 131 . Note that one of the competitive 
0-ray transitions leads to the 12-day isomer of Xe 131 . The percentages shown for 
each y ray include the competing internal-conversion transitions. 

7. Nier and Roberts [Phys. Rev., 81: 507 (1951)] find that the mass doublet 
separation Ca 40 - A 40 = 0.32 0.08 X 10"' amu. 

(a) If the ft rays of K 40 have a maximum energy of 1 .36 Mev, and no 7 rays 
accompany the negatron rays, what is the neutral atomic-mass difference 
between K 40 and Ca 40 , in amu? 

(6) What would be the maximum possible 7-ray energy following an electron- 
capture transition in K 40 ? 

(c) What is the actual kinetic energy of the neutrinos if the observed 7-ray 
energy following electron capture is 1.46 Mev? 

(d) Draw to scale a decay scheme for these observed transitions of K 40 . What 
would be the maximum energy expected for a positron 0-ray spectrum of K 40 
(none is observed) ? 



5] Mass of I\uclei and of Neutral Atoms 135 

8. Determine an experimental value for the mass of the neutrino by com- 
paring the energetics of the positron decay of N l \T m ^ = 1.200 0.004 Mev; 
no 7 rays) with the reaction C 13 (p,n)N 13 for which Q = -3.003 0.003 Mev. 
Use (n - H 1 ) = 0.782 0.001 Mev. Plot the N 13 (0 + )C 13 and C(p,n)N 1 
reactions on a single mass-energy diagram in the vicinity of 14 amu. 

9. What error, in kev, is introduced in the determination of Q from the 
threshold proton energy of the reaction H 3 (p,n)He 3 if one assumes that Jlf i/Afo 
can be represented by (a) neutral atomic masses and (6) mass numbers, instead 
of by nuclear masses? Compare with the estimated experimental uncertainty 
in the threshold proton energy. 

10. If H 3 nuclei are used to bombard hydrogen, what is the threshold energy 
for the reaction H 1 (i,n)He 1 (t = tritium = Ii 3 )? Explain physically why this is 
so vastly different from the proton threshold for H 3 (p,n)He 3 , which involves the 
same actual nuclear interaction. What is the kinetic energy, in the laboratory 
coordinates, of the neutrons which are produced just at the threshold in the reac- 
tions (a) H'C^He 3 and (6) H 3 (/?,n)He 3 ? 

11. Evaluate the (n H 1 ) mass difference from the energetics of the two 
fundamental d-d reactions (L27) 

H 2 (d,p)H 3 ; d = 4.036 Mev 
IP(d,n)He'; Q 2 = 3.265 Mev 

combined with the ft decay energy of tritium. Plot the energetics of all these 
reactions on an atomic, mass-energy diagram and show graphically the values of 
Qi, Qf, T m ^ and (n - H'). 

12. Determine the atomic-mass difference Be 7 Li 7 from the energetics of 
the reactions 

B 10 (p,a)Be 7 ; Q t = 1.150 0.003 Mev 
B 10 K)Li 7 ; Q = 2.789 0.009 Mev 

Plot the energetics on a mass-energy diagram, showing graphically the values of 
Qi, Qz, and Be 7 - Li 7 . Use (n H 1 ) = 0.782 Mev. From energetic consider- 
ations alone, what would be the maximum possible neutrino energy in the elec- 
tron-capture decay of Be 7 ? 

13. Determine the binding energy of the deuteron from the energy released in 
the nuclear reactions 

H 2 (d,p)H 3 ; Q, = 4.036 0.012 Mev 
H 2 K7)H 3 ; Q z = 6.251 0.008 Mev 

Does the value obtained conflict with the direct measurement obtained from 
H^ttjTjH 2 ? Plot the energetics of these reactions, showing graphically the 
values of Q lf Q 2 , and 5(H 2 ). 

5. Tables of Atomic Mass 

The atomic mass of a large number of the known nuclides has not 
yet been measured. All tables arc fragmentary. Compilations of the 
material available are made from time to time by various authors. In 
each instance the mass recommended for an individual nuclide depends 
upon the values adopted at the time for the substandards of atomic mass. 
Caution must be exercised in using any particular table. 

Four of the recently most used compilations of atomic masses are 
those of Mattauch (M21, M22), Bethe (B43), Sullivan (S80), and Wap- 



136 The Atomic Nucleus [GH. 3 

stra (R36). These are all pre-1950 tables and are therefore subject to 
some revisions because of changes in the mass sub standards. 

Data on mass doublet separations, nuclear-reaction energies, and 
derived atomic masses, up to December 31. 1951, will be found in the 
valuable compilations by Bainbridge (B4). 

The October, 1 954, issue of the Reviews of Modern Physics combined 
five excellent compilations of data up to early 1954, including mass 
doublet separations (D40a), nuclear-reaction Q values (V3a), ft decay 
energies (K18a), a decay energies (A34a), and mass ratios from microwave 
spectroscopy (GlSa). 

In Table 5.1 we give for future reference the self-consistent set of 
atomic-mass values derived entirely from nuclear-reaction data by Li 
et al. in 1951 and 1952 (L27, L26). A comparison with recent mass- 
spectroscopic values for those nuclides which are stable has been given 
by Li (L26). 

Table 5.2 gives post-1951 mass-spectroscopic values determined by 
Collins, Nier, and Johnson (C35, C36, C34) for most of the stable nuclides 
between S 82 and Nb 93 and by Halsted (Hll) in the same laboratory for 
many nuclides between Pd 102 and Xe 136 . By combining these masses 
with nuclear-reaction data, tables of atomic masses (in terms of A M) 
for many of the unstable nuclides from S ai to Sr 90 (C36, C34) and from 
Rh 106 to I 131 (Hll) have been compiled. 



5] 



Mass of Nuclei and of Neutral Atoms 



137 



TABLE 5.1. TABLE OF ATOMIC MASSES FOR STABLE AND RADIOACTIVE NUCLJDES 

DERIVED ENTIRELY FROM NUCLEAR-REACTION DATA (L27, L26) 
Probable errors are given in 10~ 6 amu. The reference substandard used for many 
of the reactions is Q - 1.6457 0.002 Mev for Li 7 (p,n)Be 7 , corresponding to a 
threshold energy of 1.882 0.002 Mev. 



Mass 
number 


Atomic mass 


Mass 
number 


Atomic mass 


n 1 


1.008 982 (3) 


F 17 


17.007 505 (5) 






F 18 


18.006 651 (22) 


H 1 


1.008 142 (3) 


F 19 


19.004 456 (15) 


H 2 


2.014 735 (6) 


F 20 


20.006 350 (17) 


H 3 


3.016 997 (11) 










Ne 19 


19.007 952 (15) 


He 3 


3 016 977 (11) 


Ne 20 


19.998 777 (21) 


He 4 


4.003 873 (15) 


Ne 21 


21.000 504 (22) 


He 6 


6.020 833 (39) 


Ne 22 


21.998 358 (25) 






Ne 23 


23.001 768 (26) 


Li 6 


6.017 021 (22) 






Li 7 


7.018 223 (26) 


Na 21 


21.004 286 (39) 


Li 8 


8.025 018 (30) 


Na 22 


22.001 409 (25) 






Na 23 


22.997 055 (25) 


Be 7 


7.019 150 (26) 


Na 24 


23.998 568 (26) 


Be 8 


8.007 850 (29) 






Be 9 


9.015 043 (30) 


Mg23 


23.001 453 (26) 


Be 10 


10.016 711 (28) 


Mg24 


23.992 628 (26) 






Mg25 


24.993 745 (27) 


B 9 


9.016 190 (31) 


Mg26 


25.990 802 (29) 


B 10 


10.016 114 (28) 


Mg27 


26.992 876 (30) 


B 11 


11.012 789 (23) 






B 12 


12.018 162 (22) 


Al 27 


26.990 071 (30) 






Al 28 


27.990 760 (32) 


C 11 


11.014 916 (24) 






C 12 


12.003 804 (17) 


Si 28 


27.985 767 (32) 


C 13 


13.007 473 (14) 


Si 29 


28.985 650 (34) 


C 14 


14.007 682 (11) 


Si 30 


29.983 237 (36) 






Si 31 


30.985 140 (39) 


N 13 


13.009 858 (14) 






N 14 


14.007 515 (11) 


P 31 


30.983 550 (39) 


N 15 


15.004 863 (12) 


P 32 


31.984 016 (41) 






P 33 


32.982 166 (44) 


O 15 


15.007 768 (13) 






O 16 


16.000 000 (std.) 






17 


17.004 533 (7) 


S 32 


31.982 183 (42) 


O 18 


18.004 857 (23) 


S 33 


32.981 881 (44) 



138 



The Atomic Nucleus 



[CH. 3 



TABLE 5.2. TABLE OF ATOMIC MASSES OP STABLE NUCLIDES DETERMINED 

BY MAss-srECTBOSCopic DOUBLETS (C35, C36, C34, H11) 
The substandards used are H 1 = 1.008 146 (0.3) and C 12 - 12.003 842 (0.4), 
i given in Eq. (3.15). The probable errors are in 10~ B amu. 



Mass 
number 


Atomic mass 


Mass 
number 


Atomic mass 


S 32 


31.982 236 (0 7) 


Fe 57 


56.953 59 (10) 


S 33 


32.982 13 (5) 


Fe 58 


57.952 (40) 


S 34 


33.978 76 (5) 










Co 59 


[58.951 3 (30)]f 


Cl 35 


34.980 04 (5) 






Cl 37 


36.977 66 (5) 


Ni 58 


57.953 45 (10) 






Ni 60 


59.949 01 (29) 


A 36 


35.979 00 (3) 


Ni 61 


60 949 07 (23) 


A 38 


37.974 91 (4) 


Ni 62 


61 946 81 (9) 


A 40 


39.975 13 (3) 


Ni 64 


63 947 55 (7) 


K 39 


38.976 06 (3) 


Cu 63 


62.941) 2ti (6) 


K 41 


40.974 90 (4) 


Cu G5 


64.948 35 (6) 


Ca 40 


39 975 45 (9) 


Kn 04 


63.949 55 (2) 


Ca 42 


41 972 16 (4) 


Zn 66 


65 947 22 (6) 


Ca 43 


42 972 51 (6) 


Zn 67 


66 948 15 (6) 


Ca 44 


43.969 24 (6) 


Zn 68 


67 946 86 (7) 


Ca 48 


47.967 78 (10) 


Zn 70 


69.947 79 (6) 


Sc 45 


44.970 10 (5) 


Ga 69 
Ga 71 


68.947 78 (6) 
70.947 52 (9) 


Ti 46 
Ti 47 
Ti 48 
Ti 49 
Ti 50 


45.966 97 (5) 
46.966 68 (10) 
47 963 17 (6) 
48.963 58 (5) 
49.960 77 (4) 


Ge 70 
CP 72 
Gc 73 
Ge 74 
Ge 76 


69.946 37 (7) 
71 944 62 (7) 
72.946 69 (4) 
73.944 66 (6) 
75.945 59 (5) 


V 51 


50.960 52 (5) 


As 75 


74.945 70 (5) 


Cr 50 


49.962 10 (7) 


So 74 


73 946 20 (8) 


Cr 52 


51.957 07 (9) 


SP 76 


75.943 57 (5) 


Cr 53 


52.957 72 (8) 


SP 77 


[76 944 59 (5)] 


Cr 54 


53.956 3 (20) 


So 78 


[77 942 32 (5)] 






Se 80 


79.942 05 (5) 


Mn 55 


54.955 81 (10) 


Sc 82 


81.942 85 (6) 


Fc 54 


53 957 04 (+5) 


Br 79 


78.943 65 (6) 


Fe 56 


55 952 72 (10) 


Br 81 


80 942 32 (6) 



t Brackets designate masses of stable nuclidea determined from mass- spec troacopic values for adjacent 
nuchdea, combined with disintegration data. 



51 



Mass of Nuclei and of Neutral Atoms 



139 



TABLE 5.2. (Continued) 



Mass 
number 


Atomic mass 


Mass 
number 


Atomic mass 


Kr 78 


77 945 13 (9) 


Cd 114 


113 939 97 (9) 


Kr 80 


[79 941 94 (7)] 


Cd 116 


115 942 02 (12) 


Kr 82 


81.939 67 (7) 






Kr 83 


82 940 59 (7) 


In 113 


112.940 45 (12) 


Kr 84 


83 938 36 (7) 


In 115 


114 940 40 (11) 


Kr 86 


85 938 28 (8) 










Sn 115 


114.940 14 (25) 


Rb 85 


84 939 20 (8) 


Sn 110 


115 939 27 (11) 


Rb 87 


86 937 09 (17) 


Sn 117 


116.940 52 (10) 






Sn 118 


117 939 78 (16) 


Sr 84 
Sr 86 
Sr 87 
Sr 88 


83.940 11 (15) 
85 936 84 (11) 
86 936 77 (8) 
87.934 08 (11) 


Sn 119 
Sri 120 
Sn 122 
Sn 124 


118.941 22 (12) 
119.940 59 (14) 
121.942 49 (15) 
123.944 90 (11) 






To 120 


119.942 88 (16) 


Y 89 


88 934 21 (11) 


Te 122 


121 941 93 (8) 






Te 123 


122 943 68 (39) 


Zr 90 


89.933 11 C+25) 


Tc 124 


123.942 78 (11) 






Tc 125 


124.944 60 (31) 


Nb 93 


92.935 40 (9) 


Te 126 


125 944 20 (7) 






Te 128 


127.946 49 (13) 


Pd 102 


101.937 50 (9) 


Te 130 


129.948 53 (10) 


Pd 104 


103 936 55 (11) 






Pd 105 


104 938 40 (15) 


I 127 


126.945 28 (13) 


Pd 106 


105.936 80 (19) 






Pd 108 


107.938 01 (11) 


Xc 124 


123.945 78 (7) 


Pd 110 


109.939 65 (13) 


Xe 126 


125.944 76 (14) 






Xe 128 


127 944 46 (9) 


Cd 106 


105.939 84 (14) 


Xe 129 


128.946 01 (15) 


Cd 108 


107 938 60 (11) 


Xe 130 


129.945 01 (10) 


Cd 110 


109 938 57 (13) 


Xe 131 


130 946 73 (42) 


CM 111 


110.939 78 (10) 


Xo 132 


131 946 15 (10) 


Cd 112 


111.938 85 (17) 


Xc 134 


133 948 03 (12) 


Cd 113 


112 940 61 (11) 


Xc 136 


135.950 46 (11) 



CHAPTER 4 

Nuclear Moments, Parity, and Statistics 



The concept of a nuclear magnetic moment associated with an angular 
momentum axis in nuclei was introduced in 1924 by Pauli, aw a means 
of explaining the hyperfine structure which had been disclosed in atomic 
optical spectra by spectrographs of very high resolving power. At 
that time the neutron had not been discovered, and very little was 
known about the inner constitution of nuclei. It was then impossible 
to postulate how the angular momenta of the unknown individual 
constituents of a given nucleus might combine in order to produce the 
intrinsic total angular momentum, or spin, apparently exhibited by the 
nucleus as a whole. 

In the following year, 1925, Uhlenbeck and Goudsmit extended the 
concept to atomic electrons. By assuming that each electron "spins' 7 
about its own axis and hence contributes to both the angular momentum 
and the magnetic dipole moment of its atom, they derived a satisfactory 
explanation of the anomalous Zeeman effect. The concept of electron 
spin was soon found necessary in the theoretical description of the fine 
structure of optical spectra, of the scattering of rays by electrons, and 
of many other phenomena. Empirically, it was necessary to assume 
that each electron possesses an intrinsic angular momentum, in addition 
to its usual orbital angular momentum, as though it were a spinning 
rigid body. The observable magnitude of this spin angular momentum 
is h/2. Because this is of the order of h we can infer that spin is essen- 
tially a quantum-mechanical property. No satisfactory theoretical basis 
was forthcoming until Dirac showed that the existence of electron spin 
is a necessary consequence of his relativistic wave-mechanical theory of 
the electron. In the nonrelativistic limit, the electron behaves as if it had 
a real intrinsic angular momentum of h/2. 

Analogously, the nuclear angular momentum was found empirically 
to play an important role in a variety of molecular, atomic, and nuclear 
phenomena. Chadwick's discovery of the neutron in 1932 opened a new 
era in the study of nuclear structure. The proton and neutron were 
each shown to have the same spin as an electron and to obey the Pauli 
exclusion principle. A variety of nuclear models could then be visual- 
ized. Each proton and neutron in the nucleus can be assigned values of 
orbital angular momentum and of spin angular momentum, and these 
can be combined by some kind of suitable coupling scheme to give the 
observed total nuclear angular momentum. Nuclear models of this 
type will be discussed in Chap. 11. 

140 



Nuclear Moments, Parity, and Statistics 



141 



The quantum numbers for individual particles (Appendix C, Sec. 2) 
and their addition, or coupling, rules can be shown to emerge in a natural 
way from the quantum-mechanical description of systems of particles. 
Similarly, the quantum mechanics leads to expectation values for the 
magnetic moments which are associated with spin and orbital angular 
momenta (B68). The results of these derivations can be visualized best 
in terms of a so-called vector model, which has long been used in optical 
spectroscopy to translate the quantum-mechanical results into a visualiz- 
able system (W39). In this chapter we shall review the concepts of 
angular momentum and magnetic dipole moment in terms of the vector 
model, and also the closely associated nuclear electric moments, parity, 
and statistics. In Chaps. 5 and 6 we shall discuss a number of molecu- 
lar, atomic, and nuclear phenomena which are affected by these nuclear 
properties and which provide the means of measuring the various nuclear 
moments. 

1. Nuclear Angular Momentum 

The total angular momentum of a nucleus, taken about its own 
internal axis, is readily measurable. The complex motions of the indi- 
vidual protons and neutrons within this nucleus cannot be measured 
directly. Nevertheless, it is convenient to visualize a vector model of 
the individual nuclear particles which represents the quantum-mechanical 
results and is analogous to the existing vector model for atomic electrons 
in the central field of the nucleus. 

a. Quantum Numbers for Individual Nucleons. The "state" of a 
particular nucleon is characterized by quantum numbers which arise in 
solutions of the wave equation for an individual nucleon bound in a 
nuclear potential well. The notation and nomenclature for these 
quantum numbers parallel the conventions adopted previously for 
atomic electrons bound in the coulomb field of a nucleus. 

Principal Quantum Number (n). Each bound individual particle 
has associated with it a principal quantum number n which can take on 
only positive integer values greater than zero. Thus n = 1, 2, 3, . . . . 
In a coulomb field, the first-order term for the total energy of the state is 
determined by n. This is not true for the noncoulomb fields in which 
nuclear particles are bound. The principal quantum number is the sum 
of the radial quantum number v and the orbital quantum number 2; thus 
n = v + I. 

Orbital Quantum Number (I). The orbital-angular-momentum quan- 
tum number I is restricted to zero or positive integers up to (n 1). 
Thus, I = 0, 1,2, . . . , (n 1). The magnitude of angular momentum 
of the corresponding motion is h V7(Z +1). Individual values of I are 
commonly designated by the following letter symbols as previously 
adopted in atomic spectroscopy. 



1 





1 


2 


3 


4 


5 


6 






















Symbol 


a 





d 


f 


a 


h 


i 





142 The Atomic Nucleus [CH. 4 

Magnetic Orbital Quantum Number (mi}. The orbital magnetic quan- 
tum number mi is the component of I in a specified direction, such as that 
of an applied magnetic field. It can take on any of the (21 + 1) possible 
positive or negative integer values, or zero, which lie between I and L 
Thus, I > m t > -1. More explicitly, m l = I, (I 1), . . . , 1, 0, -1, 

. . . ,(-*+!), -I- 

Spin Quantum Number (s). The spin quantum number s has the 
value is for all elementary particles which follow Fermi-Dirac statistics 
and which obey the Pauli exclusion principle. In particular, .s 1 = i for 
the proton, neutron, and electron. The magnitude of angular momentum 
of the corresponding spin is h v x ,s(s +1). 

Magnetic Spin Quantum Number (m s ). The spin magnetic quantum 
number m 8 is the component of s in an arbitrary direction, such as that 
of an applied magnetic field. It is restricted, for elementary particles 
with s = ^, to the two integer-spaced values m k = i, J. 

Total Angular-momentum Quantum Number for a Single Particle (j). 
The total angular momentum of a single particle is the summation of its 
orbital and its spin angular momenta and is represented by the intrin- 
sically positive quantum number j. The magnitude of angular momen- 
tum of the corresponding motion is h \^j(j +1). For particles with 
s = i, there, are, at most, just two permitted, positive, integer-spaced 
values of j. These are j = (I + s) and (/ s), or j = (I + I) and 
(I i). If / = 0, then j has only the value j = s = k- Thus j is 
restricted to the odd half-integer values,/ = 2, J, I 

Magnetic Total Angular-momentum Quantum Number for a Single 
Particle (m } ). The component of j in any arbitrary direction, such as that 
of an applied magnetic field, is the total magnetic quantum number ?n,. 
Like the other magnetic quantum numbers, positive and negative values, 
with integer spacing, are permitted. Thus, for 5 = i particles, 7ft, can 
have any of the (2j + 1) possible values given by m } = j, (j ]),..., 

?, i, - - - , J. 

Radial Quantum Number (v). In a noncoulomb field, such as & rec- 
tangular potential well, the principal quantum number is not a good 
index of the energy of the state. In the solution of the radial wave equa- 
tion for the rectangular potential well, there arises the so-called radial 
quantum number v which represents the number of radial nodes at r > 
in the wave function and can have only nonzero positive integer values 
, = 1, 2, 3, .... 

Isobaric-spin Quantum Number (T f ). Neutrons and protons are so 
similar in all respects except charge that much progress has been made 
through Heisenberg's concept that the proton and neutron can be repre- 
sented as the two possible quantum states of one heavy particle, the 
nucleon. This has given rise to a charge quantum number, originally 
called the "isotopic-spiri" and more recently the "isobaric-spin" quantum 
number (W47, 14). A common but arbitrary assignment for the total 
isobaric-spin quantum number T is based on a mathematical analogy 
with the two intrinsic spin states m a = + i. According to this con- 
vention, the component T f in a hypothetical " isobaric-spin space 7 ' has 



1] Nuclear Moments, Parity, and Statistics 143 

the value +1 for the neutron state and for the proton state of a 
niicleon. Then for any nucleus, T$ = ^(N Z), where (N Z) is the 
neutron excess, and T f must be a component of the total isobaric spin T 
associated with any quantized level of this nucleus. For example, in the 
isobaric triad 6 C 14 , rN 14 , B O 14 , the respective values of T f are 1, 0, 1. 
Present evidence suggests that the ground levels of the outer members 
C 14 and O 14 of this triad have total isobaric spin T = 1, while the ground 
level of N 14 is T = 0. An excited level at 2.3 Mev in N 14 appears to be 
the T = 1 level which forms a set, having multiplicity 2T + 1, with the 
ground levels of C 14 and O 14 . There is increasing evidence that total 
isobaric spin is conserved in nuclear interactions, in a manner analogous 
to the conservation of total nuclear angular momentum. 

b. Nomenclature of Nucleon States. When the character of the force 
between individual particles is known, a solution of the appropriate wave 
equation gives the energy of an individual bound particle in terms of four 
quantum numbers, such as n, /, m t , m s . According to the Pauli exclusion 
principle, which has been shown experimentally to apply to micleons, 
no two protons can have in one- nueleus the same set of values for their orbital 
and spin quantum numbers, e.g., for the four quantum numbers 

n, /, mi, m a 
or, alternative!}', for the four quantum numbers 

?i, /, ./, m 3 
or for v, 7, j, m.j 

The same condition applies to any two neutrons in one nucleus. How- 
ever, one neutron and one proton can each have the same set of values 
of these four quantum numbers because they still will differ in one 
property, namely, charge,. 

Atomic Shells, SubtsheMs, and Stales. We review here the notation of 
atomic states from which some mam features of the nomenclature for 
nuclear states have been borrowed. 

Because of the characteristics of a coulomb field, only the principal 
quantum number n enters the first-order term for the energy of an atomic 
state. In optical spectrosropy, a shell generally includes all electrons 
which have the same value of n. Each completed shell then contains a 
total of 

Z-71-1 

2(27 + I) - 2/f- = 2, 8, 18, 32, ... (1.1) 



electrons. These correspond to the (21 + 1) values of m z for each / and 
to the 2 values of m 8 for each m t . 

A subshell of atomic electrons includes all electrons having the same // 
and L Thus a completed subshell contains 2(21 + 1) electrons. The 
occupation number, or total number of electrons permitted in the 1 = 2 
subshell, is 2(2Z + 1) = 10. These are made up of (2j + 1) electrons 
from each of the.;' = I s states. Thus for j = I s = % there are four 



144 The Atomic Nucleus [CH. 4 

electrons, and for j = / + s = T there are six electrons, making the per- 
mitted total of 10 for the / = 2 subshell. 

The notation of an electronic state, then, includes the value of n, along 
with the values of I and of j. Thus an electron state having n = 3, 
I = 2,j = I 5 = 2 i = f, would be designated 



Nucleon States and Shells. In a nuclear potential well, the energy 
of a nucleon state does not depend primarily on the principal quantum 
number n, but, rather on / and v. Many, but not all, authors now write 
the radial quantum number, v = (n -- /), in the position formerly occupied 
by n, in accord with a convention introduced in 1949 by Maria Mayer 
(M24). Thus a nucleon having n = 3, I = 2, v 1, j = I s, would, 
in this notation, be designated 

Id, (1.2) 

instead of 3dg, as in the notation of atomic spectroscopy. This newer 
notation is attractive mnemonic-ally because the \d state (old 3rf) can be 
read "the first d state," the 2d state (old 4d) can be read "the second 
d state/' etc. We shall use this newer notation hereafter for nuclear 
states. 

In nuclei, the word shell does not connote constant values of n, as it 
does in optical spectroscopy. When a variety of nuclear properties (mass, 
binding energy, angular momentum, magnetic dipole moment, nemron- 
capture cross section, etc.) are plotted as a function of either the number 
of protons Z or the number of neutrons N -= (A Z) in the nucleus, dis- 
continuities are apparent when either Z or N has the value 2, 8, 20, 50, 82, 
or 126. These and possibly other so-called magic numbers are currently 
considered as representing "closed-shell" configurations in nuclei, ^Ls 
will be seen in Chap. 11, the sequence of levels by which such shells can 
be filled does not represent a simple progression in 77, /, or v. 

c. Coupling of Nucleon States. Nuclear Levels. When two or more 
nucleons aggregate to form a nucleus, the quantum state of the system 
as a whole is called! a nuclear level. This level may be the ground level 
or any one of a number of excited levels of the particular nucleus. 
Among other properties, each nuclear level is characterized by a particular 
value of the total nuclear angular momentum. 

The manner in which the values of / and s for the individual riurleons 
are added in order to form the total rmclear-angular-momentum quan- 
tum number / depends on the type of interaction, or "coupling/ 7 assumed 
between the particles. The actual individual motions of the nuclear 

t We follow in this chapter the nomenclature used by Blatt and Wcisakopf (p. 644 
of B68), in which "state" refers to a single nucleon and "level" refers to the quantum 
condition of the entire nucleus. The literature and current usage do not always draw 
this distinction. Often "level" and "state" are used interchangeably, as in "excited 
level" or "excited state," and "ground level" or "ground state." Common usage 
favors "level width" and "level spacing," but "triplet state" and "singlet state." 
We shall adhere usually to "nucleon state" and "nuclear level." 



1] Nuclear Moments, Parity, and Statistics 145 

particles must be strongly interdependent because of the small distances 
and large forces between neighboring particles. It is undoubtedly 
incorrect to imagine that the coupling scheme can be simple. 

In atomic spectroscopy, the analogous problem has been dealt with 
by defining two limiting ideal types of coupling, near or between which 
lie all actual cases. These limiting types are the Russell-Saundcrs, or 
LS, coupling and the spin-orbit, or jj, coupling. In the absence of pre- 
cise information about nuclei, and to provide convenient and familiar 
notations, these two coupling forms are also assumed for nuclei. Then 
we can ujse, the addition rules of the usual vector models of optical spectros- 
copy (p. 101 of W39, or p. 175 of H44). 

RusscU-tfaundrrs Coupling (Lti). In this coupling scheme it is 
assumed that there is a negligibly weak coupling between the orbital (I) 
and the spin (s) angular-momentum vectors of an individual nucleon. 
Instead, the individual orbital vectors I are assumed to be strongly 
coupled to one another, and to form, by vector addition, a total orbital- 
angular-momentum quantum number L for all the nucleons in the sys- 
tem. Levels of different L are presumed to have quite different energies. 
Similarly, it is assumed that the individual spin vectors s are strongly 
coupled together to form, by vector addition, a resultant total spin 
quantum number 8 for the system. For the same value of L, it is 
assumed that different values of M correspond to clearly separated energy 
levels, the go-called spin midtiphts. Finally, the resultant L and S 
couple together to form the total angular-momentum quantum number / 
for the nuclear level. 

The nomenclature of nuclear levels in LS coupling then follows the 
usual nomenclature of optical spectroscopy. For L = 0, 1, 2, 3, . . . , 
the levels are designated 8, P, D, F, . . . . For each value of L there 
are (2*S +1) possible integer-spaced values of /, provided that S < L. 
Regardless of the relative values of L and *S, the multiplicity is taken as 
(2S +1) by definition and is written as a superscript before the letter 
designating the L value. The particular / value appears as a subscript. 
For example, if L = I and S = *, the possible "levels," or "configura- 
tions/ 7 or "terms" are written 

2 P 4 and 2 P a 

Here one level might be the ground level of a nucleus, while the other is 
a low-lying excited level. The two levels taken together are a "spin 
doublet" in Kussell-Saunders coupling. 

jj Coupling. This coupling scheme is the extreme opposite of LS 
coupling. It is assumed in jj coupling that the predominant interaction 
is between the orbital (I) and spin (s) vectors of the some individual 
nucleon. These combine to form the total angular-momentum quantum 
number j for the individual nucleon, where j = I s. In turn, the total 
nuclear-angular-momentum quantum number 7 is a vector sum of the 
individual j values. Hence jj coupling is also called strong spin-orbit 
coupling. 

In jj coupling, the individual / values for different nucleons do not 



146 The Atomic Nucleus [CH. 4 

couple together; neither do the individual s values. Therefore there is 
neither an L nor an S quantum number for the level. The Russell- 
Saunders notation of term values for levels does not apply. The only 
4 'good" quantum number in jj coupling is the total nuclear-angular- 
momentum quantum number 7. 

When it is necessary to designate the separate energy levels which 
arise from the jj coupling of individual nucleon states, we may use a 
modified form of the notation adopted by White (p. 1% of W39) for.;]; 
coupling in atomic spectroscopy. Suppose an nucleon (I = 0) is to be 
coupled with a p nucleon (/ = 1). For the s nucleon j = ^, while for the 
p nucleon j = i, f. These individual./ values can he combined vectorially 
in four ways, to produce 7 = 0, 1, 1, or 2. These four levels can be 
represented by the notation 

Q 3) 

in which the resultant / value is shown as a subscript. 

In Russell-Saundors coupling, the (s,p) configuration could result 
only in P levels (L = + 1 =1), but these could be either singlet 
(S = ? - ? = 0) or triplet (S = T + I = 1) levels. The L and S values 
can be combined voctorially in four ways, to produce four levels, 7 = 0, 
1, 1, 2. The LS coupling levels for an (s t p) configuration would then be 

i7\ and *P , 'Pi, V> 2 (1.4) 

This example illustrates the generalization that the type of coupling 
which is assumed for tht configuration does not affect the total number oj 
levels produced nor the angular momenta of these levels. However, the 
coupling scheme does profoundly affect the relative and absolute energy 
separation of these separate levels. 

Intermediate Coupling. In optical upectroscopy, jj coupling is recog- 
nized as originating physically in an interaction energy between the spin 
magnetic moment of an electron and the magnetic field due to its own 
orbital motion. It therefore becomes most important for large I values. 
Light elements with two valence electrons tend to exhibit nearly pure 7,5 
coupling, while heavy elements with two valence electrons having larger 
I values exhibit nearly pure jj coupling. White (p. 200 of W39) has 
illustrated well the gradual progression from LS tujj coupling for the fine 
structure of optical levels in the group of elements C C, uSi, 3 2Ge, 5 oSn, 
8'jPb, each of which consists of completed subshells of electrons, with 
two valence electrons outside. The elements Si, Ge, Sn exhibit an 
energy separation of the fine-structure levels which is intermediate 
between LS and jj coupling. 

In nuclei, it was believed until 1949 that jj coupling would not be 
exhibited, because of the very small absolute value of the magnetic 
dipole moment of the proton and the neutron. This has proved a false 
clue. Empirically, the evidence since 1949 indicates that heavy nuclei 
exhibit nearly pure.;)" coupling (Chap. 11). The energy-level spacing in 
some of the lightest nuclei (A ~ 10) is appropriate for intermediate 



Nuclear Moments, Parity, and Statistics 



147 



1] 

coupling (J4). Pure LS coupling is seldom seen. A physical origin for 
strong spin-orbit interactions in nuclei has been sought but as yet with- 
out compelling success. 

d. Total Nuclear Angular Momentum. The total nuclear-angular- 
momentum quantum number 7 represents a rotational motion whose 
absolute angular momentum has the value 

ft V7(7 + F) 

The quantum mechanics shows that in any given direction, such as that 
of an applied magnetic field, the observable values of the time average 
of a component of this angular momentum are given by 

*/ t, /i "*\ 

mjfi (l.oj 

where the magr.etic quantum number m t can take on a series of (27 + 1) 
integrally spaced values from 7 to 7. Thus the permitted values are 

m<>= 7, (7 - 1), (7 - 2), . . . , -(7 - 2), -(7 - 1), -7 (1.6) 

The largest value of mj is /, and the largest observable component of the 
total nuclear angular momentum is In. 

Th^se relationships can be represented conveniently in the usual type 
of vector diagram, Fig. 1.1. It is 
seen that the angular-momentum 
vector, whose magnitude we de- 
note (W39, F41 ) for typographical 
convenience as 7*, whore 




(1-7) 

can take up any of (27 + 1) orien- m /=~ 

tat ions, at various angles ft with 

respect to the applied field /7, and 

subject to the restriction that *** " Vt ' ct , or **d<* the relationship 

between m r , 1, and /*, for 7 = f . The 

m,ih = (1* COS ft)h observable components of the angular mo- 

mentum am mjh til* cos jtf. 

The quantity which is colloquially 

called the "nuclear angular momentum " or sometimes less aptly the "nu- 
clear spin," is just the maximum value of m h that is, 7. 

Problems 

1. Assuming an " outer" proton or neutron to be in a circular orbit with 
I = 1 and a radius r = 4 X 10~ 13 cm, compute its velocity. Does the result sug- 
gest that nonrelativistic theory will be .satisfactory for descriptions of the heavy 
particles in nuclei? Ans.: v = 0.22 X 10 10 cm/sec. 

2. Assuming a proton to be a sphere of radius of 1.5 X 10~ 13 cm and of uni- 
form density and to have an actual angular momentum of (fc/2ir) \/s(s + 1), ( 



14.H The Atomic Nucleus [CH. 4 

where s = . compute the angular velocity in revolutions per second and the 
peripheral speed at the equator of the proton. Ans.: w = 0.96 X 10 22 revolu- 
tions/sec, 0.91 X 10 10 cm/sec. 

2. Nuclear Magnetic Dipole Moment 

Any charged particle moving in a closed path produces a mag- 
netic field which can be described, at large distances, as due to a mag- 
netic dipole located at the current loop. Therefore the spin angular 
momentum of a proton, and the orbital angular momentum of protons 
within nuclei, should produce extranuclear magnetic fields which can be 
described in terms of a resultant magnetic- dipole moment located at the 
center of the nucleus. 

a. Absolute Gyromagnetic Ratio. When a particle of mass M and 
charge q moves in a circular path, the motion possesses both angular 
momentum and a magnetic dipole moment. Using primed symbols to 
denote moments in absolute units, the classical absolute gyromagnetic 
ratio 7 is defined as 

T y, gauss" 1 sec -1 (2.1) 

where the orbital angular momentum is /' erg-sec and the magnetic 
dipole moment is M' ergs 'gauss. 

We can evaluate 7 classically for an element of mass dM grams, 
carrying an element of charge dq esu, arid moving with angular velocity 
a;, in a circular orbit whose radius is r cm. The absolute angular momen- 
tum dl f of this motion is 

dl' = dMur* erg-sec (2.2) 

Its corresponding absolute magnetic dipole moment dp is the area of the 
orbit Trr 2 times the equivalent circulating current in emu, which is 
(dq/c)(u/2w). Thus 

dp' = - wr 2 ergs/gauss (2.3) 

2iC 

The absolute gyromagnetic ratio for this element of mass and charge is 

T .*L'.l* (24) 

7 dl' 2cdM { ' 

For a distributed mass, such as a sphere, or any system of mass 
points, the total angular momentum is obtained by the usual integration 

/' = / dl' = / cor 2 dM (2.5) 

Similarly, the magnetic moment of a system involving distributed charge 
is 



- I wr z dq (2.6) 



2] Nuclear Moments, Parity, and Statistics 149 

If the ratio of charge to mass dq/dM is constant and equal to q/M 
throughout the system, then dq = (q/M ) AM, and the second integral 
becomes 

' (2 - 7) 



Thus the classical gyromagnetic ratio 7 is 



r = gausfcrl sec ~ 1 (2 - 8) 



for any rotating system in which charge and mass are proportionately dis- 
tributed, such as in a rotating uniformly charged spherical shell or a 
rotating uniformly charged sphere, or the motion of a charged particle 
moving in a closed orbit. 

b. Nuclear g Factor. If the classical relationship between the angular 
momentum and magnetic dipole moment were valid for nuclear systems, 
the absolute magnetic dipole moment /*/*, which is collinear with the 
absolute angular momentum /' = h Vl(I + 1) = ft/*, would be expected 
to have the absolute value 

;. = y l' = y hl* = -^- /* if classical (2.9) 



for the case of a spinning proton containing a uniformly distributed 
charge c arid mass M. 

However, the direct!}' measured values for the magnetic dipole 
moment of the proton, and for the gyromagnetic ratios of various nuclides, 
do not follow so simple a relationship. Indeed, a precise theory of the 
origin of nuclear magnetism is still lacking. The principal experimental 
data consist of measurements of the nuclear gyromagiietic ratios for the 
proton, neutron, and for many riuclides. What is commonly called 
either the nuclear g factor (or, in most of the pre-1950 literature, the 
nuclear gyromaynctic ratio, g) is equivalent to inserting a dimensionless cor- 
rection factor g in the right-hand side of Eq. (2.9), so that it reads (PI) 

M ;. = yi r = T/ A/* = g -A_ /* for real nuclei (2.10) 



where 7/ = g(e/2Mc) gauss" 1 see" 1 is now called the nuclear gyromagnetic 
ratio in absolute units. 

By analogy with the Bohr magneton up for atomic electrons, which has 
the value 

M/9 = -?*-. = 9.273 X 10~ 21 erg/gauss (2.1 la) 

4?rmoC 

we define a nuclear magneton HM as 

"" (ir) = 5 ' 050 x 10 ~ 24 er s/sauss 

= 3.152 X 10- 12 ev/gauss (2.11&) 



150 The Atomic Nucleus [CH. 4 

where M is the mass of a proton. Note that the nuclear magneton is 
1,836 times smaller than the Bohr magneton, because the nuclear mag- 
neton contains the proton mass instead of the electron mass. 

Then the absolute value /ij* of the nuclear magnetic dipole moment 
can be expressed in units of nuclear magnetons as the dimensionless 
quantity /i,*, where 

"" " j (2 - 12) 

Actually, neither 7* nor its collinear magnetic moment /*/* are directly 
observable quantities, but their ratio can be measured accurately in 
many different types of experiments which depend on the phenomenon 
of Larmor precession (Chap. 5). From Eqs. (2.10) and (2.12) we can 
write 

g = 7** < 2 - 13 ) 

c. Nuclear Magnetic Dipole Moment. The maximum observable 
component of 7* is 7, and because 7* and /*/* are collinear, the correspond- 
ing maximum observable component of /x/* is 

= /if (cos 0) m = n (2.14) 

This maximum component, and not /x/*, is, in fact, v/he quantity which is 
colloquially called the "nuclear magnetic dipole moment 7 '; it is denoted 
by the symbol p. We have then 

g = ^ = (2-15) 






The values of n which are found in nuclear tables are all derived from 
measurements of 7, combined with independent measurements of </, and 
represent the quantity 

M = gl (2.16) 

The units of p are generally spoken of as " nuclear magnetons/' because 
of the relationships in Eqs. (2.12) and (2.13). Actually /x, g, and 7 are 
all pure numbers. Thus g really expresses the ratio between the actual 
nuclear magnetic moment and the magnetic moment which would be 
expected if the nuclear angular momentum were entirely due to the 
orbital motion of a single proton, with angular-momentum quantum 
number 7. 

The nuclear magnetic moment is taken as positive, if its direction 
with respect to the angular-momentum vector corresponds to the rotation 
of a positive electrical charge. 

Problems 

1. Assuming all the charge on a proton to be uniformly distributed on 8 
spherical surface, 1.5 X 10~ 13 cm in radius, what should be the actual value of the 
magnetic dipole moment of the proton if the proton is a sphere of uniform den- 



3] Nuclear Moments, Parity, and Statistics 151 

sity, having an angular momentum of CyV 3)ft? If the maximum observable 
component of the angular momentum is ^h, what is the corresponding maximum 
observable component of the magnetic dipole moment, expressed in nuclear 
magnetons? Ans.: (5 \/3/6)(e&/4irfl/c); %p M . 

2. Assuming all the charge on a proton to be uniformly distributed through- 
out a spherical volume, of radius 1. 5 X J0~ 13 cm, what should be the actual 
value of the magnetic dipole moment of the proton if its angular momentum is 
(i\/3)^ and if the proton has a uniform density? If the maximum observable 
component of the angular momentum is ^h, what is the corresponding maximum 
observable component of the magnetic dipole moment, expressed in nuclear 
magnetons? Ans.: (\/3/2)(eh/4irMc); 



3. Anomalous Magnetic Moments of Free Nucleons 

a. The Spin Magnetic Moment for Atomic Electrons. In a wide 
variety of experiments the magnetic dipole moment associated with the 
orbital angular momentum of atomic electrons always has the value 
expected from simple classical considerations, as illustrated by Eq, (2.8). 
This is equivalent to saying that the atomic orbital g factor, in an atomic 
analogue of Eq. (2.10), has the value g t 1. Such is not the case for 
the magnetic moment associated with the spin angular momentum of 
electrons. From the time of the introduction, in 1925, of the concept of 
electron spin, it was clear experimentally that the magnetic dipole 
moment of the spinning electron is very closely equal to one Bohr magne- 
ton. Because s = ^ for the electron, the spin g factor g s for an electron 
appeared therefore to be g s = 2. When Dirac developed his relativistic 
quantum-mechanical theory of the electron, the value g s = 2 also emerged 
from this theory in a natural way. 

By atomic-beam methods it is possible to obtain very precise measure- 
ments of the separation of optical fine-structure levels, and such experi- 
ments first showed in 1947 (K51, L2) that the magnetic dipole moment 
of the spinning electron is slightly greater than one Bohr magneton and 
corresponds to g s ^ 2.0023. The following year, Schwinger showed 
(S21), from a reformulation of relativistic quantum electrodynamics, 
that the interaction energy between an electron and an external mag- 
netic field must include a radiative correction term representing the 
interaction of the electron with the quantized electromagnetic field. 
The detailed application of the theory showed that in the atomic-beam 
experiments the effective value of the magnetic moment due to the spin 
of the electron should be greater than one Bohr magneton by the factor 
(1 + a/2v) = 1.001 16, where a = 2Tre 2 /hc ~ TIT is the fine-structure 
constant. 

This theoretical result concerning the increased effective value of the 
spin gyromagnetic ratio for the electron in a magnetic field is in agree- 
ment with the experimental value obtained from precision comparisons 
of the frequency of lines in the hyperfine-structure spectrum of gallium, 
indium, and sodium in a constant magnetic field (K51, M7). Thus, for 
atomic electrons, the spin g factor is 

g. = 2(1 + a/271-) = 2.002 32 (3.1) 



152 The Atomic Nucleus [CH. 4 

while the orbital g factor has its classical value of g\ = 1, all the corre- 
sponding magnetic moments being in Bohr magnetons. 

These observations on the electron-spin gyromagnetic ratio are impor- 
tant for two reasons. First, many nuclear gyromagnetic ratios are 
measured in terms of high-precision direct comparisons with electronic 
gyromagnetic ratios. Secondly, the anomalous spin gyromagnetic ratio 
for the electron is reasonably well understood in terms of existing theorj". 

b. The Spin Magnetic Moment for Protons. If the spinning proton 
behaved like a uniformly charged classical sphere, then g, = 1, and its 
magnetic dipole moment should be one-half nuclear magneton, by Eq. 
(2.16). If the proton were a particle which followed Dirac's relativistic 
quantum mechanics, then g s = 2, and its magnetic dipole moment should 
be one nuclear magneton. Actually, the spin magnetic dipole moment 
of the free proton has the nonintcgral and "anomalous" value of slightly 
over 2.79 nuclear magnetons. 

Direct Observation of the Proton Magnetic Moment. The magnetic 
dipole moments of atoms, due to the spin and orbital motion of atomic 
electrons, were first demonstrated and measured directly in the epochal 
experiments of Stern and Gerlach (p. 389 of R18). The same experi- 
mental principle of deflecting a beam of neutral molecules by passing it 
through a strongly inhomogeiieous magnetic field was applied successfully 
to neutral hydrogen molecules in 1932 by Stern, Estermann, and Frisch 
(E16, E15). 

These experiments are of great fundamental importance for two 
separate reasons. First, they remain the only measurements of a nuclear 
magnetic moment in which the interpretation is independent of the gyro- 
magnetic ratio. The quantity measured is the force exerted on a neutral 
particle by a strong inhomogeneous magnetic field. This force is pro- 
portional to the classically defined magnetic dipole moment, without 
recourse to the presence or absence of angular momentum. 

Second, these experiments were the first to reveal the anomalous 
value of the proton magnetic moment. The inadequacy of the Dirac 
wave equation or of any other theory for heavy elementary particles 
became apparent because these theories are unable to predict the correct 
magnetic dipole moment for the proton. This challenging situation 
remains unrelieved by theoretical efforts up to the present. 

The essentials of the molecular-beam method for hydrogen developed 
by Estermann, Frisch, and Stern are the following: Normal hydrogen 
consists of a mixture of 75 per cent orthohydrogen and 25 per cent para- 
hydrogen. Pure parahydrogen can be prepared by adsorption on char- 
coal at liquid-hydrogen temperature. The nuclear spins, and hence the 
nuclear magnetic moments, of the two hydrogen atoms are parallel in 
the orthohydrogen molecule and antiparallel in the parahydrogen molecule. 
Therefore in parahydrogen the nuclear moments cancel each other, and 
the angular momentum and magnetic moment of the molecule are due 
to the electrons and to molecular rotation. In orthohydrogen, however, 
the nuclear moments arc parallel and reinforce each other, thus adding 
to the molecular moments. 



3] Nuclear Moments, Parity, and Statistics 153 

The deflection of a beam of molecular hydrogen in a transverse 
inhomogeneous magnetic field depends on the total molecular magnetic 
moment. The contribution of electrons and molecular rotation is deter- 
mined from the measurements on parahydrogen. In principle, these are 
then subtracted out of the deflections of orthohydrogen. The result is a 
direct measure, by differences, of the magnetic moment of the proton. 
In this way, the proton magnetic dipole moment p p was found to be 2.46 
0.08 nuclear magnetons. The result was confirmed by measurements 
on the HD molecule (E16) but is a little low in comparison with later 
studies of the proton gyromagnetic ratio by magnetic resonance methods. 
All the subsequent determinations of p p have been made by more accurate 
but less direct methods which involve measurements of the proton gyro- 
magnetic ratio, usually by observing the Larmor precession frequency 
produced by a homogeneous magnetic field [Chap. 5, Eq. (2.3)]. 

Precision Measurements of the Proton Spin Magnetic Moment. All 
measurements of nuclear g factors are currently referred to the pro- 
ton spin g factor as a reference standard. Consequently, the proton 
magnetic moment has fundamental importance, experimentally as well 
as theoretically. 

A variety of experimental methods has been used (Chap. 5). Several 
of these are thought to have an accuracy of about 0.01 per cent. How- 
ever, present results by various methods have a spread which approaches 
0.1 per cent. Each author of a table or compilation of nuclear magnetic 
moments generally chooses one or another of these individual measure- 
ments as a reference standard for the entire compilation. Caution must 
therefore be used in comparing tables by various authors and tables of 
various dates by the same authors. 

In 1953 one of the principal reference-standard values adopted lor the 
proton magnetic moment was 

Atp = 2.7934 nuclear magnetons (3.2) 

derived from comparisons by Gardner and Purcell (G9) of the Larmor 
precession frequency of protons (in mineral oil) with the cyclotron 
resonance frequency of free electrons measured in the same magnetic 
field. The observed value by Gardner and Purcell of 

ju p = (1.521 00 0.000 02) X 10~ 3 Bohr magneton (3.3) 

leads to Eq. (3.2) if the ratio of proton to electron mass is 1,836.6 (D43), 
although a more recent value for this mass ratio is 1,836.1 (D44). Equa- 
tion (3.2) was the reference standard for the table of nuclear moments 
compiled by H. L. Poss (P27) and for the earlier issues of the widely 
used cumulative tables of new nuclear data which are published as a 
quarterly supplement in Nuclear Science Abstracts (N24). Note, how- 
ever, that the parent table "Nuclear Data" (National Bureau of Stand- 
ards Circular 499) has a different basis, namely, 

Up = 2.7926 nuclear magnetons (3.4) 

This value was based upon preliminary data on the ratio of the Larmor 



154 The Atomic Nucleus [CH. 4 

precession frequency for protons to the cyclotron resonance frequency 
for free protons, measured in the same magnetic field, by Hippie, Sommer, 
and Thomas (H55) in 1949. In Eq. (3.4) a correction of 0.003 per cent 
has been added to the observed value, to compensate for the estimated 
diamagnetic effect (LI) of the atomic electrons in hydrogen. In 1950, 
the same authors (860) obtained by the same experimental method an 
improved value of 

fL p = 2.792 68 0.000 06 nuclear magnetons (3.5) 

No diamagnetic correction is included in Eqs. (3.2), (3.3), or (3.5). 

By still different methods, other groups of workers have obtained 
values of p p which vary from one another often by more than the assigned 
experimental errors. But these slight differences are probably not of 
fundamental origin, and it is more important to stress the remarkable 
degree of agreement upon some figure very close to 

PP = 2.793 nuclear magnetons (3.6^ 

as obtained by a wide variety of methods. Equation (3.6) corresponds 
to an absolute gyromagnetic ratio 77 for the proton of 

7p = -T7- = 2 - 675 X 1Q4 gauss- 1 sec- 1 (3.6a) 

IP 2Mc 

Of course, the absolute value of the spin angular momentum of the 
proton is (s')* = ^ V^s(s + 1) = /L(\/3/2), rather than h/2, and because 
g = fjL/I = &/!*, the absolute (but unobservable) magnetic dipole 
moment of the proton in the direction of s* is M** = MP v3. 

c. The Spin Magnetic Moment for Neutrons. It has been possible 
to obtain an accurate comparison of the Larmor precession frequencies, 
which are proportional to nuclear g factors [Chap. 5, Eq. (2.3)], for pro- 
tons and for free neutrons in the same magnetic field by making use of 
the phenomenon of magnetic scattering of neutrons in magnetized iron 
(A24, A31). In this way it is found that the ratio of the nuclear mag- 
netic dipole moments of the neutron and proton is (B76) 

g - = = = -0.685 00 0.000 03 (3.7) 

9p PP 

Taking Eq. (3.2) as the reference value of /i p gives 

Mn = 1.9135 nuclear magnetons (3,8) 

Thus the neutron, whose net electric charge is exactly zero, possesses an 
unknown inner constitution such that its spin angular momentum, s = -J-, 
is associated with a fairly large magnetic dipole moment. The sign of 
this magnetic moment is negative and therefore simulates the rotation 
of negative charge in the spin direction. 

Nonadditivity of Magnetic Moments. The nuclear g factor of the 
deuteron has been measured accurately with respect to that of the proton. 



4] Nuclear Moments, Parity, and Statislics 155 

Combining the results leads to /id = 0.8576 nuclear magneton for the 
deuteron. It is important to note that the neutron magnetic moment 
cannot be obtained by merely subtracting the proton moment from the 
deuteron moment. This would give 

Md - M P = 0.8570 - 2.7934 = -1.9358 nuclear magnetons 

a value which differs from the directly measured // = 1.9135 by far 
more than the limits of error of the measurements. We shall see later 
(Chap. 10) that, in the case of the deuteron, the finite difference between 
/i d and (n p + MW) can be interpreted as evidence in favor of the existence 
of a contribution of tensor (noncentral) force between nucleons in nuclei. 

Problem 

Evaluate the magnetic field, due to the intrinsic magnetic dipole moment of 
the proton, in gauss: (a) at the equator of the proton and (b) at the radius of 
the first Bohr orbit of hydrogen. Arts.: (a) ~7 X 10 1B gauss; (b) 165 gauss. 

4. Relationships between I and p 

The nuclear angular momentum and magnetic moment have been 
measured for the ground levels of more than 100 nuclides (K23, W2a) 
by the application of a variety of experimental methods. Many of these 
data are summarized in Tables 4.3 and 5.1. Some regularities have 
emerged from studies of these data. One generalization to which no 
exceptions have yet been found is that among stable nuclides 7 = for 
all even-Z even-AT nuclides and for no others. 

a. Classifications of the Experimental Data on I and JJL. For con- 
venience in systematizing many types of data, nuclides are divided into 
four classes according to whether the proton number Z and the neutron 
number N ( = A Z) are odd or even. A common arrangement of these 
four classes, and the population of each, is shown in Table 4.1, together 
with a gross summary of the experimental results on 7 and /x by classes. 

Note that even-Z even-JV is by far the most abundant class of stable 
nuclides. Among these nuclides, measurements of 7 have been obtained 
in about a dozen cases, usually by diatomic band spectroscopy (Chap. 5), 
and invariably with the result 7 = 0. This is the experimental, basis 
for a fundamental assumption which is made in a number of nuclear 
models. An even number of protons are assumed always to find their 
lowest energy level by aligning their individual angular momenta so that 
they cancel by pairs, and hence also in the aggregate. Likewise, the 
individual angular momenta of neutrons within an unexcited nucleus are 
assumed to cancel by pairs and thus to total zero for any even number 
of neutrons. 

Then an odd-Z even-AT nucleus could be visualized as one odd proton 
outside a closed "core" of even-Z even- AT. On this single-particle model, 
the core would contribute zero angular momentum and zero magnetic 
dipole moment. The nuclear 7 and /* would then be due entirely to the 
one odd proton in odd-Z even-N nuclei. Analogously, in even-Z odd-AT 



156 



The Atomic Nucleus 



[CH. 4 



nuclei, the entire nuclear / and M are attributable to the one odd neutron, 
in this single-particle model. Mirror nuclei, in which Z and N are inter- 
changed, will then have the same value of /. This applies to the ordinary 
odd-^4 mirror nuclei, which are the adjacent isobars A = 2Z 1, and 
also to the group of even-A mirror nuclei (A = 2Z 2) which are the 
outer members of symmetrical isobaric triads such as Be 10 B 10 C 10 and 
C 14 N 14 14 . 

Odd-Z odd-TV nuclei are generally unstable, except in the four cases 
!H 2 , sLi 6 , 5 B 10 j 7 N 14 for which Z = N. On the single-particle model, these 
odd-Z odd-TV nuclei consist of an even-Z even-TV core, with one proton 
and one neutron outside. The angular momenta so far measured for 

TABLE 4.1. SUMMARY OF THE FOUR "ODD-EVEN" CLASSES OF NUCLIDES AND 

THE GROSS RESULTS OF MEASUREMENTS OF THEIR GROUND-LEVEL 

NUCLEAR ANGULAR MOMENTA / AND MAGNETIC DIPOLE 

MOMENTS ^ 
Compiled from (H61) and (K23) 



Nurleon classification 


Nuclear moments 










Number 


Number of 








Mass 


Proton 


Neutron 


of 


nuclides measured 






Claim 


number 


number 


number 


known 




/ 


p 




A 


Z 


N 


stable 


















nurlidea 


Stable 


Radioactive 






I 


Odd 


Odd 


Kven 


50 


50 


11 


11 III 


Usually large, 
















usually positive 


II 


Odd 


Even 


Odd 


55 


36 


4 


1 !1 5 7 U 

a. :. -j. ;> s. 


Usually small, 


















often negative 


III 


Even 


Odd 


Odd 


4 


4 


9 


1, 2, 3, 4, 6 


Usually positive 


IV 


Even 


Even 


Even 


J65 


12 


1 





Indeterminate 



such nuclei are all nonzero. Thus, in general, the angular momenta 
of the single odd proton and neutron do not align for cancellation, as 
would a pair of protons or a pair of neutrons. 

Schmidt Diagrams. Schmidt (S14) first emphasized the guidance 
which can be obtained from plots of // against 7. Figures 4.1 and 4.2 
are modernizations of Schmidt's plots of the empirical dependence of n 
on / for nuclei containing one odd nucleon. The trend of /* against I 
is seen to be quite different when the odd nucleon is a proton (odd-Z 
even-TV, Fig. 4.1) and when the odd nucleon is a neutron (even-Z odd-TV, 
Fig. 4.2). 

b. Single -particle Model, for Odd-^4 Nuclei. If, with Schmidt, we 
assume a single-particle model, in which the total nuclear moments / and 
\JL of odd- A nuclei are due to one odd nucleon, then 7 is simply j = I s 
of this one odd nucleon. The dependence of the associated magnetic 
dipole moment on 7 can be determined in the following way. 

Let in* be the value of the magnetic dipole moment which is col- 
linear with the orbital angular momentum whose magnitude is denoted 



Nuclear Moments, Parity, and Statistics 



157 



5k 

01 

E 4 
_o> 

a 

o 
u 

I 2 

no 



Odd-Z, even-JV nuclides 



_ H 3 



1 



Co'VSc' 5 



uB7 o 1 " 1 Sh lzl 

D v .. WM 1PK 01 



Eu! JM* 
i 2 > !Mn 55 Lu' 75 



93 



oNb 



oBi 209 




-1 

-L 1 A 1_ 9 

2 2 2 2 7 

Nuclear angular momentum, 7 

Fig. 4.1 Schmidt diagram of 7 and \t. for odd-Z even-A T nuclides. The solid-line 
histogram corresponds to the Schmidt limits for each value of 7, if 7 and M were due, 
entirety to the motion of one odd proton, Table 4.2. Open circles represent nuclides 
with one proton in excess of, or with one proton less than, a "closed shell" of 2, 8, 20, 
28, 40, 50, or 82 protons. 





1 | 


| | 




Even-Z, odd-JV nuclides 




^ 2 

*J" 





^wi-H" 


c 




Zn 67 




\ 


^Hg^^ 171 ^"Os 189 





D 

u i 


Xel2 ^| dl "z3 * Hg201 


Yb" 3 . >M 25 145 M 




Sn< 11? 125 Be 


M0 97 143 g 


s 


l l 


- 1 1 




"He 30 


/I, if /->-/ 




1 1 


1 1 


_L J. 


579 


2 2 


222 



Nuclear angular momentum, / 

Fig. 4.2 Schmidt diagram of / and /* for even-Z odd-N nuclides. The histogram 
corresponds to the Schmidt limits for each value of 7, if 7 and p were due entirely to 
the motion of one odd neutron, Table 4.2. Open circles represent nuclides which have 
one neutron more, or one neutron less, than the number required to form a "closed 
shell" of 2, 8, 20, 28, 40, 50, 82, or 126 neutrons. 



The Atomic Nucleus 



[GR. 4 



158 

I* = Vl(l + 1). Analogously, denote by /*,- the magnetic dipole moment 
which is collinear with the spin angular momentum s* = V*(* -I- 1) aub 
by MJ the magnetic dipole moment which is collinear with tne total 
angular momentum j* = V/(j + 1) of the single particle. 

Figure 4.3 illustrates the geometrical relationships of the usual vector 
model. Both I* and s* precess (W39) about their vector resultant j*. 
The time-average values of the components normal to j* are zero. The 



Angular 
momentum 




Magnetic dipole moments 





Odd-neutron /* s *=-l 91 V3 




Fig. 4.3 Vector diagrams (for I = 3) illustrating the composition of angular-mom en- 
tum vectors, whose magnitudes are I* V(Z + l)i s * ~ V(s + l) f to form j* = 
\/j(j~~+ l), for the "parallel".;" = I + s and "antiparallel" case j = I s. Because 
of the anomalous values of the spin magnetic moments for the proton and the neutro^ 
<he magnetic moment p/+ for the antiparallel case, as shown graphically here, leads to 
A ,he value of M given by Eq. (4.10). 

value of j* is such that j = I + s for the so-called "parallel" alignment 
of I and s, while j = I s f or the " antiparallel " alignment of the orbit 
and spin angular momenta. Note that the quantum numbers I and s 
add as scalars, thus .7 = I s; but the corresponding angular momenta I* 
and s* do not add as scalars (j* ^ I* s*) but only as vectors 

j* = 1* s* 
In turn, j* is understood to precess about any arbitrary direction, such 



4] Nuclear Moments, Partly, and Statistics 159 

as that of an external magnetic field, in such a way that j is the maximum 
observable component of j*. Thin is analogous to the vector-model 
diagram in Fig. 1.1 for / and /*. In this single-particle model, 7 = j 
and /* = j*. 

In Fig. 4.3, the angle (l*j*) between I* and j* is given by the law of 
cosines as 

7*2 i ;*2 __ *2 

cos ff*j*) = -' -^ (4.1) 

for both of the possible cases, j = / + s and j = I s. Similarly, the 
angle between s* and j* is given by 

o*2 J- -1*2 _ 7*2 

cos ( S *j*> = -jv, - (4.2) 



The net component of the magnetic th'pole moments which is parallel to 
j* is given by 

Mj. = MI* C-OH (l*j*) + M , cos (s*j*) (4.3) 

The g factor for the single particle is, by Eq. (2.15), 

-5-J-ff () 

where /* is the observable component of the net magnetic dipole moment 
and the nuclear-angular-momentum quantum number I is the same as j 
for the single odd particle. By substituting Eqs. (4.1) to (4.3) into this 
equation, we can obtain the general relationship 



2 -'"/.-) 



in which the orbital and spin g factors are given by 

*--? 
" ~"- 



Parallel Spin and Orbit. For the "parallel" case, I = j = I + s, 
and substitution of j* 2 - j(j +1) = (/ + )(* + I) into Eq. (4.5) allows 
it to take on the simple special form 



(4 - 8) 

or IL = gl = gj + M . for 7 = Z + i (4.9) 

This represents simple additivity of orbital and spin projected magnetic 
moments and is a result to be expected on na'ive intuitive grounds. The 
antiparallel case turns out to be more subtle. 



160 The Atomic Nucleus [CH. 4 

Antiparallel Spin and Orbit. For the "antiparallel" case, 

/ = j = l - s 

and substitution of j* z = j(j + 1) = (I ?)(Z + *) into Eq. (4.5) gives 
for this special case 

711\_ 7~J-J\ 
g g \i + i/ g * \i + i/ 

or p = gl = gil - (M. - i flfi) (f^l) for 7 = Z - (4.10) 

Schmidt Limits. When we substitute into Eqs. (4.9) and (4.10) the 
g factors which correspond to single nucleons, namely, 

for protons: 0i = 1 g* = 2* = 2 X 2.79 a in 

for neutrons: g t = g s = 2^ = 2(--1.91) & ml1 ' 

we obtain the predicted relationships between 7 and M on the single- 
particle model. To each value of 7, there correspond two values of /i, 
depending on whether 7 = / + ior7 = / i. These two values of /* 
constitute the so-called Schmidt limits. They are summarized in Table 
4.2 and are shown as solid lines in Figs. 4.1 and 4.2. 

TABLE 4.2. THE SCHMIDT LIMITS, IN THE SINGLE-PARTICLE MODEL, FOR 
NUCUDES HAVING ONE ODD NUCLEON 



7 


Odd proton 


Odd neutron 


* + i 

/ l 


M - / + 2.29 

7 9 9Q 


M = -1.91 
11 1 m 


1 V 


-/ 2.29 y + ] 


"- 1 - 01 / + i 



Schmidt Groups. The 100-plus measured values of 7 and /i for the 
ground levels of nuclei are seen to fall generally between, but not on, the 
Schmidt limits. There is a tendency for the measured values to fall 
fairly clearly into two groups, each parallel to a Schmidt limit. These 
two " Schmidt groups 7 ' are presently assumed to correspond to the 
/ = I -(- ^ and the 7 = I -$ cases, in odd- A nuclides. Thus a measure- 
ment of p and 7 affords a means of "measuring" the value of I for the odd 
particle. 

For example, 2 7Co BB has 7 = 1 and p = 4.64. The high value of /* 
places this nuclide in the upper Schmidt group for odd-Z nuclides in 
Fig. 4.1. Hence 7 = Z + i, and I = 3. It is then reasonably certain 
that the odd proton in Co 69 is in an / z state and not a QI state. Such 
inferences of / are of great value in the development of theories of nucleon 
coupling and nuclear structure (Chap. 11). 

Quenching of Nucleon Spin Magnetic Moment. The two Schmidt 
groups are approximately parallel to the Schmidt limits. Their slope 



4] Nuclear Moments, Parity, and Statistics 161 

is about unity for odd-proton nuclides and zero for odd-neutron nuclides. 
Thus the groups appear physically to represent pi due to orbital motion, 
plus or minus a magnetic moment due to spin. Except for H a and He 8 , 
the total IL never exceeds what would be expected if all the / and ^ were 
due to the one odd proton (or neutron), the rest of the nucleus forming a 
closed system, or core, with 7 = and n = 0. 

The spread of actual values of \L from lines corresponding only to 
m is not as large as would be expected for /*. Thus, empirically, it 
appears that M, as measured for free protons and neutrons, may not be 
fully effective when these nucleons aggregate into nuclei. This so-called 
"quenching" of the anomalous magnetic dipole moments is qualitatively 
understandable in the language of the meson theory of nuclear forces. 

According to the meson theory, the attractive force between nucleons 
arises from the exchange of charged and uncharged mesons between 
nucleons. A free isolated nucleon then exhibits a virtual emission and 
absorption of mesons, and this meson current is the origin of the anom- 
alous magnetic dipole moment. The intrinsic magnetic moment p, of the 
free nucleon may therefore be reduced, or partially quenched, when the 
odd nucleon is bound to the nuclear core (B74). Although these con- 
cepts have a welcome plausibility, their quantitative aspects have yet to 
emerge successfully from contemporary meson theory (F27). 

Asymmetric Core. The two Schmidt groups each have a spread which 
is much greater than the observational uncertainties, and so most of the 
variations must be taken as real. Moreover, as can be seen from the 
points shown as open circles in Figs. 4.1 and 4.2, nuclides which have 
one odd nucleon above or below a "closed shell' 1 of 2, 8, 20, . . . protons 
or neutrons seem to show no overwhelming distinction from nuclides 
whose core is not blessed by a "magic number." These observations 
suggest that the core does not always have exact spherical symmetry 
I = 0, /* = but contributes at least to the magnetic moment of the 
nucleus (D25). 

The concept of an asymmetric core is strengthened also by the 
inability of a symmetrical-core model to account for the observed electric 
quadrupole moments of nuclei (Sec. 5). At least for the odd-proton 
nuclei, the single-orbit model would lead to negative electric quadrupole 
moments entirely, whereas most measured quadrupole moments are 
actually positive, and some are very large. Polarization of the core 
by the odd nucleon appears to be a principal mechanism in producing the 
asymmetric core. 

At present, we have to interpret the Schmidt diagrams of Figs. 4.1 
and 4.2 as supporting a single-particle model for odd- A nuclides, modified 
both by quenching of free-nucleon moments and by asymmetry of the 
even-even core. The quantitative aspects of these effects are not yet 
well understood in existing theories. 

Uniform Model. The total magnetic dipole moment of a nucleus is 
composed of two intrinsically different parts, an orbital moment and a 
spin moment. Margenau and Wigner (M10) have considered some 
details of a nuclear model in which the orbital angular momentum is dis- 



162 The Atomic Nucleus [CH. 4 

tributed uniformly over all the nucleons. This leads to an orbital gyro- 
magnetic ratio QL = Z/A ^ 0.4, where Z is the number of protons and A 
is the number of nucleons in the nucleus. Replacing g t in Eqs. (4.9) and 
(4.10) by g L leads to the so-called Margenau-Wigner (M-W) limits. For 
odd-proton nuclei, the Margenau-Wigner limits do not follow at all the 
empirical slope of about unity which is displayed in the Schmidt dia- 
grams. For odd-neutron nuclei the Margenau-Wigner limits predict an 
increase of p with / which is very much steeper than the empirical values. 
Thus, on a basis nf the general trends, there is no support for the uniform 
model in competition with the extreme single-particle model. 

Within a particular nucleus the volumetric distribution of the density 
of magnetic dipole moment would generally be nonuniform. Various 
methods for measuring nuclear gyromagnetic ratios can depend in differ- 
ent ways on the volumetric distribution of the magnetic moment. 
Molecular-beam methods and Larmor frequency resonance methods 
generally (Chap. 5) can treat the total moment as a point dipole. On 
the other hand, the magnitude of the hyperfme-structure separations of 
an electronic s state are proportional to the average electron density at 
the location of the nuclear magnetic moment. Bitter (BOO) has observed 
a finite difference between the relative hyperfine structure and the relative 
total moments of the isotopic nuclides Rb 87 (/=;/* = 2.75) and Rb 86 
(7 = \\ ft = 1.35) which can be traced to a dependence on the volumetric 
distribution of magnetic dipole moment, A. Bohr and Weisskopf (B90) 
were able to show that these differences are in somewhat better agree- 
ment with the simple Schmidt extreme single-particle model than with the 
uniform model. 

c. Odd-Z Odd-N Nuclei. The angular momentum 7 and the 
nuclear g factor have been measured directly for the ground levels of 
about a dozen nuclides belonging to the odd-Z odd-N class. As indicated 
in Tables 4.1 and 4.3, all these nuclides exhibit 7 > 0, even in the four 
cases in which Z = N. The simplest case, of course, is iH 2 , for which 
7 = 1. Both nucleons are surely in s$ states, with j = ? The total 7 
here is j\ + jz = 1, with the proton spin and the neutron spin aligned 
parallel to one another. 

Most nuclides in the odd-Z odd-AT class are radioactive, and they 
decay by 0-ray transitions to even-Z even-AT nuclides which presumably 
have 7 = 0. From comparisons of the radioactive half -period and decay 
energy, assignments can be made in many cases of the angular momentum 
of the odd-Z odd-N nuclide (Chap. 6). 

Nordheim's Rule. It is often possible to infer the state of the odd 
proton by comparison with a known odd-Z even- AT nuclide having the 
same Z. For example, 3 Li 7 (7 = 1) lies in the Schmidt diagram near the 
7 = I + y limit. Consequently, Z = 7 ^ = 1 , and the odd proton 
is in a p t state. Analogously, the state of the odd neutron can often be 
inferred. Then if the angular momenta of the odd proton and the odd 
neutron are called j p and j n , Nordheim (N23) has pointed out that the 
following coupling rules apply in at least 60 clear cases, though there 
are a few established exceptions. 



5] 



Nuclear Moments, Parity, and Statistics 



163 



1. If the odd proton and neutron belong to different Schmidt groups 
(that is, j p = I + ^ and j n = I , or the reverse), then / = \j f j n \. 

2. If the odd proton and neutron belong to the same Schmidt group 
(i.e., both j = I + i, or both j = I *), then 7 > \j p j n \. 

In the first case, the spin angular momenta of the odd proton and neu- 
tron are aligned parallel. In the second case, there is still a tendency 
toward parallel alignment. 

TABLE 4.3. NUCLEAR ANGULAR MOMENTUM / AND MAGNETIC DIPOLE 

MOMENT FOR SOME Qi>D-Z ODD-JNT NUCLIDES 

(As obtained by direct measurement of / and g = n/I. The first four nuclides are 
stable; the others, radioactive.) 



Nurlide 


z 


A' 


/ 


M 


H 2 


1 


1 


1 


0.857 


Li 


:; 


3 


1 


0.822 


BIO 


r> 


5 


15 


1.800 


N 1J 


7 


< 


1 


0.404 


?\ii 22 


11 


11 


;i 


1.746 


Na" 


11 


13 


4 


1.69 


Cl a " 


17 


U) 


2 


1.28 


K 


11) 


21 


4 


-1.29 


K 42 


19 


23 


2 


-1.14 


V" 


2:5 


27 


6 


3.34 


Hb B 


:?7 


H) 


2 


-J 68 


CV" 


5/1 


70 


4 


2.95 


Iji 17fi 


71 


105 


f;7 


4 



Problem 

Show that the Schmidt limit, for the C.RSO j = I s, is 



for both odd-neutron (gi 
nuclei. 



0; >x fl = 1.91) and odd-proton (gi = 1 ; /i fl = 2.79) 



5. Electric Quadrupole Moment 

If the time average of the volumetric distribution of electric charge 
within a nucleus deviates from perfect spherical symmetry, then the 
nucleus will possess finite electric multipole moments. The electro- 
static field which is produced at the position of the nucleus by the atomic 
and molecular electron configurations is generally a nonuniform field. 
The electrostatic potential energy of a nuclear electric multipole moment, 
residing in this nonuniform field, makes a contribution to the energy 
of the electronic state. Measurable effects in the hyperfme structure of 



164 



The Atomic Nucleus 



[CH. 4 



-e +e 



atomic and molecular spectra have been found which can be attributed 
to nuclear electric quadrupole moments. 

Nuclear electric quadrupole moments were not discovered until 1935 
when Schiller and Schmidt (SI 8) found them necessary in order to explain 
irregularities in the hyperfine spectra of Eu m and Eu 158 . Effects due 
to other electric multipole moments (dipole, octupole, etc.) are not ex- 
pected and have not yet been found, 
a. Classical Multipole Moments 
for Point Charges. The simplest ex- 
ample of a classical electric quadru- 
pole is the so-called axial quadrupole. 
This is composed of two electric di- 
poles, whose axes are collinear and 
antiparallel, as indicated in Fig. 5.1. 
In common with all quadrupoles, 
it produces a potential which varies 
inversely with the cube of distance, 
and its quadrupole moment has di- 
mensions of (charge) X (area). A 



Fig. 6.1 Classical quadrupoles, com- 
posed of antiparallel paired dipoles, of 
equal dipole moment ea. The axial 
quadrupole, at the left, easily can be 
shown to produce a potential, at a dis- 
tance d a along Its extended axis, of 
(f> 2a(ea)/d 3 . Its classical quadru- 
pole moment <f>d 3 in the axial direction 
is therefore 2eo 2 . 



classical octupole can be formed by two closely spaced quadrupoles, and 
so on for the higher multipoles. 

Multipole moments are also exhibited in the potential due to a single 
point charge, if that charge is not located at the origin of the coordinate 
system. Visualizing the atomic nucleus as an approximately spherical 
assembly of neutrons (charge, zero) and protons (charge, +c), we may 
specialize the general classical theory (p. 172 of S76) of electric multipole 




Fig. 5.2 To accompany Eq. (5.1) et seq. for the potential v at P, due to the charge 
+e at z, y, z. The coefficients of l/d" +1 in Eq. (5.2) are the effective components in 
the z direction of the classical electric moments of multipole order 2". 

moments. In rectangular coordinates x t y, z (Fig. 5.2) the scalar electro- 
static potential <p at the external point /*(0,0,d) on the z axis, due to a 
charge -\-e at the point (x^y,z), is 

_9\_A 

(5-1) 



where d\ = (d 2 2rd cos $ + r 2 )* is the distance from the point P to the 
charge, r = (x 2 + y 2 + z 2 )* is the distance from the origin (or mass 
centroid of a nucleus) to the charge, and cos # = z/r defines the angle 
between r and d. 



5] Nuclear Moments, Parity, and Statistics 165 

Expanding Eq. (5.1) and collecting terms in l/d n , we obtain the 
general expression 



e . er 

- + - 



(5.2) 
or, more generally, 



71-0 



where P n (cos t>) are the Legendre polynomials and n (or, more exactly, 
2 n ) is the multipole order. Thus, in Eq. (5.2) the coefficient of 1/d is the 
raonopole strength, of 1/d 2 is the z component of the dipole moment, of 
1/d 8 is the z component of the quadrupole moment, of 1/d 4 ia the z 
component of the octupole moment, etc. The first term in Eq. (5.2) is 
the ordinary coulomb potential. 

Thus even a single isolated charge, if not located at the origin of 
coordinates, exhibits a quadrupole and other moments; i.e., the electric 
field due to an asymmetrically placed proton is identical with the electric field 
which would be produced by placing, at the center of the nucleus, one proton, 
and also an electric dipole, an electric quadrupole, an electric octupole, etc. 
The total electric charge in such an equivalent structure would still be 
one proton, because the classical electric multipoles each have zero net 
charge. 

If in Eq. (5.2) we now substitute cos & = z/r, we obtain for the 
coefficient of 1/d 3 , which is the effective classical quadrupole moment 
g (2) in the direction z, as exhibited at P(0,0,d), 

q w = | (3z 2 - r 2 ) (5.4) 

2i 

The classical quadrupole moment of such a nucleus would be taken 
along the body axis of total angular momentum, /*. If a single pro- 
ton were situated at the nuclear radius R, along the body axis, i.e., if 
z = r = R, and x = y = 0, then the classical quadrupole moment would 
be, from Eq. (5.4), 

qW = eR 2 

Similarly, a single proton at the nuclear equator, z = 0, r = R, would 
have a classical quadrupole moment 



Thus if six protons were located symmetrically along the coordinate 
axes at distances + R from the center, the two protons at z = + R on the 
body axis of angular momentum would contribute +2eR z to the quad- 
rupole moment, while the four protons in the equatorial plane, at x = R 



166 



The Atomic Nucleus 



[CH. 4 



and y == 72, would contribute 4(-efl 2 /2) = -2e/2 2 . Thus, for this 
or any other spherically symmetric, distribution of positive charge in the 
nucleus, the net quadrupole moment is zero. 

If, in addition to a symmetric distribution, one or more nuclear 
protons are located asymmetrically, the nucleus will possess a net electric 
quadrupole moment. Thus both positive and negative nuclear quad- 
rupole moments are to be expected. Positive moments correspond to an 
elongation of the nuclear change distribution along the angular-momen- 
tum axis (football-shaped distribution). Negative moments correspond 
to a flattened, or oblate, distribution (discus-shaped distribution). 

b. Nuclear Electric Quadrupole Moment. The potential energy of 
an electric quadrupole when placed in a nonuniform electric field can be 
shown to be proportional to the product of the gradient d z /dz of the 
electric field and the quadrupole moment in the z direction. The inter- 
action energy between the field produced at the nucleus by the atomic or 
molecular electrons and the nuclear quadrupole moment can be measured 
accurately by several methods. It is much more difficult to evaluate the 
electronic field, and thus to obtain a quantitative measurement of the 
nuclear quadrupole moment. 

The electronic field, the quadrupole moment, and their interaction 
energy are to be calculated quantum-mechariically . Then what is usually 

called the ''quadrupole moment" Q 
receives in the quantum mechanics 
a definition which differs in some 
details from a classical definition. 
First, the quadrupole moment is not 
taken about the body axis of /* but 
about the axis of its maximum pro- 
jected component w?/ = /. Second, 
the numerical factor -* in the classi- 
cal expression Eq. (5.4) disappears. 
Third, the probability density for a 
proton at any position (x,y,z) in the 
nucleus is represented in terms of 
the square of a wave function |^| 2 . 
The quantum-mechanical charge dis- 
tribution is therefore continuous and 
can be represented by a mean charge 
density p(:r,j/,z). Fourth, the inte- 
gral over the charge distribution is 
divided by the proton charge e, which 
makes all nuclear quadrupole moments have dimensions of cm 2 only. 

Then if p is the density of nuclear charge in the volume element dr 
at the point (z,r), as illustrated in Fig. 5.3, the nuclear electric quadrupole 
moment Q is defined as the time average of 





(a) quantum- 
mechanical 



(b) classical -limit 



Fig. 6.3 The nuclear electric quadru- 
pole moment Q is evaluated in the 
quantum mechanics as an integration 
of Eq. (5.5) about the axis of m/ = /, 
which is the maximum projection of 
/* = V7~(7~+~l). In the classical 
limit, 7* > /, and the volume integral 
would be taken in the geometry shown 
at the right. 



- / p(3z 2 - r 2 ) dr = - I pr 2 (3 cos 2 1> - 1) dr 



(5.5) 



5] Nvdear Moments, Parity, and Statistics 167 

taken about m/ = 7, as /* processes about /. This is often written in 
the equivalent forms 

Q = 1 [ pr z(3 cos 2 i? - l)] av = Z(3z 2 - r 2 >. v (5.6) 



Certain other nonequivalent definitions of the nuclear quadrupole 
moment are used by some authors. Caution is necessary in comparing 
the results of various investigators, especially in the older literature. 
What we would call Q/Z is used by some as the quadrupole moment. 
Feld (F25), Bardeen and Townes (BIO), and Ramsey (R3) have presented 
very helpful comparisons of the several expressions used by various 
authors for reporting quadrupole moments. 

Relationships among Q, mi, and I. The evaluation of the quadrupole 
moment Q in the quantum state m/ I is in harmony with the conven- 
tional definitions of the magnetic moment /x and the mechanical moment 
/. In the case of Q, the effective components for other magnetic quan- 
tum numbers m r = (I 1 ), . . . , do not follow a simple cosine law, as 
they do for /x and 7, because of the cos 2 & term in Eq. (5.5). If ft is the 
angle between the body axis 7* and the z axis in space, then, as illustrated 
in Fig. 1.1, 



a 

COS = 



and it can be shown (B68) that the effective value of the quadrupole 
moment is proportional to (3 cos 2 1). Then the effective value 
Q(wi/) in the state ra/ is related to its value Q in the state m/ = / by 



Nuclei which have 7 = or 7 = -J can exhibit no quadrupole moment 
Q in the state m/ = 7. This can be seen from Eq. (5.7) or, more phys- 
ically, by noting that, in the case of 7 = i, cos ft = k/^\ X 1 = I/ A/3, 
and, from symmetry considerations, an average value of (3 cos 2 # 1) 
in Eq. (5.5) becomes zero. This does not mean that nuclei with 7 = \ 
necessarily have perfectly spherical distributions of charge about their 
body axis 7*, but only that the maximum observable component Q is 
zero. Finite electric quadrupole moments are therefore detectable only for 
nuclei which have angular momenta 7 > 1. 

Other Nuclear Electric Multipole Moments. When all nuclear electric 
multipole moments are defined quantum-mechanically in terms of the 
state mi = I, it can be shown (p. 30 of B68) quite generally that all 
electric multipoles with even values of the multipole order (quadrupole, 
2 4 -pole, 2"-pole) are zero unless 7 > n/2. Moreover, all electric multi- 
poles of odd order (dipole, octupole, 2 B -pole, etc.) are identically zero, if 
we follow the reasonable assumption that nuclei have axial symmetry 
and that the center of mass and the center of charge coincide. We refer 
here to the "static" moment, or the " permanent" moment, of the 



168 The Atomic Nucleus [GH. 4 

stationary state of the quantum-mechanical system. The electric (and 
magnetic) multipole moments which characterize radiative transitions 
between excited nuclear levels are not similarly restricted. 

c. Significance of the Experimental Data on Quadrupole Moments. 
Precise measurements of atomic and molecular hyperfine structure have 
permitted the evaluation of the quadrupole moment Q for the ground 
level of a number of nuclides. Mostly Q lies in the domain of 10~ 26 to 
10" 24 cm 2 , which is of the order of the square of the nuclear radius. Such 
moments would therefore be produced by a nonspherical distribution of 
one or a few protons at distances of the order of the nuclear radius. 

Ellipticity of Charge Distribution. In the constant-density model of 
nuclei (Chap. 2), we conventionally assume that all nuclei have a spherical 
distribution of both mass and charge. The finite quadrupole moments 
imply that some nuclei have a slightly ellipsoidal distribution of charge. 
We can relate Q semiquantitatively to this ellipticity (S15, F26, B68). 

Let the nucleus be represented, as in Fig. 5.3b, as an ellipsoid, with 
semiaxis b parallel to the z direction and semiaxis a perpendicular to z. 
If we assume that the charge is uniformly distributed throughout this 
volume, with charge density 

Ze 3Ze 
P 



dr 
then the quadrupole moment in the direction z is 

r 2 ) dr = |Z(b 2 - a 2 ) (5.8) 



- [ 



Because the semiaxes b and a will turn out to be nearly equal for real 
nuclei, it is convenient to define the nuclear radius R as their mean value 

R = ^ (5.9) 

and to measure the ellipticity in terms of a parameter 77, defined by 

, = ^^ = 2 ^^ (5.10) 

* R b + a 

The quadrupole moment of Eq. (5.8) can now be rewritten as 

Q = $nZR* (5.11) 

The quantity yZ is then a rough measure of the number of protons whose 
cooperation is required in order to produce the observed quadrupole 
moment. Alternatively, the asymmetry of the charge distribution, as 
measured by the ratio of the major and minor axes of the ellipsoid, is 
given to a good approximation by 



(5.12) 



The 
given 



e ellipticities i\ of nuclides for which Q has been measured are 
in Table 5.1. Recall that f or / = and , Q = and ij = 0. 



5] Nuclear Moments, Parity, and Statistics 169 

For 7 > 1, the usual asymmetries are seen to be of the order of a few 
per cent. This is the extent of the experimental justification for the 
common simplifying assumption that nuclei are spherical. 

It must be noted that the ellipticities calculated from Eq. (5.11) and 
shown in Table 5.1 correspond to the ra/ = / state and are therefore 
minimum estimates in regard to the actual nucleus when it is considered, 
for example, as a target in a. nuclear reaction. This is because we have 
evaluated Eq. (5.8) in the classical limit of large quantum numbers, so 
that /*>/, and ft > 0. What might be called Q*, the quadrupole 
moment about the body axis /*, will always be larger than Q. 

The Deuteron. The deuteron is included in Table 5.1 even though 
it cannot be assigned a well-defined radius. Far-reaching consequences 
are associated with the discovery (K13) in 1939 at Columbia University 
of the small, but finite, quadrupole moment of the deuteron, 

Q = 0.273 X 10- 26 cm 2 

The ground level of the deuteron could no longer be regarded as 
simply the a /Si level (L = 0, S = 1), resulting from a central force 
between the neutron and proton, because an S state must be spherically 
sjrmmetrical. 

In this simplest of nucleon aggregations, the quadrupole moment 
can be accounted for by assuming that the force between the neutron and 
proton is partly a central force and partly a noncentral, or tensor, force 
(R6, R29, 14). The very existence of a noncentral force implies that the 
orbital angular momentum L is no longer a constant of the motion, 
although the total angular momentum / remains a "good" quantum 
number. The ground level of the deuteron becomes in this model a 
mixture of 3 Si and 3 Di (L = 2, 5 = 1, / = 1) levels. (No P-state 
admixture is present because the parity of a P state is odd, while the S 
and D levels both have even parity, as discussed in Sec. 6.) 

It is found that the quadrupole moment of the deuteron can be 
attributed to an admixture of about 4 per cent *Di level with 96 per cent 
3 Si level. This same admixture just accounts for the nonadditivity of 
the spin magnetic dipole moments of the neutron and proton in the 
deuteron by introducing a small contribution from orbital motion of the 
charged proton in the 3 Di level. Then 

M* = Mn + MP - KM. + MP - *)* (5.13) 

where * = .. . .1. , ^0.039 



is the so-called proportion of D level, fa and $ D are the wave functions 
in the S and D levels, and dr is the volume element in the relative coordi- 
nates between the proton and neutron (R6, 82, F48). 

Shell Structure in Nuclei. The systematic variation of Q and 77 with 
Z was first pointed out by Schmidt (S15), who noted in 1940 that minima 
in the absolute magnitude of the nuclear quadrupole moments occur near 
Z = 50 and 82. Additional data, and the revival and improvement since 



L70 



The Atomic Nucleus 



[CH.4 



TABLE 5.1. NUCLEAR QUADRUPOLE MOMENTS Q, AND THE CORRESPONDING 
ELLJPTICITY TJ, IN THE QUANTUM STATE m/ = 7, USING Eg. (5.11) 

WITH R = 1.5 X 10-" CM 

The measured values of /, M, and Q, and the presumed state of the odd nucleon in 
the ground level, are from the compilation by Klinkenberg (K23). 



Nuclide 


Number 
of odd 
nucleona 


' 


' 


Q, 
10-" cm* 


- 


Ground 
level 


Z 


A 


Odd-Z 


Odd-N 


On 


It 




1 


i 
* 


-1.913 








1 


1H 


1 


1 




i 


+2.793 








s l 


1H 


2 


1 


1 


1 


+0.857 


+0 273 


+0 095 


(i T)I 


1H 


3 


1 




i 


+2.979 








-I 


3 Li 


6 


3 


3 


1 


+0.822 


<0 09 


<0.005 


(i T)I 


3 Li 


7 


3 




3 


+3 256 


(+)2t 


(+)0.10J 


T>\ 


5B 


10 


5 


5 


3 


+ 1 800 


+6 


+0.14 


/3 3x 


5B 


11 


5 




3 


+2.689 


+3 


+0.067 


Pi 


7N 


14 


7 


7 


1 


+0.404 


+2 


+0.027 


<i,i>I 


80 


17 




9 


5 


-1.894 


-0.5 


-0.005 


d| 


13 Al 


27 


13 




5 


+3.641 


+ 15.6 


+0.074 




168 


33 




17 


9 


+0.644 


-8 


-0.027 


<*, 


16 S 


35t 




19 


3 
15- 




+6 


+0.019 


j 


17 Cl 


35 


17 




Jd 
3 


+0 822 


-7.89 


-0.024 


** 


17 Cl 


36t 


17 


19 


2 




-1.68 


-0.005 


(t'f). 


17 Cl 


37 


17 




3 


+0.684 


-6.21 


-0.018 


d 3 


29 Cu 


63 


29 




3 


+2.226 


-13 


-0.016 


Pa 


29 Cu 


65 


29 




3 


+2.385 


-12 


-0.014 


PJ 


31 Ga 


69 


31 




i 


+2 017 


+23.2 


+0.025 


Pa 


31 Ga 


71 


31 


- - 


3 
T 


+2.561 


+14.6 


+0.015 


Pi 


32 Ge 


73 


.. 


41 


1 




-20 


-0.02 


01 




33 As 


75 


33 




a 


+ 1.439 


+30 


+0.03 


Pi 


35 Br 


79 


35 




3 


+2.106 


+26 


+0.022 


Pi 


35 Br 


81 


35 




I 


+2 270 


+21 


+0.018 


Pi 


36 Kr 


83 




47 


1 


-0.970 


+15 


+0.012 


91 


49 In 


113 


49 




I 


+5.486 


+114 


+0.055 


91 


49 In 


list 


49 




D 


+5.500 


+116 


+0.056 





t A radioactive nuclide. 

J Parentheses indicate an uncertainty. 

Os 189 is from Murakawa and Suwa (M75). 



5] 



Nuclear Moments, Partly, and Statistics 



171 



TABLE 5.1. NUCLEAR QUADRUPOLE MOMENTS Q, AND THE CORRESPONDING 

ELLIPTICITY ij f IN THE QUANTUM STATE m/ = /, USING EQ. (5.11) 

WITH R = 1.5 X 10-" CM (Continued) 





Number 












Nuclide 


of odd 
nucleons 


7 


" 


Q, 
1Q-" rm 2 





Ground 
level 


Z 


A 


Odd-Z 


Odd-N 


51 Sb 


121 


51 




5 


+3.360 


-30 


-0.013 


d ft 


51 Sb 


123 


51 




7 


+2.547 


-120 


-0.053 


01 


531 
531 


127 
129 f 


53 
53 




5 

1 


+2.809 
(+)2.618t 


-59 
-43 


-0.025 
-0.018 


4 

ffi 


54 Xe 


131 




77 


3 


+0.700 


-15 


-0.006 


ua 


63 Eu 
63 Eu 


151 
153 


63 
63 




5 
5 


+3.6 
+ 1.6 


+ 120 
+250 


+0.037 
+0.077 


(*ft 

*! 


70 Yb 


173 




103 


5 


-0 65 


+390 


+0.100 


/I 


71 Lu 
71 Lu 


175 
176f 


71 
71 


105 


7 

>7 


+2.9 
+4 2 


+590 
+700 


+0.148 
+0.174 


n'li 


73 Ta 


181 


73 




7 


+2.1 


+600 


+0.143 


9\ 


75 Re 


185 


75 




5 


+3 171 


( + 280)t 


+0.064 


d s 


75 Re 


187f 


75 




5 

2" 


+3.204 


+260 


+0.059 


d| 


76 Os 


189 




113 


3 


(+)0.7t 


+200 


+0.045 


Pi 


80 Hg 


201 




121 


1 


-0.559 


+50 


+0.010 


Pt 


83 Bi 


209 


83 




9 


+4.082 


-40 


-0.008 


k t 



1948 of the earlier shell models, allow some interesting tentative correla- 
tions to be made between the presumed shell structure of nuclei and the 
observed quadrupole moments (H54, T26, M62). With some refine- 
ments, we may use the core-plus-single-particle model, which we have 
seen correlates fairly well with the observed relationships between the 
mechanical moment 7 and the magnetic moment /n (Figs. 4.1 and 4.2). 

A core of even-Z ; which in this model has I = 0, M = 0, cannot be 
expected, in general, also to cancel out its quadrupole moments to zero. 
Certain special even-Z cores do, however, contain protons in all possible 
nij states and therefore could have spherical symmetry, and Q = 0, if the 
core were not distorted by the nucleons outside it. These are the 
so-called closed shells, corresponding in the shell model to the observed 
"magic numbers," 2, 8, 20, 28, 40, 50, 82. 

For odd-Z Qveu-N nuclides in which Z corresponds to a closed gjiell 



172 



The Atomic Nucleus 



[CH. 4 



plus one proton, we would then expect a flattened, or disklike, charge 
distribution and consequently a negative Q. Positive quadrupole 
moments can also emerge from this model. If, in odd-Z even- AT nuclides, 
Z corresponds to one proton less than a closed shell, then, by the so-called 
"partial configuration method" (F18), the "hole" in the proton shell 
behaves like a negatively charged particle. Thus for a sequence of odd-Z 



0.16 




-0.04 - 



20 



40 



60 80 

Number of odd nucleons 



100 



120 



140 



Fig. 5.4 The plotted points are the observed quadrupole moments Q divided by the 
nuclear charge Z and the square of the nuclear radius, which is taken as R = 1.5 X 
10~ 13 ^4*. The ordinates are therefore proportional to the ellipticity and correspond 
to 0.817 ~ 0.8 [(&/a) - li of Eqs. (5.11) and (5.12). Moments of odd-Z even-AT 
nuclides and odd-Z odd-JV nuclides are plotted as circles against Z. Moments of 
even-Z odd-N nuclides are plotted as triangles. Arrows indicate the closing of 
major nucleon shells. The solid curve represents regions where quadrupole-moment 
behavior seems established (T2G, P27). The dashed curve represents more doubtful 
regions. [Adapted from Townes, Foley, and Low (T26).] 

even-JV nuclides, Q would be expected to be positive for Z slightly less 
than a closed shell of protons. As Z increases and passes just beyond a 
magic number, Q would change sign and become negative. This behavior 
is clearly displayed by 4 9ln 11B (Q = +1-16 X 10~ 24 cm 2 ) and siSb 121 
(Q = 0.3 X 10~ 24 cm 2 ) as Z passes through the well-established closed 
shell of 50 protons. 

For even-Z odd-JV nuclides we can expect a systematic variation of Q 



5] Nuclear Moments, Parity, and Statistics 173 

with N only if the mean spatial distribution of the protons is somehow 
correlated with that of the uncharged neutrons. Empirically, the corre- 
lation of Q with odd-jV exists and is similar to the variation of Q with a 
corresponding number of odd protons. This suggests that the strong 
attractive forces between protons and neutrons are such that a distortion 
of the neutron distribution produces a similar distortion in the proton 
distribution. 

Q has been measured for only a few even-Z odd-# nuclides. Among 
these, i 6 S 3B (7 = 1; Q = +0.06 X 10~ 24 cm 2 ; N = 19) is a clear example 
of a positive quadrupole moment just before the closing of a neutron 
shell at N = 20. The case of 8 O 17 (/ = J; Q = -0.005 X 10~ 24 cm 2 ; 
N = Q) illustrates the negative Q observed just after the closing of a 
neutron shell. This case is especially interesting because its even-Z 
proton configuration is the closed shell Z = 8. Its core is the doubly 
closed-shell configuration Z = 8, N = 8, and so it should have spherical 
symmetry and Q 0. Even so, the odd neutron is able to distort this 
into a slightly flattened distribution of charge. The absolute magnitude 
of this negative quadrupole moment is small, however. 

This simple model leads to the following conclusions (T26) : 

1. For odd-Z even-JV nuclides, the quadrupole moment is primarily 
dependent on the number of protons. Q is always positive immediately 
before, and always negative immediately after, a proton shell is filled. 

2. For even-Z odd-JV nuclides, the sign of Q is determined by the 
number of neutrons, but the absolute magnitude of Q depends on the 
number of protons. The quadrupole moments behave in sign as though 
the neutrons were positively charged. 

3. For odd-Z odd-.V nuclides, estimation of Q is considerably more 
complex and depends in part on how the mechanical moments of the odd 
proton and odd neutron add. If these moments are essentially parallel, 
Q should be of the same sign and approximately the same magnitude 
as for a similar odd-Z even-N iiuclide (examples: B 10 , N 14 , Lu 176 ). If 
the mechanical moments of the odd proton and odd neutron are not 
essentially parallel, the magnitude of Q should be considerably reduced 
(examples: Li e , Cl 36 ). 

4. For even-Z even-JV nuclides, 7 = and hence Q = because 
/ < 1. 

The quadrupole moments for a number of nuclides are plotted in 
Fig. 5.4, where the sequence of positive and negative values of Q and 
the other features just discussed can be visualized. The apparent dis- 
tortion of the core, between closed shells, has been discussed on the 
collective model by Bohr and Mottelson (B89) and by Hill and Wheeler 
(H53). 

Problems 

1. What is the numerical value of the classical electric quadrupole moment 
of a nucleus having a finite angular momentum and containing one proton at the 
nuclear equator in addition to a spherically symmetric distribution of charge, if 
the nuclear radius is 4 X 10~" cm? What would be the quantum-mechanical 



174 The Atomic Nucleus [CH. 4 

value for this same quadrupole moment? Ans.: 0.08 X 10~ 24 cm 2 /electron; 
-0.16 X 10-"cm. 

2. Show that the quadrupole moment Q of a uniformly charged ellipsoid is 
(2Z/5K& 2 - a 2 ) as given by Eq. (5.8). 

3. Show that the ellipticity parameter 17 is actually related to the ratio of the 
semiaxes of the ellipsoid. &/a, as 



4. Show that, in the model used for Eq. (5.7), the quadrupole moment Q* 
along the body axis /* of a nucleus is related to the usual quadrupole moment Q 
of the quantum state m/ = 7 by 



27 - 1 

Compute Q* and the corresponding elliptu'ity y* for a few nuclides from Table 5.1 . 

5. If the "actual" quadrupole moment Q* about the body axis is positive, is 
there any value of 7 which permits the measurable quadrupole moment Q to be 
negative? 

If Q is positive for a particular nuclide, whose angular-momentum quantum 
number is 7, what are the magnetic quantum states ?rij for which the effective 
quadrupole moment is negative? 

6. (a) Show that the geometrical target area of a nucleus whose ellipticity 
about its body axis 7* is given by 17* = 2(6 a)/ (ft + a) is 

ff = vab = IT I- -j (1 + ^rj* + ) perpendicular to body axis 
T|| = va 2 = TT I ) (1 -jfo* + ) parallel to body axis 

\ 47T/3 / 

(6) Look up data on the fast-neutron cross section of nuclei which have large 
quadrupole moments, such as some of the rare earths, and determine whether 
their ellipticities are detectable as anomalies in the progression of fast-neutron 
cross section with mass number. 

6. Parity 

The property called "parity" is a classification of wave functions 
into two groups, those of "even parity" and those of "odd parity." 
This classification is especially useful for quantum-mechanical systems 
containing two or more particles, such as a nucleus. 

The parity of an isolated system is a constant of its motion and cannot 
be changed by any internal processes. Only if radiation or a particle 
enters or leaves the system, and hence the system is no longer isolated, 
can its parity change. We therefore refer to-the "conservation of parity" 
in the same sense and with the same rigor as the conservation of charge 
and of angular momentum. Like the angular-momentum quantum 
number 7, the parity of a nucleus i s a "good" quantum number. 

a. Definition of Parity. Wave mechanics gives a satisfactory descrip- 
tion of many nuclear, atomic, and molecular systems. For very large 



6] Nuclear Moments, Parity, and Rlalislics 175 

quantum numbers, the wave mechanics goes over into the (jqualioiiK of 
ordinary mechanics. Thus we can visualize ordinary mechanical ana- 
logues of the nuclear angular momentum, the intrinsic .spin of .single 
particles, and some other mechanical properties of nuclear systems. The 
so-called parity of a system of elementary particles, such as a nucleus, 
atom, or molecule, is a fundamental property of the motion according 
to the wave-mechanical description, but it has no :-implr analogy in 
ordinary mechanics. 

As was noted in Chap. 2, Eq. (5.43), the physical description of tlie 
system, particularly the probability of finding the particle M! the position 
and with the spin orientation given by the coordinates (.r,yy,3,.s), is pro- 
portional to the square of the absolute value of th ri wavo function, 
|^| 2 = ijnj,* _ ff* 9 where V* and ^* are the complex conjugates of V 
and ^. 

Now the probability of finding a particle or system of particles can- 
not depend, for example, on whether we are right-handed or left-handed, 
and hence W* must be the same in coordinates (x,i/,z,s) n,s in the. coordi- 
nates ( Xj y t z,s). This transformation of coordinates is equivalent 
to reflecting the particle at the origin in the (x,y*z) y isle in, nn operation 
which must either leave the wave function unchanged, or only change 
its sign, so that its squared absolute value remains unaltered in either case. 

To a good approximation, $ is the product of a function depending on 
space coordinates and a function depending on spin ori en tntion. When 
reflection of the particle at the origin does not change the sign of the 
spatial part of $, the motion of the particle is said to have even parity. 
When reflection changes the sign of the spatial part of \l/, the mot-ion of the 
particle is said to have odd parity. Thus 

iK x, y, z,s) = f(x,y,z,8) represents even parity . 

iK-z, y,-z,s) = f(x,y,z,8) represents odd parity ^ } ' 

It can be shown that the spatial part of ^, on reflection of the particle, 
does not change sign if the angular-momentum quantum number I is 
even, but it does change sign if / is odd. Hence, fur a particle with an 
even value of Z, the motion has even parity and, with an odd value of /, 
the motion has odd parity. 

For a system of particles, the wave function becomes approximately 
the product of the wave functions for the several particles, ^ = 1/^2^3 
. . . , or a linear combination of such products. Hence tlv- parity of a 
system of particles such as a nucleus depends on the parity of the motion 
of its individual particles. 

Visualizing reflection of the system as the successive reflection of paoh 
individual particle, one at a time, we conclude that a system will have 
even parity when the arithmetic sum of the individual numerical values 
fc for all its particles 2Z,- is even, and odd parity when I/ is odd. A system 
containing an even number of odd-parity particles, and any number of 
even-parity particles, will have even parity. A system with an odd 
number of odd-parity particles, and any number of even-parity particles, 
have odd parity. The symbol (+) is often used as a superscript on 



)76 The Atomic Nucleus [CH. 4 

7, for example, / = 3+, to denote even parity, and the symbol ( ) to 
denote odd parity. The intrinsic parity of the electron is defined arbi- 
trarily as even. From the properties of simple systems it has been found 
experimentally that the intrinsic parity of the proton, neutron, and 
neutrino is the same as that of the electron; hence, it is even. In con- 
trast, the TT meson is found to have odd intrinsic parity. 

All that has been said thus far applies to the nonrelativistic case, i.e., 
to heavy particles, such as protons or neutrons at energies below about 
50 Mev, and to electrons of energy less than about 0.04 Mev. The rela- 
tivistic wave mechanics has been developed for electrons but not for 
heavy particles. For electrons having an energy greater than about 
0.04 Mev the Dirac relativistic electron theory must be used. The wave 
function describing the Dirac electron is a four-component vector in 
phase space, which reduces to the simple wave function ^ for the non- 
relativistic case. For the relativistic Dirac electron, the mathematical 
concept of parity is retained, but the I values no longer determine the 
parity in the simple fashion discussed above for the nonrelativistic case. 

b. Change of Parity. Parity is conserved in interactions between 
nucleons. The parity of a system (e.g., a nucleus) can only be changed 
by the capture of photons or particles having odd total parity (intrinsic 
parity plus parity of motion with respect to the initial system) or by the 
emission of photons or particles having odd total parity. 

The selection rules for all nuclear transitions involve a statement of 
whether or not the nucleus changes parity as a result of the transition. 
Thus the notation "yes" denotes that the nuclear parity changes (from 
even to odd or from odd to even), hence that the emitted or absorbed 
particles or quanta have odd total parity. For example, an emitted a 
ray which has 1 = 1 with respect to the emitting nucleus will have odd 
parity and can be emitted only if the nuclear parity changes. Similarly, 
the selection rule "no" means that the initial and final nuclei have the 
same parity (both even or both odd). An emitted a ray which has I = 2 
with respect to the emitting nucleus will have even parity and can be 
emitted only if the nuclear parity does not change. 

c. Determination of the Parity of Nuclear Levels. The quantum- 
mechanical parity classification to which a given nuclear level belongs 
cannot be "measured" with the directness that an experimentalist feels 
in the measurement of such classical properties as charge, mass, angular 
momentum, and kinetic energy. Nevertheless, every nuclear level is 
representable only as a stationary state of a quantum-mechanical system. 
Therefore, one of the most important parameters of each level is its 
parity. 

The parity of a given nuclear level is determined by the odd or even 
character of Z! t . The evaluation of Z2 t is reliable in many instances only 
to the extent that the nuclear model employed is valid in its designation 
of the orbital quantum numbers Z,- for the individual nucleons not occur- 
ring in closed shells. 

We have seen that the core-and-single-particle nuclear model gives 
reasonable agreement with the Schmidt groups of p and / for the ground 



7] Nuclear Moments, Parity, and Statistics 177 

levels of nuclei. A very high percentage of successful predictions of I 
for ground levels and for excited levels has given much support since 
1949 to the extreme single-particle model, with jj coupling between 
nucleons. This evidence will be examined in Chap. 11. Here we may 
note that the single-particle model forms a reasonable basis for the deter- 
mination of I for an odd-nucleon and predicts 2Z, = for the core, as 
discussed earlier in connection with the Schmidt limits. On this basis, 
the parity classification of many nuclear levels can be made with reason- 
able assurance. 

Parity of Ground Levels. When / and \L have been measured, the 
Schmidt group classification determines I of the odd nucleon in odd- A 
nuclides. The interesting nuclide seBaVi 7 , with one neutron lacking from 
a closed shell of 82 neutrons, is found in its ground level to have 7 = 1, 
and p. = +0.93 nuclear magneton. Then, from Fig. 4.2, the ground level 
of this nuclide belongs to the 7 = j = I s Schmidt group. Then the 
81st neutron is taken to be in an I = ? + v = 2 or a tl d" orbit. With 
2Zt = for the core, this makes ZZ t - = 2 for the entire nucleus. The 
ground level therefore has even parity and is denoted "d s , even," or, more 
commonly, "d^." 

Parity of Excited Levels. The selection rules for every type of nuclear 
reaction and transition involve parity as well as angular-momentum 
changes. Parity and angular momentum / are two "good quantunr 
numbers" for all nuclear interactions; both are rigorously conserved. 
An explicit example of the determination of / and parity for excited 
levels in Ba 137 will be discussed in Chap. 6, Sec. 7. 

7. The Statistics of Nuclear Particles 

We have seen that the wave-mechanical concept of parity arises from 
considerations of the reflection properties of the spatial part of solutions 
^ of the wave equation. Another important property of nuclei, statistics, 
arises from considerations of the symmetry properties of wave functions, 
i.e., the effect on the wave function ^ of the interchange of all the coordi- 
nates of two identical particles. The wave functions which are solutions 
of Schrodinger's equation for a system of two or more identical particles 
may be divided into two symmetry classes, "symmetric" and "anti- 
symmetric." Transitions between these two classes are completely for- 
bidden. The symmetry class of a wave function docs not change with 
time. It is a constant of the motion. The symmetry class to which a 
particle belongs is synonymous with its statistics. The statistics, in 
turn, has a profound effect on the physical behavior of collections of the 
identical particles. Every particle in nature must obey either one of the 
two types of statistics, Fermi-Dirac (antisymmetric) or Einstein-Bose 
(symmetric), and these two have the following principal characteristics. 

a. Fermi-Dirac Statistics. The wave function of a system obeying 
Fermi-Dirac statistics is antisymmetric in the coordinates (three spatial 
and one spin) of the particles. This means that if all the coordinates 
of any pair of identical particles are interchanged in the wave function, 



178 The Atomic Nucleus [GH. 4 

the new wave function representing this new system will be identical 
with the original except for a change in sign. The probability density 
W* is, of course, unaltered. Thus if ^(zi, . . . ,z,-, . . . ,xy, . . . ,z n ) 
is the wave function of a system of n identical particles obeying Fermi- 
Dirac statistics, and z, stands for all the coordinates of the particle i, 
the new wave function $(x\ } . . . ,Xj, . . . ,z t , . . . ,z n ) resulting from 
interchanging the particles i and j will be given by 

. . . ,Zi, . . . ,x jt . . . ,z n ) (7.1) 

It can be shown (p. 491 of B87) that antisymmetry of the wave func- 
tions restricts the number of particles per quantum state to one. This 
is equivalent to saying that the Pauli exclusion principle holds for Fermi- 
Dirac particles, since two particles may not occupy the same quantum 
state. 

Inferences from experiments show that nucleons (both protons and 
neutrons), electrons (both + and ), /* mesons, and neutrinos are 
described only by antisymmetric wave functions and therefore have 
Fermi-Dirac statistics. Direct experiments on the relative intensity of 
the successive lines from the rotational levels of diatomic homonuclear 
molecules show that H 1 , Li 7 , F 19 , Na 23 , P 31 , C1 3B obey the Fermi-Dirac 
statistics. Generalizing these observations with the aid of Sec. c below, 
we can expect that all nuclei of odd mass number have Fermi-Dirac statistics. 

b. Einstein-Bose Statistics. A system whose wave function is sym- 
metric is said to follow Einstein-Bose statistics. Interchange of two 
identical Einstein-Bose particles leaves the wave function for their system 
unaltered. In the notation of Eq. (7.1), this condition is expressed 
analytically as 

. . . ft, . . . ,x n ) 

= iKzi, . . . &, . . . ,xj, . . . ,z n ) (7.2) 

Einstein-Bose particles do not follow the Pauli exclusion principle. 
Two or more such particles may be in the same quantum state; in fact, they 
may be said to prefer joint occupancy of identical position and spin 
coordinates. 

It is known from collision experiments that photons and a particles 
obey Einstein-Bose statistics. From diatomic band spectra, H 2 , He 4 , 
C 12 , N 14 , O lfl , S 32 are known to obey Einstein-Bose statistics. Generaliz- 
ing, we can expect that photons and all nuclei of even mass number have 
Einstein-Bose statistics. The IT meson, which is associated with the 
binding forces between nucleons, is an Einstein-Bose particle. 

c. Statistics, Mass Number, and Angular Momentum. The two gen- 
eralizations regarding the statistics of nuclei of odd and even mass num- 
ber are easily demonstrated, provided that all nuclear constituent par- 
ticles have Fermi-Dirac statistics. Consider two identical nuclei located 
near points a and b and each composed of Z protons and N neutrons. 
(Reasons will be summarized in Chap. 8 for believing that neutrons and 
protons are the only constituents of atomic nuclei.) The wave function 



7] Nuclear Moments, Parity, and Statistics 179 

describing this system of two nuclei will include the coordinates of each 
of these 2(Z + N) particles. We can conceptually interchange the posi- 
tion of the two nuclei by individually exchanging the identical particle 
constituents between the two nuclei until all have been exchanged. Each 
such individual exchange of a particle from nucleus a with its twin from 
nucleus b will simply change the sign of the wave function. If the total 
number of particles (Z + N) in each nucleus is odd, the complete inter- 
change of the two nuclei through this step-by-step process will result only 
in changing the sign of the wave function. Hence any nucleus which 
contains an odd number of constituent particles will have Fermi-Dirac 
statistics. If protons and neutrons are the only constituent particles in 
nuclei, such a nucleus must also have an odd mass number. In an exactly 
similar manner, nuclei of even mass number, containing an even number 
of nucleons, will provide a step-by-step exchange having an even number 
of stages. Because interchange of two such nuclei leaves the wave func- 
tion unaltered, these nuclei must have Einstein-Bose statistics. 

That the neutron is an elementary particle obeying Fermi-Dirac 
statistics is seen most directly from the experimental fact (band spectra) 
that the deuteron obeys P]in stein -Rose statistics and consists only of one 
proton and one neutron. Because the proton has Fermi-Dirac statistics, 
the neutron must also. 

Nucleons obey Fermi-Dirac statistics and also have a spin of i- 
Pauli (P9) has shown, from the relativistically invariant wave equation, 
that elementary particles -with any odd half -integer spin (s = i, , . . .) 
must necessarily obey the Fermi-Dirac statistics, and further that ele- 
mentary particles with any arbitrary integral spin (s = 0, 1, 2, . . .) 
must obey the Einstein-Bose statistics. This is a far-reaching, funda- 
mental generalization. 

Due to the conservation (vectorially) of angular momentum, all nuclei 
having an odd number of nucleons also have an odd half-integer total 
angular momentum (7, 7, ?, . . .). Similarly, all nuclei having an even 
number of nucleons have even half-integer total angular momentum 
(0, 1, 2, 3, . . .). In tabular form we have, then, 



Mass number 


Angular 
momentum 


Statistics 


Odd . ... 


/ - i, i I. - 


Fermi-Dirac 


Even 


/ - 0, 1, 2, ... 


Einstein-Bose 









d. ft Rays and Atomic Electrons Are Identical Particles. An ingen- 
ious and definitive application of statistics, through the Pauli exclusion 
principle, has been carried out by Goldhaber (G26) in order to prove the 
complete identity of the electrons arising in ft decay and ordinary atomic 
electrons. If ft rays differed in any property, such as spin, from atomic 
electrons, then the Pauli principle should permit the capture of slow ft 
rays into bound atomic states (K, L, M , . . . shells) even though the 
corresponding states are filled with atomic electrons. Goldhaber has 



180 The Atomic Nucleus [GH. 4 

shown that no anomalous X rays are emitted when the soft rays of C u 
are absorbed in lead. Hence these rays obey the Pauli principle, have 
spin i, and must be identical with atomic electrons. 

e. Fermions and Bosons. Time often tends to compress the language 
in which physical ideas are expressed. Fermi-Dirac statistics is now 
frequently called simply Fermi statistics, and Einstein-Bose statistics is 
shortened to Bose statistics. Time has also brought spontaneous trans- 
formations in the corresponding adjectives, as denned by the identities 

(Fermi-Dirac particle) * (Fermi particle) fermion 
(Einstein-Bose particle) > (Bose particle) > boson 

Thus a ic meson (s = 0) is a type of boson, while a p meson (s = i) is a 
type of fermion. Nucleons, electrons, and neutrinos are fermions. 

Problem 

A variety of experimental evidence has shown that electrons (e^,0^) and 
nucleons (p,n) have Fermi-Dirac statistics, while photons (7) have Einstein- 
Bose statistics. With this information, determine the spin and statistics for the 
neutrino (v), antineutrino (v), M meson (/*), and TT meson (w), from the following 
observed processes: 

(a) ft decay of neutron, n * p + P~~ + v. (Half -period agrees with theory 
of allowed transitions on Gamow-Teller selection rules; hence 0~ and v probably 
emitted with parallel spins.) 

(6) Decay of p mesons, p*^-* e* + v + v. 

(c) Decay of charged TT mesons, T + * p + 4* w, and ir~ > n~ + v. 

(d) Decay of neutral ir mesons, ir > 2% 



CHAPTER 5 



Atomic and Molecular Effects of Nuclear Moments, 
Parity, and Statistics 



The influence of the added energy due to the nuclear moments and 
the influence of the statistics of the nucleus are felt in a variety of molecu- 
lar and atomic phenomena. These effects give rise to a variety of 
experimental methods for the determination of the absolute values of 
nuclear ground-level moments. 

1. Extranuclear Effects of Nuclear Angular Momentum and 
Statistics 

a. Number of Hyperfine-structure Components. The main features 
of atomic spectra, including the so-called fine structure (which is associ- 
ated with electron spin), have been described adequately in terms of the 
energy states of atomic electrons in a central electrostatic field, of strictly 
coulomb nature, due to the charge on the small massive atomic nucleus. 
When this nucleus is given the added property of quantized angular 
momentum /, we have seen that a magnetic dipole moment p will also 
be associated with it, because of the motion of internal electric charges 
in the nucleus. The interaction of this small nuclear magnetic moment 
with the electrons not appearing in closed shells, particularly with a 
penetrating s electron in the group of valence electrons, gives rise to a 
multiplicity of slightly separated energy states for this electron. This 
closely spaced group of energy levels of the atom is called a hyperfine- 
structure multiplet. The energy separations in this multiplet can be 
measured in a number of cases by molecular-beam magnetic-resonance 
methods, which we shall discuss later. When such a penetrating electron 
undergoes a transition to a state having a much smaller coupling with the 
nuclear moment, the hyperfine structure (hfs) of the resulting optical 
transition may be particularly clear and readily resolved by means of a 
Fabry-P6rot interferometer. 

The magnitude of the hyperfine-structure separations, and hence the 
possibility of observing them, depends on the magnitude of p. 

The number of hyperfine states depends only on 7 and on the elec- 
tronic angular-momentum quantum number J. J and 7 couple, in a 
manner which is completely analogous to LS coupling in atoms, to pro- 

181 



182 



The Atomic Nucleus 



[GH. 5 



<lujf the total angular-momentum quantum number F. Thus F can 
tuke on the series of integer-spaced values from 7 + J to |7 J\ m The 
total number of possible values of F is the multiplicity, or number, of 
hypM-(ine hi at us und is (27 + 1) when / < J, or (2J + 1) when I > J. 
Tlu* oiirresjxi.'iding vector diagram is shown in Fig. 1.1. 

We M<r iliiit the number of hyperfine components in an atomic term 
is generally different for different terms in the same atom. J = is 



/'\ 




c ::>/ 

Fig. 1.1 Vivfw- diagram Ulustrating the coupling of the resultant electronic angular 
mumr'.'.tniu J* --- \ J(J ~\- i) with the nuclear angular momentum /* V7(/ + 1) 
to furru 1 1 ii' ffi*;i! angular momentum f* -\/F(F + 1), about which both /* and /* 
prectss v V.'.iJ, . Y!u- li^ure is drawn for the special case J T, / 1, which gives a 
byperimc jiiiiUijik-l rout. lining (he three components F ^, f, . i 

always sin^K,, but if / is at least as large as / then the nuclear moment 
imirjUL'ly i Id ermines the number of hyperfine levels as 

hyper/me multiplicity = (27 + 1) if I < J (1.1) 

b. Relative Separation of Hyperfine Levels. The magnetic field pro 
ducofl by ilio atuniic ok'ctrons i.s of the order of 10* to 10 7 gauss at the 
position ui tlin nucleus in the alkali atoms Li, Na, Rb, Cs, which have 
one wl'-iii'o rlK-inm. Although nuclei have very small magnetic dipole 
is. thr nia^noMc interaction energy between them and such huge 
IM'P- i-Nou^h to be measurable easily and constitutes the hyper- 
Lftiu-p splitting yf atomic levels. 



r>':;v of ti dipole in a magnetic field we can write 



\V 



cos 



(1.2) 



'c IV 



(7V*) 
In i 
have f uj. 
become* 



lu.i^nruc interaction energy 

nuchiur mugnetic dipole moment! Eq. (2.10), Chap. 4 
nia^iiC'lic fiold, parallel to J*, produced by atomic electrons 
arij^if; but ween /* and /*, Fig. 1.1 

//./* is proportional to J*. From Eq. (2.15) of Chap. 4, we 
(//u//*, where g ^ p/I is the nuclear g factor. Then Eq. (1 .2) 



W = aI*J* COB (/ V*) 



(1.3) 



1] Atomic and Molecular Effects of Nuclear Moments 183 

where a is the so-called interval factor of hyperfine structure. The inter- 
val factor is proportional to g and involves constants of the J state of the 
atom which we will evaluate later, Eq. (1.10). 

We see from Fig. 1.1 and the cosine law of trigonometry that 



cos (7V*) 



_ J*2 



27V* 



(1-4) 



Substituting in Eq. (1,3), the nuclear magnetic interaction energy of 




Fig. 1.2 Graphical illusl ration of the interval rule of hyperfine separations, baaed on 
Eq. (1.3), for the special case 1 = 1, / = -. The atomic level J = fis split, because 
of the nuclear angular momentum / and an associated /u (assumed positive here), into 
(21 H- 1; hyperfine levels. These are characterized hy the total-angular-momeritum 
quantum numbers F = , |, ^ and are displaced in energy by the amounts W given by 
Eq. (1.6). The relative separations AW are (5a/2): (3a/2) = 5:3, as given by Eq. 
(1 7). (Adapted from White, p. 355 of W39.) 

ordinary hyperfine structure nan be rewritten as 



? (F** - 7* 2 - .7* 2 ) 



+ 1) - 7(7 + 1) - J(J + 1)1 



(1.5) 



which is called the interval rule of hyperfine structure. Now, as F takes 
on its allowed values of (7 + .7), (7 + 7-1), (7 + J - 2), . . . , 
1 7 7 [, the corresponding values of W become 



for F = (I + J), 
for F = (I + J - 1), 
for F =(/ + /- 2), 



W^ = aU 

W t = a[U - (7 + J)] 

W 3 = a[IJ - (I + J) - (I + J - 1)] 

The energy spacing between successive hyperfine levels is then 

= Wt - Wi = a(I + .7) 

= FT, - W 4 = 0(7 + 7-2) 



(1.6) 



(1.7) 



184 



The Atomic Nucleus 



fen. 5 



Thus the relative separation between successive hyperfine levels is propor- 
tional to the larger of the values of F for the two levels. 

For example, if the largest value of F happened to be, say, 5 (such as 
for 7 = f, ./ = ), then the relative separations of the hyperfine levels 
would be in the ratio 5:4:3:2:1. In optical transitions to other atomic 
levels having a negligible hyperfine splitting, the relative separation of 
successive lines in the hypertine-structure spectrum displays this same 
set of ratios, thus giving rise to the familiar "flag" pattern of optical 
hyperfine spcctroscopy. 

This interval ride of hyperfine structure is illustrated in Fig. 1.2. In 
many cases, an independent determination of / can be made from 





Fig. 1.3 Vector diagram of the Zee- 
man effect in hyper fine structure. 
In a very weak external magnetic 
field, 7* arid /* remain coupled tc 
form F*, about which both preccas. 
F* precesses about the direction of 
the external magnetic field H and 
has (2F + 1) magnetic substates 
m r = F, (F - 1;, (F - 2), . . . , 
-F. 



Fig. 1.4 Vector diagram of the 
Paschen-Back effect in hyperfine 
structure. In a weak external mag- 
netic field //n, /* and J* become 
decoupled, and each preccsses inde- 
pendently about -//n, with independ- 
ent magnetic quantum numbers, m/ 
and tnj. 



measurements of .^e relative separation of three or more hyperfine 
levels, ./ being inferred from other evidence. The absolute separations 
depend upon the interval factor a and hence are proportional to the 
nuclear g factor g = p/I. 

c. Zeeman Effect in Hyperfine Structure. If an external magnetic 
field HQ is now applied to the atom, the magnetic energy given by Eq. 
(1.2) changes, because the total magnetic field at the nucleus is now due 
to both the internal atomic field 7//* and the applied field 77 . A variety 
of effects can occur, depending on the magnitude of HQ. 

If HQ is very small, then 7* and J* will remain coupled to form F*, 
while F* will precess about the direction of 77 , as shown in Fig. 1.3. 
Then F* can take up any of a series of possible orientations such that its 
projection in the direction of the external field is given by its magnetic 



Atomic and Molecular Effects of Nuclear Moments 



185 



quantum number m F . Thus each hyperfine level F is broken up into 
(2F + 1) magnetic substates, with magnetic; quantum numbers m F = F, 
(F - 1), (F - 2), . . . , -F. This is the Zecman effect of hyperfine 
structure. 

Paschen-Back Effect. As the external field 77 is increased, the fre- 
quency of precession of F* about 77 increases. Because of the small 
absolute value of the nuclear magnetic dipole moment, the coupling 
between /* arid J* is weak, and the frequency of their precession about 
F* is not large. At sufficiently large external fields, the frequency of 
precession of F* about 77 exceeds that of /* and J* about F*. Then 7* 
and 7* become decoupled, and each becomes space-quantized inde- 
pendently in the direction 7/o, with independent magnetic quantum 




Hyperfine Zeeman Pashen-Back 

ground state hyperfine hyperfine 

multiple! structure structure 

zero field very weak field weak field 

Fig. 1.6 Spcctroscopic diagram of the magnetic sublevcls of a 2 j atomic slate, due 
to a nuclear angular momentum / 1 with an associated positive magnetic dipole. 
moment. 

numbers mi and mj. This state of affairs is illustrated in Fig. 1.4 and is 
usually called the Paschen-Back effect of hyperfine structure. 

Each level of a given mj (which corresponds to the ordinary Zeeman 
effect of fine-structure spectra) is further split into a number of substates 
corresponding to the (27 + 1) values of m h that is, T?Z/ = 7, 7 1, 
I 2, . . . , 7. This number of substatcs (27 + 1) is the same for 
all terms in an atom and thus constitutes a very direct method for deter- 
mining the nuclear angular momentum, merely by counting up the num- 
ber of line components. This elegant method was first used by Back and 
Goudsmit for determining 7 = J for bismuth. 

The shifts in the energy of the magnetic sublevcls from \ery weak 
fields to weak fields are illustrated in Fig. 1.5 as they occur in optical 
spectroscopy. 

Transitions between the magnetic sublevels follow the selection rules: 
AF = 0, 1 ; Am/- = 0, 1 in the Zeeman hyperfine-structure region, 



186 The Atomic Nucleus [CH. 5 

and Am/ = 0, 1, or Araj = 0, 1, in the Paschen-Back hyperfine- 
structure region. 

Breit-Rdbi Formula. The region of intermediate fields (order of 1 to 
1,000 gauss) has become of particular importance in recent years because 
the atomic-beam magnetic-resonance method (K12) makes it possible to 
measure the energy separation of the magnetic sublevels in the ground 
state of many atoms. For atoms whose electronic angular momentum is 
/ = i, the energy behavior in the Zeeman, Paschen-Back, and inter- 
mediate domains of hyperftne structure is given in closed form by a 
formula due to Breit and Rabi (B116, T8), which can be written in the 
form 



Q> | ri I ** rr I 1 I Tfftjr I 9 1 /I 0\ 

- + m r g^fl - ( 1 + OT , . J + s 2 ) (1.8) 



2 V 
x 2 } 



where a = hyperfine-strutrture interval factor, Eq. (1.3) 
m f = magnetic total quantum number 

g = n/I = nuclear g factor, Eq. (2.15), Chap. 4 
MM = ch/4irl\fc = 5.05 X 10~ 24 erg/gauss = nuclear magneton 
H = magnetic field intensity, gauss 
&W = h Ay = a(l + *) = hyperfine separation, Eqs. (1.7) and 

(1.11) 

x = (gw - giL M )H/*W ~ ZuJf/AW 
gj = Land^ atomic g factor = +2(1 + a/2ir) Bohr magnetons for 

2 Si state 
up = eh/4trm^c = n M (M/mo) = 9.27 X 10~ 21 erg/gauss = Bohr 

magneton 

The + is to be used f or F = I + J = I + i, and the - for F = I - i. 
The ordinary hyperfine splitting, at zero field, is given b^ the third term. 
With H = 6 and x = 

W t +i - W^ = Al^ = a(I + i) (1.9) 

in agreement with Eq. (1.7). Expansion of the square-root term in Eq. 
(1.8) will yield a term linear in x, which defines the Zeeman splitting and 
is the dominant field-dependent term when x 1. The higher terms in 
x, which become significant in "weak fields," x < 1, define the Paschen- 
Back effect. If these higher terms are neglected in "very weak fields, 11 
x < 1, the Breit-Rabi formula then reduces to one for the Zeeman 
splitting of hyperfine structure. 

The second term in Eq. (1.8), mFgniiH, represents a portion of the 
direct interaction energy between the nuclear magnetic dipole moment 
^ = gl and the external field. Because of the small value of the nuclear 
magneton /ijf, this energy is of the order of 1,000 times smaller than the 
Zeeman splitting. However, the high accuracy now attainable with the 
atomic-beam magnetic-resonance method permits Eq. (1.8) to be used 
to determine the hyperfine-structure separation ATT of the ground state 
of certain atoms and both the magnitude and sign of the nuclear mag- 
netic dipole moment (D9). 



1] 



Atomic and Molecular Effects of Nuclear Moments 



187 



Note that x is a dimension less parameter proportional to H . It is the 
ratio of two energies, namely, the magnetic energy of the whole atom in 
the external field and the zero-field hyperfine-structure splitting. Thus 
x <3C 1 defines magnetic splittings which are much less than the zero-field 
hyperfine structure?. But this is just what we mean by the Zeeman 
domain. It is when the Zeeman splitting becomes of the same order of 
magnitude as the zero-field hyperfine structure (separation of levels of 
different F) that the onset of the Paschen-Back effect occurs. A "very 



Zeeman Intermediate Paschen 
region region Back reg. 





Fig. 1.6 The variation with magnetic field of the energy levels which make up the 
ground level "Sj of an atom with J = and / = (as in hydrogen, at left) or / 1 
(as in deuterium, at right). The nuclear magnetic dipole moment /i has been taken 
here as positive. The zero-field, very-weak-field (Zeeman hyperfine structure), and 
weak-field (Paschen-Back hyperfine structure) quantum numbers are marked on each 
curve. The curved show quantitatively the continuous variation of W with H, Eq. 
(1.8). The curves for 7 = 1 represent the same physical situation as shown in the 
conventional spectroscopic diagram of Fig. 1.5. 

weak field 11 is thus defined as one for which x <K 1, yielding the Zeeman 
levels. We see that the definition of a "very weak field" varies from 
atom to atom because of the different values of AW. 

Figure 1.6 is a representative plot of the variation of the energy 
of the magnetic sublevels with applied field H } as given by Eq. (1.8). 

Interval Factor of Hyperfine Structure. We noted in Eq. (1.3) that 
the hyperfine-structure interval factor a is proportional to g = ft/I and 
to the magnetic field which the atomic electrons produce at the position 
of the nucleus. It can be shown that this field is proportional to the 



188 The Atomic Nucleus [CH. 5 

average value of 1/r- 3 , where r represents the electron's radial distance 
from an assumed " point" dipole located at the center of the nucleus. 
It is therefore very sensitive to the wave functions chosen to represent 
the probability density of an electron in the vicinity of the nucleus. 
Fermi (F31) and others (G38, C52) have shown that for a single s elec- 
tron ( 2 Sj state, / = i, as in the hydrogens and the alkali metals) the inter- 
val factor is 

- ! g^w -J-T O- 10 ) 

tj 71 CLjf 

where ^ 7l (0) = wave function at. zero radius for x electron with principal 

quantum number n 
a H = AV^V-wo = 0.529 X 10~ 8 cm = radius of first Bohr 

orbit for hydrogen 

and the other symbols have the same meaning as in Eq. (1.8). Then 
the energy difference between the F = I + ? and / \ hyperfine levels 
is, from Eq.-(1.7), 



= a (l 



Air = h A, = a l + = (27 + 1 W*hMO)|> (1.11) 



where AV is called the by per fine-structure separation. 

A number of refined theoretical evaluations of ^ ri (0) have been made, 
including, among other correction terms (K23, T8), the effect of the 
decrease of electron probability density at the center of the nucleus 
because of the finite volume of the nucleus and the assumed uniform 
distribution of charge in the nucleus (C52). With these correction terms 
included, the theoretical values of the hyperfi Tie-structure interval factor 
a mid of the hyperfi lie-structure separation Av are in excellent agreement 
with the very accurate experimental values of AJ>, and independently of 
/i//, which are obtainable by atomic-beam magnetic-resonance methods 
(T8). 

Diarnagnctic Correction. Whenever AV and ,u// are measured by a 
magnetic-resonance method, the added magnetic field intensity at the 
nucleus is slightly less than the externally applied field H as measured 
in the laboratory. This redaction is due to the diamagnetism of the 
atomic electrons. The induced field which is produced at the position 
of the nucleus by the induced Larmor precession of the atomic electrons 
is proportional to the external applied field H. Therefore the correction 
cannot be evaluated experimentally. Theoretical evaluations have been 
made by Lamb and others (LI, D37, R3), using various degrees of 
approximations for the electron wave functions. The Fermi-Thomas 
atom model leads to the simple relationship (LI) 



" 3.19 X lO^Z*) (1.12) 

where the term in Z* is the ratio of the induced field at the nucleus to the 
external applied field. Hartree and Hartree-Fock wave functions have 
been used to obtain more accurate values for individual elements. The 
correction runs from 0.0018 per cent for atomic hydrogen to 1.16 per 



1] Atomic and Molecular Effects of Nuclear Moments 189 

cent for aranium. The corrected values of g and /i are always larger 
than the raw observed values. 

d. Relative Intensity of Hyperfine Lines. The probability of excit- 
ing each of the magnetic substates m p is assumed, with good experimental 
justification, to be the same, i.e., the several magnetic substates are 
said to have the same statistical weight. If the externally applied mag- 
netic field is reduced toward zero, the energy differences due to the 
magnetic interaction vanish. Therefore at zero magnetic field all the 
(2F +1) magnetic substates are superposed. Then, in the absence of 
an external field, each ordinary hyperfine state has a relative statistical 
weight of (2F + 1). 

In the hyperfine multiplet at zero field each state is characterized by 
a different statistical weight (2F +- .1). Hypcrfine-structure lines in 
optical spectroscopy originate from transitions between two hyperfine 
multiplets (with the additional selection rule AF = 0, 1 allowed; > 
forbidden), the statistical weight of each level being determined by its 
F value. Hence the relative intensity of the lines in a hyperfine spectrum 
depends on the angular momenta J and 7 and not on the nuclear mag- 
netic dipole moment. The relative-intensity relationships are analogous 
to those of fine-structure multiplets (p. 206 of W39). 

e. Alternating Intensity in Diatomic Molecular Band Spectra. The. 
relative intensity of each spectral line in the rotational band spectrum 
of a homonuclear diatomic molecule (for example, H'H 1 , C 12 C 12 , N 14 N 14 , 
O 16 O 16 , etc.) is determined by the statistical weight of the states involved 
in the transition. 

Consider a homonuclear diatomic molecule. If / is the total intrinsic- 
angular-momentum quantum number of each nucleus, then the total 
nuclear-angular-momentum quantum number T of the diatomic molecule 
can have any of the values T = 27, 21 1, , . . , 0. It can be shown 
that the values 27, 27 2, ... of T belong to one of the two types of 
rotational states (symmetric or antisymmetric in the space coordinates 
of the nuclei) and that the values 27 1, 27 3, ... belong to the 
other type. Each state with total nuclear angular momentum T con- 
sists of 2T + 1 magnetic substates which coincide in the absence of an 
external magnetic field. Each of these substates has an equal chance of 
occurrence, so that the frequency of occurrence, or statistical weight, of 
the T state is 2T + 1 times that of a state with T = 0. If the statistical 
weights 2T + 1 for all the 27, 27 - 2, . . . values of T are added and 
compared with the total statistical weights for all the 27 1, 27 3, 
. . . values of T, it is found that the sums are in the ratio (7 + l)/7. 

Now transitions between these two types of rotational states (sym- 
metric to antisymmetric or vice versa) are almost completely forbidden 
(the mean life for such a transition is of the order of months or years), 
and transitions between states of the same type (e.g., symmetric) can 
occur only when accompanied by an electronic transition. Hence, homo- 
nuclear diatomic molecules do not have any pure rotational (or rotation- 
vibrational) spectra. Alternate lines in the rotational fine structure of 
the electronic spectra arise from transitions between states belonging to 



190 The Atomic Nucleus [CH. 5 

one of the symmetry types, depending on the electronic states involved. 
For example, the first, third, fifth, . . . lines may be from the symmetric 
rotational states, and the second, fourth, sixth, . . . lines from the anti- 
symmetric states. Accordingly, successive lines have an intensity ratio 
of (7 + I)//. This is the ratio of symmetric lines to antisymmetric 
lines for nuclei obeying the Einstein-Bose statistics (for which / isO, 1,2, 
. . .). The ratio of the relative intensity of the symmetric lines to the 
antisymmetric lines for nuclei obeying the Fermi-Dirac statistics (for 
which / is*, i . . .) is //(/ + I)/ 

Thus, regardless of which type of statistics is obeyed by the nuclei 
in a homonuclear diatomic molecule, the average ratio of the intensity 
of the more intense to the less intense family of lines is always 

0-13) 



It is important to notice that the total nuclear angular momentum 
determines uniquely the relative intensity of successive lines in the rota- 
tional band spectrum of molecules composed of two identical atoms. 
Neither the pure rotational nor the rotation-vibrational bands are 
emitted, but the effect can be observed in either the electronic bands or 
the Raman spectra. A nuclear angular momentum of zero leads to an 
infinite intensity ratio for successive lines, i.e., alternate lines are missing. 

The nuclear angular momentum has been obtained from band- 
spectrum studies for a number of nuclei, including H 1 , H 2 , He 4 , Li 7 , C 12 , 
C' 13 , X 14 , X 15 , O 1B , F 1S , Xa 23 , P 31 , S 32 , Ci". A convenient review of the 
theory of alternating intensities of band spectra, together with illustrative 
data for F 19 , had been compiled by Brown and Elliott (B133). 

f. Specific Heat of Diatomic Gases. Two forms of the hydrogen 
molecule exist. In orthohydrogen the spins of the two protons are parallel, 
while in parahi/drogcn the proton spins are antiparallel. The statistical 
weighting of rotational states and recognition of the absence of transi- 
tions between the two types of hydrogen under ordinary conditions were 
necessary in order to explain the specific heat of hydrogen at very low 
temperatures (D24). A proton spin of \ accounts for the observed 
specific heat. 

The ortho and para forms exist for all diatomic molecules whose 
atoms do not have zero spin. The general considerations applied to 
hydrogen are also applicable to other molecules. 

Problems 

1. Expand the Breit-Rabi formula, for the Zeeinan separations in hyper- 
fine structure, into a power seiies in H, as far as quadratic tcrmw. Discuss the 
physical significance of each term. To the same approximation, obtain a for- 
mula for the energy of transition between mp F and m F = F + 1 in the 
F = 7 + / levels. Discuss the dependence of this transition energy on ju and / 
and especially whether it depends on the sign of /i. 

2. Calculate and plot the energies W/\AW\ vs. magnetic field (x ^ to 3) for 
the Zeeman effect of hyperfine structure for the case of negative nuclear magnetic 



2] Atomic and Molecular Effects of Nuclear Moments 191 

dipole moment /*. Take / = ^ = ^ and 1 . Compare the result with the 
curves of Fig. 1.6 for positive p, and state the general consequences of the sign 
of M- 

3. Calculate the hyperfine splitting Av expected for the ground state of the 
hydrogen atom, in units of (a) cycles per second, (6) cm" 1 , and (c) h AP in ev. 
Ans.: 1,420 megacycles/sec; 0.0473 cm' 1 ; 6 X 10~ 6 ev. 

4. Show that the "center of gravity" of a spectral line is unaltered by its 
hyperfine splitting in zero field, if each hyperfine-structure level is given a weight 
of (2F + l);i.e., 

+ \)W = 



Verify this relationship for the hyperfine structure in hydrogen and deuterium 
(Fig. 1.6). 

5. In Fig. 1.6, for 7 = ^, identify the following two transitions: 

! = (p = i, mF = 0) -> (F = 1, m F = -1), 
and v z = (F = 1, m F = 1) <-> (F = 0, m F = 0) 

Show from the Breit-Rabi formula that, if the frequencies v\ and v t are mea- 
sured in the same magnetic field //, then the hyperfine separation Ai> for zero field 
is given directly by Ay = vi v\. 

2. Extranuclear Effects of Nuclear Magnetic Dipole Moment 

a. Absolute Separation of Hyperfine-structure Components. We 

have seen that the nuclear angular momentum / determines the number 
of hyperline levels and also their relative separations. The absolute 
magnitude of these separations, however, is proportional to the interval 
factor a of Eqs. (i.3) and (1.10) and therefore depends upon both the 
magnitude and sign of the nuclear magnetic moment and on factors 
related to the electronic states and the probability of the electron being 
near the nucleus. Also, the effects of perturbations from other electronic 
states, as well as the presence of a nuclear electric quadrupole moment, 
may alter a. 

To minimize these perturbations, spectroscopic observations are usu- 
ally made on states having large hyperfine structure, as these are least 
perturbed. In spite of the uncertainty in computing p from observations 
of the optical hyperfine structure, many of our values of nuclear magnetic 
moment come from hyperfine-structure separations. 

It must be emphasized that the absence of detectable optical hyperfine 
structure in some atoms may be due to the smallriess of M and cannot be 
taken as definite evidence for zero mechanical moment. 

b. Absolute Separation of Atomic -beam Components. The number 
of components into which an atomic beam is split by a magnetic field 
depends upon the nuclear mechanical moment. The separation of these 
(27 + 1)(2J + 1) components, however, is determined by the magnitude 
of the field gradient and the hyperfine-structure separation factor a for 
the normal slate of the neutral atom, hence by the nuclear magnetic 
moment. 

By the atomic-beam deflection method, Kellogg, Rabi, and Zacharias 
(K14) measured in 1936 the hyperfine-structure separation A? for the 



192 The Atomic Nucleus [CH. 5 

ground state of the hydrogen atom H 1 , from which they obtained 

M P = 2.85 0.15 

nuclear magnetons for the proton. The essential agreement between 
this value and the directly measured value of n p = 2.46 0.08 obtained 
from the deflection of orthohydrogen molecules in an inhomogeneous field 
by Stern and his collaborators (Chap. 4, Sec. 3) gave the first truly direct 
confirmation of the origin of atomic hyperfinc structure in nuclear magnetic 
moments. 

From accurate measurements of the deflection patterns, Rabi and his 
coworkers (Rl, K14, M47) have succeeded in obtaining the magnitude 
and sign of the nuclear magnetic moment of a number of atoms, some of 
whose hyperfine-structure separations are too small to be measured by 
optical methods. The atomic-beam deflection method has been used 
successfully on atoms having a single valence electron, i.e., the hydrogens 
and the alkalis H 1 , H 2 . Li, Na, K, Rb, and Cs. It has generally been 
superseded by the magnetic-resonance method as applied to both atomic 
and molecular beams. 

c. Larmor Precession Frequency. Larmor showed in 1900 from 
classical electrodynamics that any gyromagnetic system which has angu- 
lar momentum and a collinear magnetic dipole moment will be set into 
precessional motion when placed in a uniform magnetic field. 

If the absolute angular-momentum vector /' makes an angle ft with 
the direction of the magnetic field 77, the Larmor precession is such that 
/' describes the surface of a cone having H as an axis, with the frequency 
v which is given by 

where v = frequency, cycles/second 

H = magnetic field intensity, gauss 
7 = absolute gyromagnetic ratio, gauss" 1 sec ' 

// = absolute magnetic dipole moment, ergs/gauss 

/' = absolute angular momentum, erg-sec 
The kinetic energy added to the system by the Larmor precession is 

W = v!H cos ft (2.2) 

The absolute value of the nuclear magnetic dipole moment which is 
collinear with the nuclear angular momentum is given by Eq. (2.10), 
Chap. 4. Substituting in Eq. (2.1) we have for the Larmor frequency 
of a nucleus 

1 / 1 / I I A -* m~ \ T-if 

(2.3) 



where g = p/I is the usual nuclear g factor as defined in Eq. (2.15), Chap. 
4. Substituting numerical values, we obtain 

= 7620 cycles/ (sec) (gauss) (2.4) 



2] Atomic and Molecular Effects of Nuclear Moments 193 

Thus lor nuclei the Larmor frequencies will be of the order of 10 8 /7, or 
about a megacycle per second for H = ] ; 000 gauss. 

The kinetic energy of the Larmor precession is easily obtained by 
eliminating n'H between Eqs. (2.1) and (2.2) and is 

W = 2irvl' cos ft = hvl* cos ft = mjhv (2.5) 

where ra/ = 7* cos ft is the magnetic quantum number for the projection 
of 7* on H in the usual vector model (Fig. 1.1, Chap. 4). 

We see that quantization of the classical Larmor theorem leads to a 
precession energy which can have only a series of (27 + 1) discrete 
values, as m/ takes on the integer-spaced values from +7 to 7. We 
note the very interesting facts that the Larmor precession frequency v 
is independent of m t , so that the precession energies have a uniform spac- 
ing equal to hv. Thus the Larmor frequency v, which such a processing 
system would radiate classically, is exactly equal to the Bohr frequency 
condition for the emission or absorption of electromagnetic radiation in 
transitions between adjacent levels. 

d. Radio -frequency Spectroscopy. If nuclear magnets can be sub- 
jected to radiation at their Lannor frequency while they are in a constant 
magnetic field, a nucleus in a lower magnetic energy state may absorb a 
quantum of energy from the radiation field and make a transition to its 
next higher magnetic level. It turns out that this can be accomplished 
and detected experimentally in several different ways and that the 
resonances for absorption of energy at the Larmor frequency are sharp. 

Indeed, transitions between molecular levels as well as atomic levels 
can be induced in this way, The Larmor precession frequency of the 
effective gyromagnetic system has only to match the Bohr frequency 
condition, namely, that hv be the energy separation between the levels 
concerned in the transition. 

The frequency range which is involved can be obtained from Eq. (2.3), 
which shows that for electronic systems 

_L ^ ^ ^ i megacycle/ (sec) (gauss j (2.6) 

H h 

and for nuclear systems 

-^ <**> - 1 kilocycle/ (sec) (gauss) (2.7) 

H h 

At ordinary laboratory magnetic field intensities of the order of 1 to 
1,000 gauss, these frequencies lie in the radio-frequency domain. Studies 
of nuclear, atomic, and molecular properties by Larmor resonance meth- 
ods have therefore come to be characterized as radio-frequency spectroscopy. 
Three principal experimental arrangements are in current use: (1) 
molecular-beam magnetic-resonance method (Rabi et al., 1938), (2) the 
nuclear paramagnetic-resonance absorption method (Purcell et al., 1945), 
and (3) the nuclear resonance induction method (Bloch et al., 1945). 



194 



The Atomic Nucleus 



[CH. 5 



Molecular-beam Magnetic-resonance Method. Magnetic-resonance 
methods and radio-frequency spectroscopy got their start (R2, K52) with 
the method of "molecular beams/' a generic term which now includes 
beams of neutral atoms as well as of neutral molecules. This technique 
has evolved into one of the most versatile, sensitive, and accurate meth- 
ods for studying the h^perfine structure and Zeeman levels of atoms and 
molecules. 

A schematic representation of one modern form of molecular-beam 
apparatus is shown in Fig. 2.1. With it, / and p were first measured 
directly for the nuclear ground levels of the radioactive nuclides Na 22 , 
Cs 185 , and Cs 187 (D9), and /, /i, and Q were measured for both the stable 
chlorine isotopes Ol 36 and Cl 87 (D8). A beam of neutral atoms diffuses 
at thermal velocities from the oven and passes successively through 
three magnetic fields. The first and last are inhomogeneous fields, whose 



Differential 
pumping chambei 



Multistage 
beryllium-copper 
electron multiplier. 




shield 



Distances from oven in cm 



556 



^60 wedge 936 
mass spectrometer 
magnet pole face 



Fig. 2.1 Schematic diagram of a modern molecular-beam apparatus capable of deter- 
mining /, M, and Q on very small samples of material. (From L. Darts, Jr., Massa- 
chusetts Institute of Technology, Research Laboratory of Electronics Technical Report RR, 
1948.) 

purpose is to deflect and then refocus the beam. As used in this experi- 
ment, the final refocusing field was arranged to refocus only those atoms 
which had undergone a Zeeman transition involving a change of sign of 
their magnetic moment while passing through the centrally located homo- 
geneous magnetic field H. In this homogeneous field H, the Larmor 
precession frequency v can be determined as the frequency / of a small 
additional radio-frequency field, directed normal to H, which produces the 
sought-f or Zeeman transition, and thus permits the final focusing magnet 
to bring the beam onto the detecting elements. At / = v, a sharp 
resonance peak in the transmitted beam is observed, and thus the Larmor 
frequency is determined. 

The detecting elements consist of a narrow hot tungsten ribbon, on 
which surface ionization of the originally neutral atoms takes place. The 
ions thus formed are then sorted for mass by passing them through a 



2] Atomic and Molecular Effects of Nuclear Moments 195 

single-focusing mass spectrometer. Finally, the accelerated ions impinge 
on the first plate of a Be-Cu multistage electron multiplier, where they 
are detected as "counts" in the output circuit. 

The sensitivity is so high that only 4 X 10~ 10 mole of Na 22 was used 
up in obtaining the final measurements of the nuclear moments 7 and /* 
of the 3-yr radionu elide Na 22 . It is believed that measurements on other 
radionuclides can be accomplished with as little as 10 13 atoms, which 
should now permit studies of a great many shorter-lived radioactive 
species. 

With genetically similar molecular-beam apparatus, a number of 
nuclear, atomic, and molecular constants have been determined. The 
details of this work will be found in Physical Review and in various sum- 
maries and reviews (III 2, K12, R3). 

Nuclear Paramagnetic-resonance Absorption Method. Nuclear ^-factors 
can be determined in bulk solid, liquid, or gaseous samples by two closely 
related methods: " nuclear resonance absorption" and " nuclear resonance 
induc.tion." The nuclear resonance absorption method stems from the 
work of Puroell, Torrey, and Pound (P37) who first demonstrated the 
.attenuation of 29.8 megacyrles/sec electromagnetic radiation by 850 cm 3 
of paraffin in a radio-frequency resonant cavity, when an external magnetic 
field of 7,100 gauss was impressed on the paraffin at right angles to the 
magnetic vector of the electromagnetic field. These conditions corre- 
spond to the Larmor precession frequency of the proton. 

It can be shown that the absorption of radiation by the protons is 
largely canceled by stimulated emission. The net absorption effect is a 
small one and is attributable to the Boltzmann distribution, which favors 
a slightly greater population in the lower-energy levels. As an illustra- 
tive numerical case (PI), for 1 million hydrogen atoms in thermal equi- 
librium at room temperature, and in a field of 20,000 gauss, an a.verage of 
only seven more protons are in the lower magnetic state than in the upper 
state. This slight asymmetry accounts for the net nuclear paramagne- 
tism. By utilizing a radio-frequency bridge circuit, and modulating the 
magnetic field at a low frequency, the nuclear resonance absorption. can 
be clearly and very precisely measured. 

The nuclear resonance absorption method (and also the nuclear 
induction method) each require moderately large samples, running at 
present in the neighborhood of ~ 10 18 nuclei. The methods are therefore 
applicable to stable or very long-lived nuclides. In order to eliminate 
the effects of electric quadrupole interactions, and to minimize the inter- 
actions of nuclei with their neighbors, these methods have so far been 
confined to nuclei with / = i (which have no observable quadrupole 
moment) or to cubic crystals. 

The techniques and results have been summarized in several excellent 
review articles (Pi, P28), and further details may be found in the current 
periodical literature. 

Nuclear Resonance Induction Method. The "nuclear resonance induc- 
tion 1 ' method developed by Bloch and co workers (B73, B75) also is 
applicable to matter of ordinary density. As in the nuclear resonance 



196 The Atomic Nucleus [CH. 5 

absorption method, a small sample (~ 1 cm 3 ) is placed in a strong uni- 
form magnetic field H, about which the nuclear magnets process at the 
Larmor frequency v which is proportional to their nuclear g factor 

J7 = M//- 

A radio-frequency field, whose frequency is /, is applied with its 
magnetic vector at right angles to H . At the resonant frequency / = v, 
changes occur in the orientation of the nuclear moments, corresponding 
to transitions between the magnetic substates, in accord with Eq. (2.5). 
In the resonance absorption method, these changes are observed by their 
reaction on the radio-frequency driving circuit. In the resonance induc- 
tion method, these changes are observed directly by the elcc.tr omotive 
force which they induce in a receiving coil, which is placed with its axis 
perpendicular to the plane containing // and the driving field /. The 
success of this very direct detection method gives a sense of immediate 
physical reality to the concept of space quantization of the Larmor 
precession of nuclei. 

The thermal relaxation process, by which energy is exchanged between 
the nuclear magnets and the " lattice" of thermally vibrating atoms and 
molecules, has received extensive theoretical and experimental study 
(B73, B75, B79). The so-called spin-lattice relaxation time> as observed 
in both the nuclear resonance absorption and resonance induction 
methods, ranges from the order of 10~ b sec to several hours for various 
materials. 

Values of the Larmor frequency for many nuclides relative to that 
of the proton in the same magnetic field can be obtained by the nuclear 
resonance absorption method, and by the closely related nuclear reso- 
nance induction method, with a precision of the order of 0.02 per cent. 
The corresponding relative values of the nuclear g factors g = p/I and 
the nuclear magnetic dipole moments p are not as accurately known, 
because of uncertainties in the corrections for the diamagnetism of the 
atomic electrons, Eq. (1.12). 

For details, the reader will be well rewarded by the study of the 
original papers of Bloch (B73) and his colleagues (B75) and of later 
descriptions of routine determinations of 7 and /z, such as those of Tl, Sn, 
Cd, and Pb (P35). 

e. Conversion of Parahydrogen and Orthodeuterium. Under ordi- 
nary conditions there are no transitions between the ortho and para 
states of either hydrogen or deuterium. Moreover, relatively pure para- 
hydrogen and relatively pure orthodeuterium can be prepared by adsorp- 
tion on charcoal at liquid-air temperatures. In these pure substances 
transitions leading to the equilibrium mixture of ortho and para materials 
can be induced by an inhomogeneous magnetic field. Such a field is 
supplied by the presence of the paramagnetic oxygen molecule. 

The rate of conversion of para- to orthohydrogen and ortho- to para- 
deuterium depends only on the equilibrium concentrations and the 
mechanical and magnetic moments of the proton and deuteron. By 
observing the relative speeds of conversion for parahydrogen and ortho- 
deuterium, the ratio of the nuclear magnetic moments of proton and 



13] 



Atomic and Molecular Effects of Nuclear Moments 



197 



deuteron is found (F8) to be MP/MJ = 3.96 0.11. This observation 
probably contains some unknown source of error, as the ratio obtained is 
definitely higher than is obtained by magnetic-resonance methods 
= 2.793/0.857 = 3.26. 



Problems 

1. Derive the Larmor precession frequency and kinetic energy from classical 
electrodynamics. 

2. Consider the hyperfine-structure separation A v as an energy difference due 
to Larmor precession of the nucleus in the magnetic field II j* produced by the 
atomic electrons. Evaluate and plot Hj* in gauss against Z for hydrogen and 
the alkali metals. 





Ai* 


M, 




Principal quantum 


Nuclide 


av, 
megacycles /sec 


nuclear 
magnetons 


/ 


number n of 
valence electron 


HI 


1,420.5 


2.793 


i 

2 


1 


H* 


327.4 


0.857 


1 


1 


Li' 


228.2 


0.822 


1 


2 


Li* 


803.5 


3.256 


2 


2 


Na" 


1,771.6 


2.217 


3 


3 


K 39 


461.7 


391 


3 
'2 


4 


Rb 


3,035 7 


1.353 


5 


5 


Rb 


6,834 


2.750 


3 
V 


5 


Cs 133 


9,193 


2.577 


7 


6 



NOTE: Values of bv are from P. Kusch and H. Taub, Phys. Rev., 76: 1477 (1949). 
Ans.: hydrogen, 0.289 X 10 G gauss; . . . ; cesium, 3.54 X 10 G gauss. 

3. Extranuclear Effects of Nuclear Electric Quadrupole Moment 

a. Deviations from the Interval Rule for Hyperfine-structure Separa- 
tions. The relative separations of hyperfine-structure components are 
predicted by the interval rule of Eq. (1 .5), which is derived from an energy 
term proportional to cos (/*,/*). Two types of deviations from the 
interval rule have been observed. The first is due to perturbations in the 
individual hyperfine-structure levels when the energy difference between 
the parent term and an adjacent electronic state is comparable with the 
hyperfine-structure separations. 

Observations by Schiiler and Schmidt of the hyperfine structure of 
europium (SI 8) first revealed deviations which were different for the 
two isotopes Eu 1B1 (/ = ) and Eu 163 (7 = i) and therefore could not be 
explained as perturbations. However, the introduction of an additional 
interaction term which depends on cos 2 (/*./*) satisfactorily accounts 
for these and many subsequent observations on other elements. The 
physical interpretation of this energy term implies, through Eq. (5.7), 
Chap. 4, that the deviations are due to a nonspherical distribution of 



198 The Atomic Nucleus [CH. 5 

positive charge in the nucleus, having the characteristics of a nuclear 
electric quadrupole moment. 

The observed quadrupole deviations from the interval rule give the 
product of the nuclear electric quadrupole moment and the average 
charge distribution of the electronic states involved. Where these elec- 
tronic charge distributions are not known accurately, the value deter- 
mined for the nuclear quadrupole moment will reflect this uncertainty. 
Accordingly, much more reliance is to be placed on the ratio of the quadru- 
pole moments for two isotopes of the same element, as Eu Jhl and Eu 158 , 
than on the absolute value for either of them. The electronic, charge 
distribution, being the same in both isotopes, does not affect the value 
for the ratio of the quadrupole moments. 

Only certain atomic states are affected by the nuclear electric quadru- 
pole moment. Electronic states having ,/ = | have no quadrupolo effect. 
Thus s and p$ electrons do not show the quadrupole deviation in the 



F=3 




Fig. 3.1 Hyperfine splitting of an atomic energy level J = -y , if 7 = | (e.g., in U 8 ";. 
Left, the single level, in the, absence of magnetic, dipole p and electric quadrupole 
moment Q. Center, normal hypcrfine structure, with n finite and Q absent. Right, 
pure electric quadrupole splitting. [From McNatty (M38).] 

hyperfine-structure spacings. Other states should be influenced approxi- 
mately in proportion to their fine-structure doublet separation, leading 
to large deviations for the low p$ and d electrons of the heavy elements. 

According to Casimir (C6), the ordinary interval law of hyperfine 
structure, Eq. (1.5), is to be replaced by 

w - a c + b 
w ~ 2 c + 8 

where C = F(F + 1) - /(/ + 1) - J(J + 1) 

a = hyperfine-structure interval factor, Eqs. (1.3) and (1.10) 
b = electric quadrupole factor, proportional to Q 

The effects of the electric quadrupole moment Q compared with those 
of the magnetic dipole moment /* are illustrated in Fig. 3.1. 

An example of the determination of /, M, and Q from optical hyperfme 
structure is shown in Fig. 3.2. 

Deviations from the magnetic levels expected in the Zeeman effect 
of hyperfine structure (Fig. 1.6) can also be interpreted in terms of 
the nuclear electric quadrupole moment (F28, D8). In this way, the 



3] Atomic and Molecular Effects of Nuclear Moments 199 

deuteron was first shown (K13) to have a quadrupole moment 
Q = +0.273 X 10- 26 cm 2 

b. Hyperfine Structure of Molecular Rotational Spectra. In poly- 
atomic molecules, the energy of interaction between the nuclear electric 
quadrupole moment and the gradient of the molecular electric field at 
the nucleus gives rise to a hyperfine structure in transitions between 
molecular rotational levels. This interaction energy depends on the 
relative orientation of the nuclear angular momentum I and the angular 
momentum of molecule rotation. The relative intensity and relative 

n* 192 

76 OS 







Os 189 







5 190 


a,6,c r d 


* ? 




! C 


B 1H 

b Os 186 a 
4 1 1 


-0.208 -0.0310 / / 0.148 \ 0.330cm- 1 
+-V 0.0589 4.1220 M1.1927 cm" 1 



Fig. 3.2 Hyperfine structure and iaotope shift, as measured with a Fabry-Pe'rot 
clalon, in the X4 7 260 line of singly ionized osmium. The lines due to the even-Z 
even-JV isotopes, 186, 188, 190, 192, are single, corresponding to nuclear angular 
momenta of 7=0, and display an approximately constant isotope shift of about 
0.03 cm" 1 per mass unit, The line due to Os 189 is split into four hyperfine com- 
ponents, marked a, fe, r, d, whose relative spacings show deviations from the interval 
rule. For Os 189 , Murakawa and Suwa (M75) interpret their measurements of this 
and other osmium lines as: /: multiplicity = (27 + 1) = 4, 7 = ; /*: from absolute 
separations of a, 6, c, d, /* = 0.7 0.1; Q: from deviations from the interval rule, 
Q = +(2.0 0.8) X 10-" cm*. 

spacing of the lines depend on 7 and provide an unambiguous measure 
of/. 

The absolute values of the energy differences in the hyperfine-struc- 
ture pattern are the product of a function of the quantum numbers of 
molecular rotation and of nuclear angular momentum, multiplied by the 
energy of the "quadrupole coupling." The "quadrupole coupling" is 
defined in different ways by different investigators, and caution must be 
exercised in comparing their reported results. Helpful comparisons of 
the conventions used by different investigators have been compiled by 
Feld (F25) and by Bardeen and Townes (BIO). The "quadrupole cou- 
pling" may usually be interpreted as [cQ(d*l r /dz z )] } where e is the elec- 
tronic charge, Q is the quadrupole moment defined as in Eq. (5.6), Chap. 
4, and d 2 U/dz 2 is the second derivative of the electric potential U due to 
all the electrons in the molecule, taken in the direction of the symmetry 
axis of the molecule. 

Microwave Absorption Spectroscopy. For heavy molecules, the sepa- 
ration of successive rotational energy levels corresponds to frequencies 



200 The Atomic Nucleus [GH. 5 

of the order of 10 4 megacycles/sec, and hence wavelengths of the order of 
1 cm. These frequencies lie in the so-called ''microwave" domain where 
enormous advances in technique have been accomplished as a consequence 
of radar developments. Thus techniques are available which provide 
both accurate frequency measurements and high resolution, and these 
have been applied in the now rapidly expanding field of "microwave 
spectroscopy." 

The pure rotation spectra of IC1 36 and Id 87 were the first to be 
investigated by the methods of microwave spectroscopy. In Id 86 the 
transition between the molecular-rotation quantum numbers 3 and 4 has 
a frequency of about 27,200 megacycles/sec. The exact frequency 
depends not only on the rotational quantum numbers but also on the 
nuclear angular momentum 7 and the nuclear electric quadrupole moment 
Q. Thus each transition between any two successive rotational quantum 
numbers, e.g., 3 4 in Id 35 , is actually split into a large number of 
identifiable lines which give quantitative information on / and Q (but 
not fi). These individual lines have separations of the order of 10 to 
100 megacycles and can be resolved and measured with an accuracy of 
0.1 megacycle/sec or better. Representative experimental results, and 
their interpretation in terms of 7 and Q for N 14 , O 18 , S 33 - 34 , C1 3B - 37 , Br 79 - 81 , 
and I 127 , will be found in the thorough work of Townes, Holden, and 
Merritt (T27). 

These measurements are accomplished by observing the absorption 
of microwaves, in Id, as a function of frequency. Microwaves of about 
1-cm wavelength are passed through a waveguide 16 ft long which con- 
tains IC1 vapor and acts as a 16-ft absorption cell. The minimum detect- 
able absorption lines have absorption coefficients of about 4 X 10~ 7 cm" 1 , 
and the corresponding differences of the order of 0.02 per cent in over-all 
transmission can be determined by the use of calibrated attenuators or 
by a balanced waveguide system. The techniques and results of micro- 
wave spectroscopy have been reviewed by Gordy and coworkers (G34, 
G37). 

One of the outstanding achievements by this method was the dis- 
covery from the microwave-absorption spectrum of boron carbonyl, 
H 3 BCO, that 7 = 3 for the ground level of B 10 (G36, W19), whereas the 
value 7 = 1 had long been erroneously assumed for B 10 by analogy with 
the only other stable nuclides which contain equal numbers of protons 
and neutrons, namely, H 2 , Li 6 , and N 14 . 

In a small number of cases it has been possible to place the absorption 
cell in a magnetic field of the order of 2,000 gauss and thus to observe 
Zeeman, or magnetic, splitting, superimposed on the electric quadrupole 
hyperfine structure of rotational transitions. In this way it is possible 
to determine some nuclear magnetic dipole moments p, by microwave- 
absorption methods (G35), but the accuracy is not competitive with the 
magnetic-resonance methods of radio-frequency spectroscopy. 

Because isotopic molecules, such as IC1 3B and IC1 37 , have an appreci- 
able difference in rotational moment of inertia, there is a relatively large 
frequency separation between analogous groups of lines from the two 



3] Atomic and Molecular Effects of Nuclear Moments 201 

similar molecules containing different isotopes of the atom in question. 
The microwave-absorption data possess such high resolution and accuracy 
that they permit the determination of atomic mass ratios (for example, 
C1 35 /C1 37 ) and isotopic abundance ratios with an accuracy comparable with 
that obtained from mass spectroscopy (L35). 

Molecular-beam Electric-resonance Method. The electric moments of 
molecules can be measured also by a molecular-beam resonance method, 
in which all the fields are electric instead of magnetic. These electrical 
resonance methods of molecular-beam spectroscopy were first applied to 
the fluorides of the alkali metals CsF (H71, T28) and RbF (H72). The 
nuclear electric quadrupole moment Q can thus be measured, as well as 
the molecular electric dipole moment, moment of inertia, and internudear 
distance. 

Problem 

Show that electronic states having / = or J have no quadrupole effect, even 
if the nuclear quadrupole moment is finite. 



CHAPTER 6 

Effects of Nuclear Moments and Parity on Nuclear 
Transitions 



Differences of angular momentum and parity between nuclear levels 
produce profound effects on the relative probability of various competing 
nuclear transitions. Studies of these transitions provide the experimental 
basis for the determination of relative values of nuclear angular momen- 
tum and parity. These purely nuclear effects are especially useful for 
evaluating excited levels in nuclei, as well as some ground levels. 

The total energy, total angular momentum, and parity of any isolated 
set of nuclear particles are always conserved in all nuclear interactions 
and transformations. Any changes of nuclear angular momentum arid 
parity which may occur in a nucleus must therefore be found associated 
with an emitted or absorbed particle. 

The probability of any type of nuclear transformation depends on a 
number of factors, the best understood of which are: (1) the energy 
available, (2) the vector difference 1^ 1 B between the angular momen- 
tum of the initial and final levels, (3) the relative parity of the initial 
and final levels, (4) the charge Ze of the nucleus and ze of any emitted 
particle, and (5) the nuclear radius. 

From measurements of the relative probability of various nuclear 
transformations, quantitative inferences can often be made concerning 
the difference in angular momentum and parity between tw r o nuclear 
levels. The angular distribution of reaction products, and of successive 
nuclear radiations, is also markedly dependent upon angular-momentum 
and parity considerations. In these two general ways relative values of 
nuclear angular momentum and parity can be determined. Conversion 
to absolute values usually is made by reference to ground-level values of / 
and parity, \vhich have been determined through the measurement of 
hyperfine structure or other extranuclear effects. In these ways, nuclear 
transformations provide a means of studying the moments of short- 
lived excited levels, which are generally inaccessible to methods of radio- 
frequency spectroscopy, band spectroscopy, and microwave absorption. 

Nuclear transformations of substantially every type are impaired 
if the change in angular momentum is large and are easiest and most 
probable for transformations in which I A = IB or I B 1. 

The vector change in angular momentum 1 A IB can have any abso- 
lute value from \I* I B \ to \I A + IB\, depending on the relative spatial 

202 



y the minimum possible value, namely, 

A/ = \I A - T B \ 

Drin cipal exceptions to this general rule occur when I A = IB in the 
D 7-ray emission and will be summarized in Table 4.2. 

onserratioTi of Parity and Angular Momentum 

few of the reactions of Li 7 , when bombarded by protons, will serve 
amples of the effectiveness of parity conservation in prohibiting 
nuclear reactions, even though an abundance of energy is available. 



_18J8 



-0.096 




19.9 Mev 7-2" 1 



19J8 / 



17.63 




2.94- 



\ 



17.242 



Li 7 +p 



3.0 5 
2.22 I 
1.882-J 



0.44 




2a. 



Be 8 



1 Some of the known energy levels of Be 8 and reactions involved in their 
,ion and dissociation (A10). 

gure 1.1 depicts a few of the known (A 10) resonance reactions, 
)He 4 , Li 7 (p,rc)Be 7 , and Li 7 (p,7)Be 8 , followed by the fission of Be 8 
wo a particles, Be 8 > He 4 + He 4 . The scales and manner of plot- 
re the same as those used in Chap. 3, Figs. 4.1 and 4.8. In addition, 
ependence of reaction cross section on proton bombarding energy 
shown above the ground level of (Li 7 + p). The numerical values 
are shown, for convenience, in laboratory coordinates, alongside 
?rtical energy scale which, of course, is actually plotted in center-of- 
coordinates. 



204 The Atomic Nucleus [CH. 6 

We note particularly here the sharp resonance at E p = 441 kev, whose 
measured width is only 12 kev. The excited level of the compound 
nucleus Be 8 , formed of (Li 7 + p) at this particular bombarding energy, 
has an excitation energy of 17.63 Mev above the ground level of Be 8 . 
The ground level of Be 8 disintegrates spontaneously into two He 4 nuclei 
with a half-period of less than 10~ u sec arid a decay energy of about 96 
kev. Yet when Be 8 is in its excited level at 17.63 Mev it is completely 
unable to disintegrate into two He 4 nuclei. This experimental fact can 
be understood and accepted only in terms of the quantum-mechanical 
ideas of parity and statistics. 

The dissociation of Be 8 into two a particles gives a final system com- 
posed only of two identical a particles. In the wave function of the 
final system, the interchange of these two identical particles must leave 
the sign of the wave function unaltered, because a particles have Ein- 
stein-Bose statistics. Thus the final wave function is symmetric. Inter- 
changing the two a particles, which are spinless (7 = 0), is in this case 
equivalent to reflecting the spatial coordinate system through the origin, 
and hence this reflection also must leave the sign of the wave function 
unchanged. Thus the parity of the final system must be even. 

To possess even parity, the relative motion of the two or particles 
must have even orbital angular momentum, I = 0, 2, 4, . . . . Due to 
conservation of angular momentum, any level in Be 8 which can break up 
into two a particles must also have even angular momentum. Parity 
conservation requires, in addition, that the level must have even parity. 
Hence only such levels in Be 8 as the ground level (/ = 0, even parity; 
denoted I = + ) and the excited levels at 2.94 Mev and 19.9 Mev [both 
7 = 2+ from independent evidence (A10)] can dissociate into two a 
particles. Evidence from the scattering cross section and other consider- 
ations shows (W10) that the excited level at 17.63 Mev is / = 1+. Its 
odd angular momentum completely excludes it from breaking up into two 
a particles. It has no other alternative but the emission of 7 radiation, 
which can carry changes of both angular momentum and parity, in 
order to arrive at some lower level having even parity and even angular 
momentum. The resulting 17.6-Mev 7 radiation ranks among the 
highest known energies for radiative transitions in nuclei. 

Another resonance level of Be 8 , at 19.18 Mev, is known to have odd 
parity and does not emit a rays. This level lies above the separation 
energy for a neutron; hence it can and does emit a neutron in accord 
with the reaction Li 7 (p,n)Be 7 . 

2. Penetration of Nuclear Barrier 

We have seen in Appendix C, Fig. 10 and Eq. (103), and Chap. 
2, Eq. (5.79), that the mutual orbital angular momentum I of two 
nuclear particles corresponds to an energy which is not available for 
penetration of a coulomb barrier and is known as the centrifugal barrier, 
(h/2ir) 2 l(l + l)/2Mr 2 . Thus barrier transmission is simplest, and reac- 
tion cross sections are largest, when the formation of the compound 



3] Nuclear Effects of Nuclear Moments 205 

nucleus and its subsequent dissociation both correspond to s-wave inter- 
actions (I = 0). 

For a-ray disintegrations in the heavy elements, numerical substi- 
tution in Eq. (103) of Appendix C shows that the transition probability 
varies about as e~ Q - ll(l+1) . Thus even an angular-momentum change of 

I = A/ = 5 

produces only about a 20-fold reduction in the a-r&y decay constant. 
This feeble effect is completely swamped by the much larger effects due 
to slight variations in nuclear radius arid by shell effects relating to the 
probability of formation of the a ray (PI 5). Hence the fine structure 
which is present in some a-ray spectra (ThC, ThC', RaC, RaC', etc.) 
cannot be used for quantitative evaluation of the angular momenta of 
excited levels of the nuclei involved, as was once thought possible. 
Nevertheless, Gamow's interpretation of the fine structure of a-ray 
spectra in terms of changes in nuclear angular momentum was of great 
historical value. It led him later to initiate analogous considerations 
regarding 0-ray transformations, where nuclear angular momentum plays 
a predominating role. 

In the lighter elements, whose coulomb barriers are lower, the cen- 
trifugal barrier can exert more profound effects. 

3. Lifetime in Decay 

a. Allowed and Forbidden Transitions. The concept of allowed and 
forbidden nuclear transitions first came to prominence in the empirical 
correlation known as the Sargent diagram, Fig. 3.1, between the half- 
period and the energy of decay. In 1933 only the 0-ray emitters of the 
uranium, thorium, and actinium families were known. These were 
shown by Sargent (S4) to fall into two groups. For a given energy 
(E mn of the 0-ray spectrum), one group 1 as about 100 times the half- 
period of the other group. As a specific illustration that some parameter 
besides disintegration energy plays a predominant role in the lifetime for 
ft decay, we may compare the energy and lifetime of RaB and RaE. 
These have 



Parent nuclide 


fl mRX| Mev 


Half-period 


Class 


RaB ( = 8Z Pb 214 ) 
RaE ( = B3 Bi) 


0.7 
1.17 


27 min 
5.0d 


Allowed 
Forbidden 



Even though the RaE decay is more energetic, its half -period is more than 
200 times that of RaB. In 1934, in the light of Fermi's newly proposed 
theory of /3 decay, Gamow (G4) proposed that nuclear angular momentum 
was responsible for the existence of the two (groups. He suggested that 
the shorter-lived group are "permitted," or "allowed" transitions, with a 
selection rule (A/ = 0, no), while the longer-lived group are "not-per- 
mitted," or "forbidden" transitions, with (A/ = 1, yes). 



206 



The Atomic Nucleus 



[CH. 6 



With the discovery of artificial radioactivity in 1934 and the subse- 
quent study of several hundred 0-ray emitters, the situation is known to 
be much more corrplicated than it seemed in the beginning. In par- 
ticular, the clear separation into two groups has vanished. Newlj r found 
radionuclides are scattered over the entire area of the Sargent diagram, 
beneath an upper envelope, or ceiling, on the decay constant X for a par- 
ticular value of ,. The forbidden transitions are subdivided into 
first-forbidden, second-forbidden, third-forbidden, etc.., and each of these 
classes is further subdivided into "favored" and "unfavored" transitions, 
in consonance with Wigner's (W47, K39) theory of supermultiplets. A 



S-6 



-8 



-10 



T v 

J min 
-Ihr 



AcB 




-Iday 



RaC 



0.5 Mev IMev 

I I 



3 Mev 



-1 



Fig. 3.1 Original form of the Sargent diagram for some of the naturally occurring 
0-ray emitters. The maximum energy of the 0-ray spectrum E mt * is plotted against 
the partial decay constant X for the principal branch of the decay. The half-period 
7*1 = 0.693/X is also shown. [From Sargent (S4).] 



vast amount of experimental and theoretical work has enlightened the 
topic of ft decay, but some fundamental problems remain unsolved. 

b. Comparative Half-periods in ft Decay. Two forms of empirical 
classification of ft emitters have emerged so far. One is based on the 
relationships between E mn and the half-period, and the other on the 
shape of the /3-ray spectrum. Both classifications are in acceptable 
agreement with present theories of the lifetime and shape of forbidden 
0-ray spectra (Chap. 17). 

The classification by "comparative half-periods," or "ft value," was 
introduced by Konopinski (K37) in connection with his theory of for- 
bidden 0-ray transitions. It permits comparison of the observed half- 
periods "" after due allowance is made through the function "/" for 
differences in nuclear charge Z and the energy of the 18 transition. 



3] Nuclear Effects of Nuclear Moments 207 

From the theory of allowed (3 spectra, there emerges the relationship 



I TO 

where J = half-period in seconds 

\P\ 2 = nuclear matrix element for transition 
Wo = (E mAX + wi r z )/mor 2 = total energy of ft transition 
Z = atomic mini her of decay product 
TO = universal time constant determined by electron and neutrino 

interaction with nucleons 

The somewhat complicated analytical form of the so-called "Fermi func- 
tion" /(,B 7 o) is discussed in Chup. 17. This function /(Z,Tf 7 ) is very 
strongly dependent upon H 7 n, varying approximately as H"?,. Graphs 
(F22) and a useful nomogram (Mfil, HOI) of f(Z,W {} ) are now available 
in the literature. The nuclear matrix element \P\ 2 can be visualized 
physically (Chap. 17) as representing the degree of overlap of the wave 
functions of the transforming nucleon (n > p + /9~ + P, or p --> n + /3+ 
+ v) in its initial and final state. |/ J | 2 is of the order of unity for allowed 
transitions. 

Rewriting Eq. (3.1) in the form 

- _ universal constants f . 

ft- ^ (3.2) 

shows that all allowed 0-ray transitions should have the same ft value, 
except for minor variations in \P\ 2 . This is found to be the case. Indeed , 
the transitions between the mirror isobars (Z = N 1) constitute a 
special class of super all owed, or allowed and favored , transitions , for which 
// lies between 1,000 and .1,000 sec (K39). The so-called allowed and 
unfavored transitions have larger / values, mostly in the domain of 5,000 
to 500,000 sec. Remembering that, even for allowed transitions, / and t 
individually vary by factors of the order of 10 B (t = 0.8 sec for Ke fi ; 
t = 12 yr for H 3 ), the constancy of ft within a factor of about 100 repre- 
sents an exceptionally good accomplishment for the theory. In view of 
the large numbers involved in the ft values, it is often more convenient 
to use only the exponents, i.e., the "log/Z" value, as is done in Table 3.1. 
The first-forbidden transitions lie generally around 10 fi to 10 K sec, or 
log // = (j to 8. 

c. Selection Rules for Decay. In the theory of forbidden /3 decay 
(K39, K41, K37) the transition probability can be expanded in a rapidly 
convergent series of terms characterized by successive integral values for 
the angular momentum of the electron-neutrino field with respect to the 
emitting nucleus. The largest term represents the allowed transitions; 
the successively smaller terms represent the forbidden transitions. The 
selection rules follow from inspection of the character of each term. The 
ultimate nature of the nucleon-clectron -neutrino interaction (Y3) is not 
yet perfectly understood. Comparisons between experiment and various 
theoretical formulations at present narrow 7 the choices to two possible 



208 



The Atomic Nucleus 



[CH. 6 



sets of selection rules, known as "Fermi selection rulea" and the "Gamow- 
Tellcr (G-T) selection rules." 

The Gamow-Teller rules (G8) emerge when the intrinsic spin of the 
transforming nucleori is introduced into the Hamiltonian which describes 
the transformation. This couples the electron-neutrino spin directly 
to the nuclconic spin. The Fermi rules (F34) emerge when this is 
omitted. In allowed transitions, Ferrni rules imply that the electron- 
neutrino pairs are emitted with antiparallel .spins (singlet state), while in 
Gamow-Teller rules they are emitted with parallel spins (triplet state) 
in the nonrelativistic limit (p. 679 of B68). At present, CJamow-Teller 
rules appear to describe most cases of decay, especially among all but 
the lightest elements, but there are a few instances where Fermi rules 
or a mixture of Fermi and Gamow-Teller rules are required to describe 
the observations (B6t), S33) (C 10 and O 14 , allowed and favored, / = > 0, 
no, transitions). These selection rules are summarized in Table 3.1. 

TABLE 3.1. SELECTION RULES FOR ft DECAY, ACCORDING TO THE 

CLASSIFICATION BY KONOPINSKI (K37) 

For classification on the basis of the "order" A/ of the transition, see Blatt and 
Weisskopf (p. 705 of BOS). 







A/ 




/^11 






Approximate 


Class 
of 
transition 


Parity 
change 


Gamow-Teller 
(tensor and axial 
vector) 


Fermi 
(scalar and 
polar vector) 


log ft 
values 
(M27) 


Allowed 


No 


0, 1 





3-6 






(but not -> 0) 






First-forbidden 


Yes 


0, 1, 2 


0, 1 


6-10 






(not > 0, ^ > ?, 


(not - 0) 








0<-> 1) 






Second-forbidden 


No 


2, 3; 0->0 


1, 2, 


>10 






(not <-> 2) 


(not <-> 1) 




nth-forbidden 




n, (n + 1) 


(n - 1), n 




n odd 


Yes 








n even 


No 









The Gamow-Teller rules are characteristic of a tensor or an axial-vector 
interaction, while the Fermi rules represent a scalar or an ordinary vector 
interaction between the transforming nucleon and the electron-neutrino 
field (K37, L5, S43). 

It will he noted that the selection rule on a parit3 r change in the trans- 
forming nucleus is the same for either type of selection rules and alter- 
nates in the successive degrees of forbiddenness. This corresponds to the 
emission of the electron-neutrino pair with successively larger values of 
orbital angular momentum. 

d. Shell Structure. From measurements of the /3-ray energy and 
half-period, log ft values are now available for over 140 odd-4 radio- 
nuclides and over 100 even- A nuclides. These provide empirical evi~ 



3] 



Nuclear Effects of Nuclear Moments 



209 



dence on the degree of forbiddenness for the various transitions. With 
the help of the selection rules, this evidence can be interpreted in terms 
of the parity and angular-momentum differences between the parent and 
daughter nuclides. Many absolute reference values are available from 

TABLE 3.2. ILLUSTRATIVE ALLOWED AND FORBIDDEN /S TRANSFORMATIONS 
Correlation of /3-decay log ft values with assignments of odd-nucleon states on the 
sin glo-p article shell model with jj coupling, and with Gamow-Teller selection rules 
(selected from M27, K39, and K18a compilations for odd-^4 nuclides). The nucleon 
states marked * have been determined from Schmidt group classification, following 
direct measurement of I and /*. When the odd nucleon is a neutron, the neutron 
number is given as a subscript beneath the mass number of the nuclide. The energy- 
level diagram and decay scheme for the two transitions of Cs 137 are given in Chap. 6, 
Fig. 7.1. 



ft transition 


Mev 


Log 
ft 


Odd-nucleon states 


Parity 
change 
of the odd 














Initial > final 


A/ 


A/ 


nucleon 


Super- O n\ * iH 1 


0.78 


3.1 


*J* -> 5J* 








No 


allowed jn^oji 


1.72 


3.4 


UB ~~* UP 








No 


Allowed 1.SJ5 -+ i 7 Cl aB 


17 


5 


us ~~* d& 








No 


[ 0- 














* 3 2 GeJ 3 B -> aaAs 7 ' 


1.1 


5.0 


j)L ^ Pi 


1 





No 


^-forbidden ygNiJJ > zeCu 66 


2.10 


6.6 


/,- 


1 


2 


No 


(9- 


2 27 


6.7 


/!-* 





1 


Yes 


0- 














First- 36 KrgJ -> 37 Rb 87 


3 6 


7.0 


Ul ~~t 7) a 


1 


1 


Yes 


forbidden B9 ^" g 


140 


8 5 


d,- PJ * 


2 


1 


Yes 


, t Cs'-..Ba!;' 


51 


9.6 


01 ~^ feAA 


2 


1 


Yes 


Second- ft~ 














forbidden 6 6 Cs 137 -> ^eBai? 7 


1.17 


12.2 


01 "^^ flj 


2 


2 


No 


fhird- r 














} forbidden 3 7 Rb 7 -> 3B Sr B 4 J 


0.27 


17.6 


PB ^^ (7fi 


3 


3 


Yes 


Fourth- P- 














forbidden 4In llB > B oSn 6B 


63 


23.2 


fli ^ Sj 


4 


4 


No 



the directly measured values of / and /x, which can be interpreted in 
terms of total angular momentum I, orbital angular momentum / of the 
add nucleon, and parity, with the help of the Schmidt groups and the 
single-particle model (Chap. 4, Figs. 4.1 and 4.2). 



210 



The Atomic Nucleus 



[CH. 6 



The results of these multilateral correlations (M27, N23, K39) have 
been most encouraging. They have greatly strengthened the case for 
the single-particle model, with,;)' coupling, and closed shells at 2, 8, (14), 
20, (28), 50, 82, and 126 nucleons. 

The careful *tudy of a few examples will give the flavor of the correla- 
tions, Table 3.2 contains well-verified cases in which the log ft values 
are quite reliable and the nuclear 7, 7, and parity of the odd nucleon have 
been determined for either the initial or final .state (or both). The other 



Odd -A 



CD Allowed 
1st forbidden 
/-forbidden 
2nd and higher 
forbidden 



Allowed 

1st forbidden 
Bi /-forbidden 
CD 2nd and higher 
forbidden 




Fig. 3.2 Frequency histograms of log ft values for odd-^4 and even- .4 radionurlides. 
{Compiled by D. R. Wil rW52i.| 

state in the transition is that predicted by the shell model withj[; coupling 
(Chap. 11). 

In all these cases Gamow-Teller selection rules agree with the differ- 
ential assignment between the initial and final states. Fermi selection 
rules would agree in some cases and not in others. The preference is 
clearly in favor of Gamow-Teller selection rules in these cases and in 
most, but not all, other cases. 

In the superallowed group, the initial and final odd nucleons always 
belong to the same shell. In the allowed but unfavored group, the odd 
toucleon is usually in a different shell before and after the transition. 
The overlap of initial and final nucleon wave functions therefore cannot 



4] Nuclear Effects of Nuclear Moments 211 

be as complete as for the superallowed group, and the matrix element 
|P| 2 is of the order of 0.01. 

Note that in the allowed but unfavored transitions there is no distinc- 
tion between the transition probability, as measured by log ft, for 
A/ = and for A/ = 1 . The controlling condition appears to be AZ = 0, 
that is, zero orbital momentum change in the transforming nucleus, hence 
also no parity change, and zero orbital angular momentum for the 0-ray 
electron-neutrino pair with respect to the emitting nucleus. Analogously, 
the first-forbidden transitions all have A/ = 1, second-forbidden, AZ = 2, 
etc. 

The so-called l-forbidden transitions (N23, M27, K39J constitute at 
present a, small subgroup, closely related to the allowed unfavored transi- 
tions, and obeying the Gamow-Teller selection rules on A/ for allowed 
transitions but having A/ = 2 on the single-particle shell model. These 
/-forbidden transitions (A/ = 1, no, A = 2) have log/ ~ 5 to 9, hence 
half-periods which are usually somewhat longer than would be expected 
for ordinary allowed transitions (A/ = or 1, no, A/ = 0). 

Figure 3.2 collects the presently evaluated log ft values (M27, N23) 
in a frequency histogram. This figure is equivalent to sighting up a 
modern Sargent diagram in the direction of the line of allowed transitions, 
so that the scale of energy disappears. The spike at log// = 3.5 0.2 
contains the superallowed transitions, with no over-all trend from II 3 to 
Ti. 4:i The allowed but unfavored transitions are mainly in the band 
log// = 5.0 0,3, with very few values above 6. The first-forbidden 
transitions are mostly above log // = 6 and tend to cluster between G 
and 10. 

I. Radiative Transitions in Nuclei 

In the early days, even the origin of 7 radiation was a mystery. 
It was first pointed out by C. D. Ellis and by Lisc Meitncr in 1922 that 
the complex 7 rays from some radioactive substances, such as RaB, 
exhibit simple additive rules, hvi + hv 2 = hv- A , and hence that 7 rays 
represent radiative transitions between quantized energy levels in nuclei, 

After the development of the quantum theory of radiation, it became 
possible to classify 7 radiations into electric and magnetic multipole radia- 
tions of various orders and to associate these with the difference in angular 
momentum and parity between the two states involved in the transition. 

The quantum theory of radiation borrows the classical representation 
of a radiation source as an oscillating electric or magnetic moment. The 
complicated spatial distribution of the corresponding electric charges and 
currents is represented by spherical harmonics of order 1, 2, 3, ... , 
and the names dipole, quadrupole, octupole, . . . are applied both to 
these equivalent nuclear moments and to the resulting radiation. Con- 
vergence of the expansion (p. 248 of Sll) is assured by the fact that the 
wavelength of 7 radiation (X = \/2ir ~ 200 X 10~ 13 cm for 1 Mev) is 
generally much larger than the dimensions of the nucleus in which the 
radiation originates, that is (R/\) <3C 1. 



212 The Atomic Nucleus [CH. 6 

a. Angular Momentum and Multipole Order of 7 Radiation. It can 

be shown (p. 802 of B68) that the angular momentum of 7 radiation with 
respect to an emitting (or absorbing) system is determined by the same 
type of quantum numbers I, m as the angular momentum of a material 
particle. For photons, Z can have only nonzero values. Thus the angu- 
lar momentum of a quantum of light is h VZ(Z + 1), and its projection 
on any arbitrary axis is mh with maximum component Ih. The prob- 
ability of emission (or absorption) decreases rapidly as I increases, 
roughly as (R/\Y l - 

Angular momentum is conserved between the 7 ray and the emitting 
(or absorbing) system, so that I is the vector difference between the 
angular momentum of the initial and final nuclear levels, or 

I=\!A- IB\ (4.1) 

Thus, between levels I A and 1 B , I can have any nonzero integer value 
given by 

A/ - \I A - IB\<1< IA + IB (4.2) 

The magnetic quantum number m of the radiation is the difference 

between the magnetic-angular-momenturn quantum numbers of the 
levels I A and IB, or 

m = m B m A (4.3) 

In practice, I is usually confined by relative transition probabilities to 
I = A/. In exceptional circumstances, a measurable fraction of the 
transitions may have I = A/ + 1, in competition with I = A/. If eithe 1 ^ 
I A or IB is and the other is nonzero, then there is only one possible value, 
namely, I = A/. This simplifying circumstance is met, for example, in 
all transitions to the ground level of even-Z even-JV nuclei. 

The multipole order of 7 radiation is 2 l ; thus I = 1 is called dipole 
radiation, I = 2 is quadrupole, etc. 

One of the consequences of the transverse nature of an electromag- 
netic wave is that it contains no I = multipole. Hence, from con- 
servation of angular momentum as represented in Eq. (4.2), 7-ray transi- 
tions between two levels I A = I B = are absolutely excluded. 

b Electric and Magnetic Multipoles. Parity of 7 Radiation. For 
each multipole order two different waves are possible. These are called 
the "electric" and "magnetic" multipole radiations. For each value of 
Z, a quantum of electric and one of magnetic radiation have the same 
angular momentum but differing parity. The parity of an electric multipole 
is the same as that of a material particle having the same I. Thus any 
electric multipole has even parity when I is even and odd parity when I is 
odd. Magnetic multipole radiation has the opposite parity, i.e., odd 
parity when Z is even and even parity when I is odd. This can be sum- 
marized as 

parity of electric multipole = (1)' 
parity of magnetic multipole = (!)' 

where + 1 means even parity and 1 denotes odd parity. 



4] 



Nuclear Effects of Nuclear Moments 



213 



c. Selection Rules for 7-Ray Emission (or Absorption). The prob- 
ability of any transition from the state V A to ^ B in a system of particles 
is proportional to the integral J^ A q^a dr, where ^ is the complex conju- 
gate of VB, dr is a volume element, and q depends on the nature of the 
transition. Thus, for electric dipole transitions, q is the effective electric 
dipole moment Ze,z t - and changes sign on reflection, i.e., when x becomes 
x. For electric quadrupole transitions, q is the effective electric 
quadrupole moment, symbolically Se^z, 2 , and does not change sign with 
reflection of the particles in the origin of coordinates. 

The value of a definite integral cannot possibly change by an altera- 
tion of the system of coordinates. Hence, if for electric dipole radiation 
4(Se0t)J changes sign with reflection, its integral over all space must 
be identically zero. Consequently if V A represents a state having 
even parity, ^ B (and ^) must represent a state of odd parity, or vice 
versa, to allow a finite transition probability. That is, the parity of the 

TABLE 4.1. SELECTION RULES AND SYMBOLS FOR 7 RADIATION 



Classification 


Symbol 


I 


Parity change 
in nucleus 


Electric dipole 


El 


1 


Yes 


Magnetic dipole 


Ml 


1 


No 


Electric quadrupole 


E2 


2 


No 


Magnetic quadrupole 


M2 


2 


Yes 


Electric octupole 


E3 


3 


Yes 


Magnetic octupole 


M3 


3 


No 


Electric 2 z -pole 


EZ 


I 


No for I even 








Yes for I odd 


Magnetic 2 '-pole 


Ml 


I 


Yes for I even 








No for I odd 



final state must be opposite to that of the initial state for emission of 
electric dipole radiation. Conservation of parity in the system as a 
whole (i.e., nucleus and quantum of radiation) then requires that for 
electric dipole radiation the photon must have odd parity with respect 
to the system it leaves (or the system it enters in an absorption or excita- 
tion process). 

Similar considerations for the emission of electric quadrupole and 
magnetic dipole radiation show that for each of these radiations to be 
possible the parity of the final state of the nucleus must be the same as 
that of the initial state. 

The selection rules for emission (or absorption) of 7-ray photons 
are those combinations of I and parity which give nonvanishing values 
of the transition probability. These are summarized in Table 4.1. 

d. 7-Ray Emission Probability. Although we saw in Chap. 4 that 
the "static" electromagnetic moments of nuclei are generally confined 
to a magnetic dipole and an electric quadrupole, this restriction does not 
apply to the "dynamic" electromagnetic moments which are involved 



214 



The Atomic Nucleus 



[CH. 6 



in the y-ray transitions. These electromagnetic effects have their origin 
in the motions of individual protons, in the intrinsic magnetic moment of 
neutrons and protons, and probably also in an " exchange current" which 
would be associated with the exchange of charge between neutrons and 
protons in connection with the exchange forces between nucleons. 

The absolute values of the theoretically predicted transition prob- 
abilities are proportional to these electromagnetic multipole moments, 
and the estimation of these moments depends strongly on the nuclear 
model which is assumed in the calculation. The most recent of these 
theories, and one which agrees with experimental results in some areas 
where earlier theories failed (G27), was developed in 1951 by Weisskopf 
and is based on the single-particle shell model of nuclei. In the single- 
particle theor}', a y-ray transition is associated with a change in the 
quantum numbers of only one nucleon. 

Electric Multipoles. Weisskopf (W23, BG8) has shown that for elec- 
tric multipole transitions of order 2', the partial mean life r el (reciprocal 
of the partial decay constant) for the emission of a 7 ray whose energy is 
hv is given very approximately (perhaps within a factor of 10 to 100) as 



(4.5) 



where R = R^A* is the nuclear radius and S is a statistical factor given 
by 

2(1+ 1) / 3 \ 2 

I)] 2 V + 






X 3 X 5 (21 
From Eq. (4.6) the numerical values of l/S are as follows: 



1 


1 


2 


3 


4 


5 














l/S 


4 


2 1 X 10 2 


1.6 X 10 4 


1.9 X 10' 


3.2 X 10" 



Note that l/S increases by roughly 10 2 for an increase of I by unity. 
The energy W of the nuclear transition can be expressed alternatively 
in terms of the frequency v or the rationalized wavelength X of the radia- 
tion as 

--*- (ft/raoc) _ 1,240 X 10~ 13 cm 
" ~v ~ W ~ (W/mQC z ) ~ ~W~(in Mev) 



(4.7) 



Equation (4.5) refers to a transition in which a proton jumps from an 
initial state /i = j\ = I + ^ to a final state 7 2 = jz = ?(lz = 0), and it 
is regarded by Weisskopf as representing a minimum value of the theo- 
retical mean life of the level against y-ray emission. 

Equation (4.5) can be put into the convenient equivalent form 



= I [" 137 __ 



70 ] 2 < +1 |"_2 ] 1.29 X 

tF(inMev)J [A*\ S 



sec 



(4.8) 
(4-9) 



54J 



Nuclear Effects of Nuclear Moments 



10 

10 8 

10 6 

10* 

10 2 

1 

10' 2 



- 

io- 8 



10 

10' 



-' 



12 



10 



-" 



io- 



16 



10 



- 18 



-\\ 



,, 



\\ VvN 



Vv\ \ 



. \V 




1 1 



1 1 1 1 



\ 



\\\ \ 



\\\ 



215 

100 yr 
lyr 

Iday 
Ihr 

1 min 
Isec 



0.01 M2 0.05 0.1 - 2 O- 1 1 

Energy of Y ray in Mev 



10 



Fig. 4.1 The minimum estimated partial mean life r e i of nuclear levels for deexci- 
tation by the emission of electric multipolc 7 radiation, of order 2', on the uingle- 
particle model, Eq. (4.9). For each value of /, the lower curve gives r ol for A = 200, 
while the upper curve is for A = 20. For electric multipole transitions, the values 
plotted here are not remarkably different from the predictions of the liquid-drop 
model, as plotted by Moon (M54). For magnetic multipoles, sen Eq. (4.10). 

where the ratio of the classical electron radius 



ro = f.. 6 L\ = 2.82 X 10 l3 
\mocV 



cm 



to the nuclear unit radius R is taken as (r /Ro) ^ 2, and 
= 1.29 X IO- 21 sec 



Figure 4.1 expresses Eq. (4.9) in graphical form. For each value of l t 
the variation of T B I with mass number A is indicated by curves for 
A = 20 and A = 200. Note that the smaller nuclei are expected to 



216 The Atomic Nucleus [CH. 6 

have greater mean lives against 7 decay. A "typical" mean life of 
~ 10~ n sec is found for a nuclear level which can be deexcited only by 
emission of a 1-Mev electric quadrupole, or E2, 7 ray. 

Magnetic Multipoles. The electric multipole radiation originates in 
periodic variations in the charge density p in the nucleus. The magnetic 
radiation originates in periodic variations in the current density, which 
is of the order (v/c)p, where v is the velocity of the charges. Therefore, 
for the same multipole order Z, the probability of emission (or intensity) 
of magnetic 2 l radiation usually will be much smaller than the prob- 
ability of electric 2 l radiation, by the order of (v/c) 2 for dipole radiation. 
This concept, contained in Weisskopf's theory of 7 decay, replaces the 
erroneous estimates of pre-1951 theories according to which the ratio of 
magnetic to electric radiation of the same multipole order depended 
primarily on the ratio of the size of the emitting system to the wavelength 
of the emitted radiation, that is, (72 A). 

For magnetic multipole 7 radiation, on the single-particle model, 
Weisskopf (W23) estimates that the ratio of the squares of the magnetic 
and electric moments will be of the order of (Mi m /Qi m ) 2 ~ (h/McR) z , 
where M is the mass of a nucleon. Because the rationalized wavelength 
of a nucleon X ~ h/Mv must be of the order of the nuclear radius R, 
the ratio (h/McR) 2 is of the order of (v/c) z . The transition probabilities, 
and reciprocal partial mean lives, are proportional to the multipole 
moment for the transition. Therefore, for magnetic multipole transitions, 
the mean life r mRB of the upper level is longer than for an electric multi- 
pole having the same I, W t and A , by a factor which is given very roughly 



i= ~ -i r -J? r ~ 4.44* 

r ri W\(h/Mc)\ 



where the factor 10 arises because of the intrinsic magnetic moments of 
the nucleons and we have used R c^ 1.4 X 10~ u cm and 

-^ = 0.211 X 10- 13 cm 
Me 

Notice that the ratio in Eq. (4.10) is independent of both the transition 
energy W and the multipole index I. The curves in Fig. 4.1 may there- 
fore be used for estimating the partial mean life T mag of a level against 
magnetic multipole 7 radiation, simply by multiplying each curve by 
4.4A*. This is a factor of about 30 for A = 20 and of about 150 for 
A = 200. 

Predominant Transitions. In all theories, the probability of 7 emis- 
sion per unit time decreases very rapidly with increasing I, the depend- 
ence being roughly as (R/X) 21 . The multipole radiation which is actually 
observed in a transition from an initial nuclear level, with angular- 
momentum quantum number I A , to a final level I B therefore will corre- 
spond primarily to the smallest value of I which is consistent with con- 
servation of angular momentum, Eq. (4.2), and parity, Eq. (4.4). Often 
this will be simply I = \I A I B \ = A/. 



4] 



Nuclear Effects of Nuclear Moments 



217 



The transitions which correspond to the smallest value of I can be 
enumerated by reference to the conservation laws as embodied in the 
selection rules, combined with the principle that 2'-pole radiation is 
much more probable than 2 (f+I) -pole radiation, and that the intensity of 
magnetic 2'-pole radiation is smaller than the intensity of electric 2'-pole 
radiation by a considerable factor (though not as much as in pre-1951 
theories ; especially for low energies). 

TABLE 4.2. THE TYPE OF MULTIPOLE RADIATION EMITTED IN TRANSITIONS 

BETWEEN NUCLEAR LEVELS WHOSE ANGULAR-MOMENTUM QUANTUM 

NUMBERS ARE IA AND IB AND WHOSE PARITY Is THE SAME 

(No CHANGE) OR WHOSE PARITY Is OPPOSITE (YES) 

The types of multipolc radiation are indicated by the usual symbols as given in 
Table 4.1. Transitions A/ = are possible between levels I A ~ IB only if both ha\ e 
nonzero values, so that \m\ > 1, Eq. (4.3). Transitions are .absolutely excluded. 
The rules are the same for emission and for absorption of photons by a nuclear system. 



Parity 


A/ = I/A - /.| 


Spin 
change 
in 


Parity 
change 


Pre- 
dominant 


Weak 
admixture 






nucleus 


in 
nucleus 


radiation 


of 




Even (not zero) 





No 


EA/ 


M(A/ + 1); absent if 












I A or IB = 


Favored 


Zero 





No 


Ml 


E2; absent if 












I A - IB - \ 




Odd 





Yes 


EA/ 


M(A/ + 1); absent if 












I A or IB = 




Even (not zero) 


1 


Yes 


MA/ 


E(A/ + 1); absent if 












/A or IB - 


Unfavored 


Zero 


1 


Yes 


El 


M2; absent if 




Odd 


1 


No 


MA/ 


E(A/ + 1); absent if 












I A or IB = 



The results are summarized in Table 4.2, which shows also the classi- 
fication which can be made into parity-favored transitions (no "spin flip") 
and parity-unfavored transitions for which, in the single-particle model, 
one unit of the angular momentum contained in I has to arise from 
the reversal of the direction of intrinsic spin of the odd nucleon; thus 
AS = 1 within the emitting nucleus. 

For example, the transition /j > d^ involves a change of only the 
orbital motion of the odd nucleon and would be " parity-favored," 
whereas /j >/ s would involve a "spin flip" (AS = 1) and would be 
"parity-unfavored." 



218 The Atomic Nucleus [CH. 6 

e. Mixed Transitions. Admixtures of the predominant multipole 
with the weaker competing multipole shown in the right-hand column of 
Table 4.2 are expected, in the theory, to be of the same order of magnitude 
only when the predominant radiation is a magnetic multipole. The 
fact that the competing electric multipole has to be one order higher than 
the magnetic, i.e., 2 (/ + n instead of 2 l in order to conserve parity, may 
be only partially compensated in suitable cases by the factor of the order 
of O'/r) 3 between the squares of the effective "dynamic" magnetic and 
electric moments for the transition. 

When the predominant radiation is an electric multipole, the compet- 
ing magnetic multipole is at a double disadvantage. Then the radiation 
can be expected to be substantially the pure electric multipole. 

Experimentally, the only mixed transitions found have been Ml + E2, 
in a classification of over 90 isomeric 7-ra'y transitions by (ioldhaber and 
Sunyar (G27). 

f . Forbidden Transitions. In ordinary optical spectroscopy the selec- 
tion rules for allowed transitions arc simply those for the electric dipole 
/ = 1, yes, which correspond to the usual selection rules for electronic 
transitions in atoms: A./ = 0, 1, (not 0); Am = 0, 1, yes. The 
so-called forbidden transitions involve all the other electric and magnetic 
multipoles. These all involve longer lifetimes for the excited level. 
Under ordinary laboratory conditions of temperature and pressure, these 
longer-lived atomic levels generally lose their excitation energy in col- 
lisions with other atoms and thus are deexcited by nonradiative collision 
processes. The forbidden electron transitions only show up strongly 
where the pressures are much lower than those attainable on earth, such 
as in the nebulae. Thus the nebular spectrum lines which were once 
attributed to "nebulium 11 were shown by Bowen to be forbidden transi- 
tions in ionized oxygen atoms (B104). 

In nuclei, analogous thermal collision processes are not accessible for 
nonradiative deexcitation. Hence the forbidden radiative transitions are 
observed. Indeed, they are so ordinary that they usually are not referred 
to as forbidden; in fact, electric quadrupole radiation, or E2, is probably 
the most common type of nuclear y radiation. 

Nuclei do have available several types of deexcitation which do not 
involve the emission of 7 radiation. The most common of these is 
internal conversion. Here the nuclear excitation energy is given, in a 
nonradiative process, to a penetrating electron as discussed in the next 
section. In > transitions, which are truly and absolutely forbidden 
in any radiative process, all the transitions must proceed by nonradiative 
processes, usually internal conversion within the nuclear volume. 

5. Internal Conversion 

The transition from an excited level of a nucleus to a lower level of the 
same nucleus can also be accomplished without the emission of a photon. 
The energy W involved in the nuclear transition can be transferred 
directly to a bound electron of the same atom. This energy transfer is a 



5] 



Nuclear Effects of Nuclear Moments 



219 



direct interaction between the bound atomic electron and the 3ame 
nuclear multipole field which otherwise would have resulted in the emis- 
sion of a photon. All nuclear 7-ray transitions are accomplished in 
competition with this direct coupling process, which is called internal 
conversion. 

The nuclear energy difference 
W is "converted" to energy of an 
atomic electron, which is ejected 
forthwith from the atom with a 
kinetic energy E l given by 



i = W - 



(5.1) 



where 5, is the original atomic bind- 
ing energy of the electron. Figure 
5.1 shows the spectrum of conversion 
electrons which are ejected from the 
K, L, and M shells of indium by 
internal conversion of the 392-kev 
transition in In 113 . After the ojec- 
tion of the photoelectron, the atom 
emits the energy 5 t as characteristic 
X rays or as Auger electrons. 

Equation (5.1) rests simply on 
the law of conservation of energy. 
No photon is involved. The process 
of internal conversion haw intervened 
and won over a competing radiative 
transition. However, Eq. (5.1) is 
identical in form with Einstein's pho- 
toelectric equation, if W is replaced 
by the energy hv of the unsuccess- 
ful photon. This fact led to many 
years, even decades, of misinterpre-. 
tation of the mechanism of internal 
conversion. Equation (5.1) began 
its important career as a purely em- 
pirical relationship in 1922, when 
C. D. Ellis (E8) and Use Meitner 
(M39) independently showed that 
it held for what was then called 
the "line spectrum of rays" (now 
called conversion electrons) from 
RaB (Ellis) and ThB, RaD, RdTh, 




400 



Fig. 6.1 Internal -con version electron 
spectrum for the U92-kev transition in 
In 111 , [draws, Lanyrr, and Moffat ((Ml i.| 
The energy differences for the electron 
groups from the A', L, M shells corre- 
spond to the differences in binding en- 
ergy of these atomic shells in In. Con- 
version in the LI, L\\, . . . and M\ t 
Afu, . . . subshells are not resolved in 
this particular work. The observed 
ratio of conversion in the K shell to 
that in the (L + M) shells is 4.21 for 
this transition, which has been identi- 
fied as an M4 transition (magnetic 
2 4 -pole; A/ = 4, yes) in agreement with 
single-particle shell model predictions of 
P\ -* 9\ (G25). [The interlocked decay 
schemes of the isobars 4?Ag 113 , uCd 111 , 
win 111 , and 6 Sn lla have been correlated 
by Goldhaber and Hill (G25).] 



and Ra (Meitner). These experi- 
ments were the first to prove the presently accepted views on the ori- 
gin and nature of 7 rays. Up until that time y rays were thought to 
be bremsstrahlung (continuous X rays) associated with the passage of ft 
rays through the electron configuration of the emitting atom. Meitner, 



220 The Atomic Nucleus [CH. 6 

especially, disproved this view by showing that line spectra of electrons 
were associated with some a-ray emitters, which possess no continuous 
/3-ray spectrum. Both Ellis and Meitner resolved the conversion electron 
groups from the LI, Ln, I/m and M\ } Mu, Mm subshells, as well as Ni and 
shell conversions, and proved the rigorous validity of Eq. (5.1). 
Meitner especially emphasized the fact that the K L difference has to be 
the same as the K a X-ray photon energy and hence is a direct measure 
of the atomic number of the atom in which the nuclear transition takes place. 
This proved, for example, that the conversion electrons and the y rays 
associated with the ft rays of ThB (g2Pb 212 ) in fact are emitted from the 
decay product ThC (saBi 212 ), hence that the 7 decay follows the ft decay 
in this radionuclide. 

In the complete absence of any theory of the probability of 7-ray 
transitions, the similarity of Eq. (5.1) with Einstein's photoelectric equa- 
tion became the basis for Ellis's and Meitner's interpretation of the line 
spectrum of electrons as due to an "internal photoelectric effect. " This 
model presumed that the nucleus first emits a photon but that this 
photon is absorbed photoelectrically in the inner electron shells without 
ever escaping from the emitting atom. This model was all right ener- 
getically because it does lead to Eq. (5.1) ; ne*-*prtheless this model is quite 
incorrect. Its disproof lies in the agreement between experimental and 
modern theoretical values of the "internal-conversion coefficient." The 
simplest decisive situation is the > transition which proceeds readily 
enough by internal conversion within the nuclear volume although the 
emission of photons by the nucleus is completely forbidden. 

a. Internal-conversion Coefficient. After the development of the 
quantum mechanics, Taylor and Mott (T10) first clearly pointed out in 
1933 that the theoretical probability of the "internal photoelectric effect" 
was generally negligible compared with that of the "direct-coupling" 
mechanism of internal conversion. The quantum mechanics was able to 
provide a theory for the relative probability of internal conversion by 
direct coupling as compared with the probability of photon emission. 

Let the decay constant X Y represent the probability per unit time for 
the emission of a photon, whose energy is W = hv, by a radiative nuclear 
multipole transition. Let the decay constant X e represent the probabilit} r 
per unit time that this same nuclear multipole field will transfer its 
energy W to any bound electron in its own atom. Then the total 
internal-conversion coefficient a is defined as 

Xe ./Ya /r o\ 

a = = w (5.2) 

XT Ny 

where experimentally N e and N y are the numbers of conversion electrons 
and of photons emitted in the same time interval, from the same sample, 
in which identical nuclei are undergoing the same nuclear transformation 
characterized by the energy W. The total transition probability X is 

X - X 7 + X. = X T (1 + a) (5.3) 

and the total number of nuclei transforming is JV 7 + N e . 



5] Nuclear Effects of Nuclear Moments 221 

The theoretical value of the internal-conversion coefficient depends on 

W, the energy of the transition 

Z, the atomic number of the transforming nucleus 

I, the multipole order of the transition 

parity-favored (electric multipole) or parity-unfavored (magnetic 

multipole) 
atomic shell (K, Li, I/n, . . . , J/j, Jl/n, . . .) in which conversion 

takes place 

Happily, the conversion coefficient for each atomic shell does not depend 
on the value of the nuclear electric or magnetic multipole moment for the 
transition, because this moment enters both X and X T and cancels out 
when only the relative transition probability a is sought. 

A potentially confusing residue of the disproved "internal-photo- 
electric-effect " model of internal conversion is the occasional reappear- 
ance of the pre-1933 definition of internal-conversion coefficient, which 
we may call a p , where a p = N e /(N c + N y ). Then a = a p /(l a p ). 
The possible values of a T are only < a p < 1, in comparison with 
< a < oo. Expressions such as "80 per cent converted" mean 
a p = 0.8, hence a = N e /N y = 0.8/0.2 = 4, not a = 0.8. 

b. K -shell Conversion. Equation (5.2) represents the total internal- 
conversion coefficient a, which in fact is made up of the sum of individual 
coefficients acting separately for each atomic subshell. In present experi- 
mental work, such as Fig. 5.1, conversion in the separate subshells is 
usually not resolved; the L T , Ln, Lm conversions therefore often are 
lumped as L conversion. Accordingly, the shell-conversion coefficients 
become important in theory and experiment, where 

a = a K + CL L + a* + - (5.4) 

and a K is the shell -con version coefficient for both K electrons, a L is the 
shell-conversion coefficient for all L electrons, and so forth. 

Exact theoretical values of a K , a L , . . . cannot be expressed in 
closed form. A number of approximate formulas have been developed. 
Although their usefulness is limited, they have served for many years 
as rough guides. One helpful example is the relativistic treatment by 
Dancoff and Morrison (D2) whose result, for a transition energy W 
which is small compared with m c 2 , and neglecting the binding energy 
of the K electron, reduces to 



W ) 

for two K electrons, and m c 2 W ^> B K . Equation (5.5) applies only 
to electric multipoles, of order 2', with I = 1, 2, 3, . . . . For magnetic 
multipoles (parity-unfavored, as shown in Table 4.2), the TiC-shell internal- 
conversion coefficient for raoc 2 )> W y> B K reduces to 

(5-6) 



222 



The Atomic Nucleus 



[CH. 6 



Both Eqs. (5.5) and (5.6) involve the Born approximation in their 
derivation. Therefore their validity is further restricted by the usual 
Born condition [Z/137(v/c)] 1, where v is the emission velocity of the 
conversion electron. 

Qualitatively, Eqs. (5.5) and (5.6) bring out several essential points. 
They correctly imply a strong increase of a K with increasing multipole 
order, hence with increasing angular-momentum change A/ in the nuclear 
transition. Also a K increases strongly as Z increases and as W decreases. 




0.1 0.15 o.2 0.3 0.4 0.6 1 -5 

Nuclear transition energy, in Mev 

Fig. 5.2 7v -shell internal-conversion coefficients (<UK)*\ and 
/ = 1 to 5. [Fro?n tables by M. E. Rose et al. (R32, R31).] 



3 4 



for Z = 40 and 



The conversion coefficients for electric and for magnetic multipoles vary 
in a slightly different way with I and W, and usually, but not always, 
(a A -)ma K > (c* A ')ei for the sa,mG W B,i\d I (= A/). Sec Fig. 5.2. 

Internal conversion was well known in the natural radioactive nuclides 
long before the discovery of the artificial /3-ray emitters in 1934. The 
strong Z dependence (~ Z 3 ) made the observation of internal conversion 
in the new low-Z artificially radioactive bodies seem unlikely. Alvarez 
(A23) obtained the first experimental evidence of internal conversion in 
artificial radionuclides in connection with his experimental proof of the 
existence of electron-capture transitions, such as in Ga 67 . With the 
gradual improvement of experimental methods, internal conversion has 



5) 



Nuclear Effects of Nuclear Moments 



223 



become a process of first-rank importance in the study of the angular 
momentum and parity of nuclear energy levels (G25, M33). 

Exact theoretical values of (a A Oai and (a*)* for 1 < Z < 5, 10 < Z 
< 96, and 0.3m c 2 < W < 5w c 2 have been obtained with an automatic 
sequence relay calculator by M. E. Rose and coworkcrs (R32, R31). 
These exact numerical values cover a domain of about 10 8 7 from a K ^ 10~ 6 
for small Z, small Z, and large W, to ~ 10 2 for large Z, large I, and small W. 

Figure 5.2 shows the strong dependence of (a A -)ei and (a K ) m * g on the 
transition energy, and on the multipole order, for the particular case of 
Z = 40 (zirconium). The shapes of the curves are qualitatively similar 
for other values of Z. Quantita- 
tively, Fig. 5.3 depicts the increase 
of O.K with Z for four illustrative 
cases. 

c. L-shell Conversion. If the 
transition energy is adequate, that 
is, W > B K , conversion is usually 
more probable in the K shell than 
in the L shell, because the A' elec- 
trons have the greater probability 
of being near the nucleus. Approx- 
imate calculations of a. L have been 
made by Hebb and Nelson and 
others (H27, G16). Exact calcula- 
tions by M. E. Rose et al., includ- 
ing the effect of screening by the K 
electrons, are in progress (R32). 

K/L Ratio. It is well estab- 




20 



lished both experimentally and the- 
oretically that a L depends on I, W, 
and Z in a markedly different way 
than does a K . Then the so-called 
" K/L ratio" a K /a L becomes also a 
function of W, Z y and the multipole 



40 60 80 
Atomic number Z 

Fig. 6.3 Variation of A'-shell internal- 
conversion coefficients with atomic, num- 
ber for two common types of multipole, 
TC2 (A/ = 2, no) and M4 (A/ =4, yes), 
and for two representative values of the 
transition energy. [From tables by AT. 
E. Kosectal. (R32, R31).| 



order of the transition. This im- 
portant point was first emphasized by Hebb and Nelson (H27), who also 
made approximate calculations of ot L and of a K /cx L under the same simpli- 
fying assumptions as those of Eqs. (5.5) and (5.6). 
Experimental determinations of the K/L ratio 



N L 



(5.7) 



where NK and NL are the relative number** of K -conversion electrons and 
of L-conversion electrons, are much simpler and more reliable than abso- 
lute measurements of either a* or a fj . Figure 5.1 illustrates the direct- 
ness with which a/c/a/, can be determined. Such methods provide one 
of our best procedures for determining the angular-momentum difference 
A/ between levels in the same nucleus. This approach has both exoeri- 



224 



The Atomic Nucleus 



[CH. 6 



mental and theoretical advantages. Experimentally, the difficult deter- 
mination of N 7 is made unnecessary because it cancels out. On the 
theoretical side, the uncertainties surrounding the estimation of the 
nuclear multipole moments are also circumvented, because these moments 
cancel out in the definition of the total internal-conversion coefficient, 
Eq. (5.2), and of the shell-conversion coefficients, Eq. (5.4). 



4 




Ba 



10 



15 



20 



25 



30 



35 



40 



45 



50 



Fig. 6.4 Empirical values of the ratio of conversion electrons K/(L + M) * ax/ 
(OIL + ttAf) for transitions which have been otherwise identified as M4 (A/ = 4, yes). 
E is the transition energy in kcv; Z is the atomic number. [From Graves, Langer, and 
Moffat (G41).] 

Two generalizations can be made. First, as I = A/ increases, OIL 
becomes more pronounced in comparison with a K . Thus, for the same 
W and Z, 

decreases as A/ increases (5.8) 



Second, as I = A/ increases, the decrease in a K /a L is more pronounced for 
electric 2 z -pole transitions than for magnetic 2 z -pole transitions. Thus, 
for the same W, Z, and A/, 



(5,9) 



a/./ ei 



The experimental values of the K/L ratio range between 10 (large W, 
small A7, small Z) and 0.1 (small W, large A/, large Z). 



5] Nuclear Effecls of Nuclear Moments 225 

Pending completion and testing of the exact theoretical values of 
a L and OLK/OLL, empirical values of the K/L ratio have been accumulated 
(G27, G41) on over 60 transitions, including El to E5 and Ml to M4, in 
which A/ and parity can be established from other types of experimental 
evidence or can be inferred from the single-particle shell model. An 
example of this procedure, for the case of Cs 137 > Ba 137 , will be discussed 
in Sec. 7. Figure 5.4 is a representative compilation of empirical ratios 
for M4 transitions in which a L and OL M are treated as unresolved. 

d. Pair Internal Conversion. The creation of a positron-negatron 
pair in the external field of a nucleus is energetically possible whenever 
more than the rest energy 2m c 2 (= 1.02 Mev) of two electrons is avail- 
able. If the nuclear excitation energy W exceeds 2m c 2 , then nuclear 
deexcitation can occur by an additional process, related to internal con- 
version, in which an electron is lifted from an occupied negative energy 
state into the continuum of possible positive energy states. The result- 
ing "hole" in the negative energy sta-tes is the experimentally observed 
positron, while the electron in a positive energy state is the negatron 
member of the observed positron-negatron pair. 

The energy W of the nuclear transition then appears as a positron- 
negatron pair, whofae total energy is 

W = E+ + E- + 2moc* (5.10) 



where E+ and E- are the kinetic energies of the positron and the negatron. 

Momentum is conserved between the nucleus and the electron pair. 
Although the pair internal-conversion process (or, synonymously, internal 
pair formation) can take place anywhere in the coulomb field of the 
nucleus, the probability is greatest at a distance from the nucleus which 
is of the order of (Z/137) 2 times the radius of the K shell of atomic elec- 
trons (J5). 

The energy distribution of E+ tends to be symmetric with EL for 
small Z. For large Z there is a strong preponderance of high-energy 
positrons (J5, R34) as a consequence of the action of the nuclear coulomb 
field on the pair. 

The angular distribution is strongly peaked in favor of small angles, 
# > 0, between the directions of the emerging positron and negatron 
(R34, R30). 

In the deexcitation of a nuclear level for which W > 2m Q c 2 1 the 
processes of ordinary (atomic) internal conversion and of pair internal 
conversion compete with and supplement one another, and both compete 
with 7-ray emission. The absolute probability of internal pair formation 
is greatest where the probability of ordinary internal conversion is least, i.e., 
for large W, for small Z, and for small I. For Z ~ 40 and W ~ 2.5 Mev, 
the two processes are of roughly equal importance, the pair internal- 
conversion coefficient being between about 1.0 X 10~ 3 pair per photon 
for an El transition, and 1.8 X 10~ 4 for an E5 transition. The absolute 
value of the pair-conversion coefficient is almost independent of Z; 
indeed, it decreases slightly with increasing Z. Convenient graphs of the 
pair internal-conversion coefficient, from W 1.02 Mev to 10.2 Mev, 



226 



Tlie Atomic Nucleus 



[CH. 6 



15 



d. 



I 



MI,Z=B4 




and for both electric and magnetic multipoles from I = 1 to 5 inclusive, 
have been published by M. E. Rose (R30). Figure 5.5 shows illus- 
trative values of the pair internal-conversion coefficient. 

Experimentally, the radioactive 
nuclides provide only a few 7-ray 
transitions whose energy is great 
enough (say, > 2 Mev) to make pair 
internal conversion an important 
process in competition with ordi- 
nary atomic internal conversion. 
Among these, Alichanowetal. (A 13, 
LI 2) first reported that RaC emils 
about, three positrons per 10 4 7 rays 
in its 1.7fi-Mov and 2.2-Mev transi- 
tions, and that the 2.02-Mev 7-ray 
transition which follows the /? decay 
of ThC" emits about four positrons 
per 1C) 4 7 rays, in agreement with 
the theory of Jaeger and Hulme 
and the known E2 character of this 
transition. Pair internal-conver- 
sion coefficients have been measured by Slatis and Siegbahn (S4(>) for the 
following well-known 7-ray transitions: 

TABLE 5.1. PAIR INTERN AL-CONVJCHSJON COEFFICIENTS 



l 



Fig. 5.5 



2 3 

Transition energy W, in Mev 

Representative theoretical 
curves and experimental points? for the 
pair internal-conversion coefficient. 
Note the strong contrasts with the ordi- 
nary internal-conversion coefficient, as 
plotted in Figs. 5.2 arid 5.3. [From Rlatis 
(846).] 



Parent 
radioimclide 


Transition 
energy VF, 
Mev 


Number of 
positrons 
per 7 ray 


Multipole 
character 
of transition 


ThC" (H,T1 20B ) 


2.62 


4 3 X 10 4 


E2 


Co 80 


1.33 


Detectable 


E2 


Co 60 


1.17 


Detectable 


E2 


Mn" 


2.13 


4.(i X 10 4 


? 


Mn" 


1.81 


5 6 X 10~ 4 


? 


Na 24 


2.7G 


SOX 10~ 4 


E2 




i 




Na 24 


1 38 


3 X 10~ 4 


E2 



Figure 5.6 shows the energy spectra of the positrons which compete with 

0,-> 

the 2.76-Mcv and 1.38-Mev 7-ray transitions in Na' 24 > Mg 24 , as deter- 
mined in a careful experimental study of the shape and relative intensity 
of the positron and negatron spectra by Bloom (B80). 

At even higher energies, the pair internal-conversion coefficient is 
rather insensitive to the multipole order of the transition (R30). Hence 
there has been little incentive to study internal pair formation in the 
higher-energy transitions which accompany some nuclear reactions. 

e. > Transitions. Internal Conversion within the Nuclear Vol- 
ume. For any Z, there is a contribution to the matrix element of internal 



5] 



Nuckar Effects of Nuclear Moments 



conversion by the region within the nucleus, r < R. This contribution 
is usually negligible in comparison with the region outside the nucleus, 
r > R. But in the special case where 7 = for both the initial and final 
states there is no electromagnetic field outside the nucleus (because I = 0) 
and consequently no internal conversion in the extranuclear region. 
Then the energy transfer to an atomic electron (e.g., to an 5 electron 
from the K shell or L shell) can take place only inside the nucleus. Thus, 



1.39 Mev 



4.14 Mev 



1.38 Mev 




0.5 1.0 1.5 f 20 

Kinetic energy of positrons, in Mev 1-74 Mev 

Fig. 5.6 Energy spectra of the positrons produced by pair internal conversion, in 
competition with the 2.76-Mev and 1.38-Mev 7-ray transitions which follow the ft 
decay of Na 2 ' 1 . The area under these curves, when compared with the area under the 
associated negatroii spectrum of ft rays and pair conversion negatrons, gives for the 
pair internal-conversion coefficients 7.1 X 10~ 4 for the 2.76-Mev transition and 
0.6 X 10~ 4 for the 1.38-Mev transition. [From S. D. Bloom (B80).] The predomi- 
nant mode of decay of Na 24 is shown in the inset. Both the - f transitions are E2 
(based on K-shell internal-conversion coefficients, pair internal-conversion coefficients, 
and T-T angular correlation); the ft transition is allowed (log .ft = 6.1). A competing 
feeble (0.003 per cent) 4. 17- Mev, second-forbidden ft transition is omitted in the 
figure, as is also a 0.04-per cent crossover E4 -y-ray transition. 

in > transitions, ordinary single quantum y-ray emission is absolutely 
forbidden. Internal conversion can take place, but the probability per 
unit time is small because the region in which the energy transfer can 
take place is restricted to the interior of the nucleus. The > transi- 
tions are distinguished experimentally by the emission of conversion 
electrons and the complete absence of 7-ray emission. Thus the internal- 
conversion coefficient is infinite. 



228 The Atomic Nucleus [CH. 6 

Emission of a Single Nuclear Internal-conversion Electron. The mean 
life r e _ for a > transition, if both levels have the same parity, is 
approximately (B68) 



where all symbols have their customary meaning, and W B e is the 
kinetic energy of the emerging conversion electron. Equation (5.11) 
says that for A ~ 64, Z ~ 30, and W ~ 1 Mev, the mean life for K-shell 
conversion within the nuclear volume in a 0, no, transition is ~ 5 
X 10- 9 sec. 

Experimentally, the classical case of a >0, no, transition is the 
1.412-Mev excited level in RaC' ( 8 4Po 214 ), which was first studied by 
C. D. Ellis (E9). This level is one of some 12 known excited levels in 
RaC', all of which are produced in the ft decay of RaC, and all of which 

emit long-range a rays (see Chap. 16) in the transitions of RaC' > RaD. 
In addition to a rays, 7 rays and conversion electrons are observed from 
all these levels except the one at an excitation energy of 1.412 Mev. 
This one level, in its transitions to the ground level of RaC', emits con- 
version electrons but no 7 rays. The mean life of the level is markedly 
increased by the prohibition of 7 transitions. Although the ^-branching 
ratios of RaC are not fully studied, it is noteworthy that the 1.412-Mev 
level in RaC' emits more than twice as many long-range a rays as all the 
other 11 excited levels put together. 

A second experimental example of the > 0, no, transition appears 
to be the 0.7-Mev level in 32 Ge 72 . This level is produced in an ~ 1 per 
cent branch in the ft decay of 3 iGa 72 . Conversion electrons are observed, 
but no 7 rays, and the level has a mean life of 0.3 /zsec as measured by 
delayed coincidence techniques (G25). The ground level of azGe 72 is 
presumed to be 7 = 0, even, because of the even-Z even-N composition 
of this nucleus. Because of the absence of 7 rays, the 0.7-Mev meta- 
stable level is also assumed to be 7 = 0, even. At present, this is the 
only known case of nuclear isomerism which is attributable to the slow- 
ness of > transitions. 

Nuclear Pair Formation. When the transition energy W exceeds 
2moc 2 , deexcitatiori by positron -negatron pair production can occur in the 
> 0, no, transitions. Exactly as in the case of nuclear /ST-electron 
conversion, the pair can be produced only within the nuclear volume, 
because there is no multipole field external to the nucleus. These 
"nuclear pairs 1 ' can be distinguished experimentally from pairs produced 
outside the nuclear volume by the ordinary process of pair internal con- 
version. The nuclear pairs have a much narrower energy distribution 
(05, 03). 

The one clear and carefully studied case of nuclear pair formation in a 
> 0, no, transition is the 6.04-Mev-excited level in O 16 . Because of the 
low Z and high W, no single-electron "nuclear" internal conversion has 
been observed in this instance. The level is produced by the F 19 (p,a)0 18 



6] Nuclear Effects of Nuclear Moments 229 

reaction, has a measured mean life of about 7 X 10~ n sec (D36), and 
emits positron-negatron pairs whose total kinetic energy is 

6.04 - 1.02 = 5.0 Mev 

The moderately extensive literature on this transition has been summar- 
ized by Bennett et al. (B34) and by Rasmussen et al. (R9). Figure 5.7 
shows the narrow momentum distribution of the positron members 
of the nuclear pairs which are produced in this 6.04-Mev > 0, no, 
transition. 

> 0, Yes, Transitions. The tw<5 types of nuclear internal conver- 
sion which we have just discussed apply only to > transitions in 

Mev 
0.20.4 0.81.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 

1 X 




4,000 8,000 12,000 

Bp in gauss-cm 



16,000 



Fig. 6.7 Momentum spectrum of the positrons from the "nuclear pair" formatioi 
transition in O 1B , The dotted curve shows the broader and flatter distribution which 
would be expected if these pairs had been produced in the external field of the O 1A 
nucleus by pair internal conversion of dipole radiation (compare Fig. 5.6). [From 
Rasmussen, Horny ak, Lauritsen, and Lauritsen (R9).] 

which there is no change in parity. A > transition between levels 
of different parity cannot occur at all by internal conversion. However, 
the mean life for such a transition is not infinite, as deexcitation can be 
accomplished by various processes involving the emission of two simul- 
taneous radiations, such as two quanta, or one quantum and one internal- 
conversion electron. The probability of these processes is extremely 
small, and they do not compete effectively except for the 0, yes, 
transition, for which both y radiation and internal conversion are abso- 
lutely forbidden. 

6. Nuclear Isomers 

The existence of iso topic isobars (same-Z, same-A), with clearly dis- 
tinguishable properties such as different radioactive half-periods, was 
anticipated in 1917 when Soddy proposed that such nuclei be called 



230 



The Atomic Nucleus 



[CH. 6 



isomers if and when found. The prediction that some nuclei would be 
found which have one or more long-lived excited levels was first made on 
sound theoretical grounds by Weizsiicker (W27), who pointed out in 1936 
that 7-ray decay of levels whose excitation energy W is small should be 
delayed by an easily measurable amount if the angular-momentum 
change A/ is large. 

a. Long-lived Metastable Levels. Experimentally, the first case of 
nuclear isomerism was discovered in 1938, when Feather and Bretscher 
(F14) unraveled the interlocked decay schemes of UXj, UX 2 , UZ, 
and UII UTh 234 , 9 jPa 234in , fl ,Pa 234 , and 92 U 234 ) and showed that UX 2 
(7 7 J = 1.14 min) is only a long-lived excited level of UZ (T^ = 6.7 hr). 



AU 




i i i i i i i i i 


8 


- 


.... Qdd-JV 


"I 




i 


I 




i 


mber of isoi 

* 





ii 
ii 

u ::: 
: ;: 


' 1 ! 

1 '''" 

'! ! ! 


3 


i 



i ! j!i !|l ji 


2 


1- 


|| Jljjjj ;! 













i! ijlii! i mi n 
ii Illlli i iiii n n 
!! Hill! ! !!!! !! '! 



20 40 60 80 100 

Number of odd nucleons, Z or N 



120 



140 



Fig. 6.1 The frequency distribution of odd-A isomeric pairs displays "islands oi 
isomerism" (F18) in which the odd-nucleon number is less than 50, or less than 82. 
The solid bars represent the number of odd-Z even-AT cases, while the cven-Z odd-JV 
isomeric pairs are shown dotted, {The data arc from the tabulation by Goldhaber and 
Hilt (G25).] 

As experimental techniques have improved, a large number of isomers 
have been found and studied in artificial radionuclides and in the stable 
nuclidcs. 

Figure 6.1 shows the frequency of isomerism in odd-^4 nuclides as a 
function of the number of odd nucleons. The so-called islands of isom- 
erism appear here as two principal groups, in which the number of odd 
nucleons is just below the magic numbers 50 and 82. 

By 1951 Goldhaber and coworkers (G27, G25) could classify the 
nuclear properties of 77 isomers for which the half-period of the excited 
metastable level is between 1 sec and 8 months. About half these are 
M4 transitions (A7 = 4, yes), and the remainder are M3, E3, and E4. 
Most of them occur in odd- A nuclides. The correlation of nuclear 



6] Nuclear Effects of Nuclear Moments 231 

angular-momentum values with the single-particle shell model (Chap. 11, 
Sec. 2) is excellent. 

This systematic classification of known isomers, by multipole type, 
has provided important empirical evaluations of the decay constant X Y 
for 7-ray emission and of the internal-conversion coefficients or, a K , and 
a K /&L. Comparison with Weisskopf's theory for X Y , and Rose's calcu- 
lations of a K , shows acceptable agreement, whereas pre-1951 theories were 
valid only in a few limiting cases. 

b. Short-lived Metastable Levels. By the usual definition, an iso- 
meric level is one whose half -period is " measurably " long. The develop- 
ment of experimental techniques utilizing scintillation counters in delayed 
coincidence circuits has made accurate measurements of half-periods 
possible in the microsecond domain. Thus some Hi cases whose half- 
periods lie between 10~ 5 and 10~ a sec have been added to the lists of 
studied "isomers" (G27). This shorter-lived group is made up of Ml, 
M2, and E2 transitions. Experimental techniques for exploration of the 
millisecond domain await development and systematic utilization. 

c. Half-period for Isomeric Transitions. The half-period would be 
that of 7 decay if the excited nucleus were stripped of its atomic electrons. 
Internal conversion shortens the half-period by providing an alternative 
mode of deexcitation. Then, from Eq. (5.3), the half-period T for a 
simple isomeric transition becomes 

In 2 0.693 0.693 , ft1 . 

r * = IT = = T fU 



where r T is T BI or r maR of Fig. 4.1 and Eqs. (4.9) and (4.10). If there is 
branching, i.e., several modes of decay of the metastable level, then X in 
Eq. (6.1) is to be replaced by the sum of the decay constants for all 
competing modes. Because both a and T\ are measurable quantities, 
studies of the isomeric transitions give values of X T for direct comparison 
with the theory of 7 decay. 

d. Experimental Identification of Multipole Order of a 7 Transition. 
The qualitative effects of the change in nuclear angular momentum 
A/ = \I A I B \ can now be summarized. For fixed energy W and charge 
Z, with increasing A/, 

X T decreases a A increases 

X e decreases a L increases (6.2) 

TI increases a^/ai. decreases 

The two general experimental approaches for the evaluation of A/ involve 
measurement of 

a or CLK/OLL (6.3) 

or X T through T h and Eq. (6.1) (6.4) 

Theory and empirical correlations with A/ are now available for the 
quantitative appraisal of most cases (G27, G25). As an illustration, the 
2.6-min isomeric level in Ba 137 is worked out in the following section. 



232 



The Atomic Nucleus 



[CH. 6 




7. Determination of Angular Momentum and Parity of Excited 
Levels from /J- and ^-Transition Probabilities 

We illustrate the conventional methods for determining 7 and parity 
for excited levels, and also for the ground levels of some radionuclides, by 
detailed consideration of the Cs 137 > Ba 187 transitions. Ba 187 is the 
stable end product of the decay of the important radionuclide BsCs 137 , 
whose long half-period (33 yr) and accurately measured monoenergetic 
y rays (0.6616 Mev) make it a very useful and common substance. The 
decay scheme is shown in Fig. 7.1. The ground level of Ba 137 is 7 = 1, 
M = +0.93, therefore df, as was shown in Chap. 4, Sec. 6. 

a. ft Decay of Cs 137 . Two modes 
of ft decay are in competition. The 
high-energy ft transition directly to 
the ground level of Ba 137 occurs in 
only about 8 per cent of the disin- 
tegrations of Cs 137 . This transition 
therefore has a partial half-period of 
33 yr/0.08 ~ 400 yr. This is an ex- 
ceptionally long life for a ft transition 
involving 1.17 Mev. The log ft value 
is 12.2 and corresponds in the theory 
of ft decay (Table 3.2) to a second-for- 
bidden transition, for which the selec- 
tion rule involves no parity change. 
The shape (L5) of the 0-ray spec- 
trum is characteristic of A7 = 2, 
no, in agreement with this assignment 
(Chap. 17, Sec. 3). The angular momentum of the ground level of Cs 137 
therefore has the same parity (even) as the ground level of Ba 187 , and 
an angular momentum 7= + A7 = -2-. 

The low-energy ft transition of Cs 187 has a measured decay energy of 
0.51 Mev and a partial half-period of 33 yr/0.92 ~ 36 yr. The shape 
(O6, L8) of the /3-ray spectrum corresponds in the theory of forbidden 
ft decay [Chap. 17, Eq. (3.22)] to an angular-momentum change of 2, 
accompanied by a change in parity (A7 = 2, yes). This is a first- 
forbidden transition according to Gamow-Teller selection rules and is in 
quantitative agreement with the long half-period, log ft = 9.6. Then the 
excited level at 0.66 Mev in Ba 187 must have odd parity. Its angular 
momentum should be that of Cs 137 2, that is, 7 = 2 = TT, or |. 

b. 7 Decay of Ba 137 . Decision regarding 7 for the 0.66-Mev excited 
level in Ba 137 can be made by several methods. In the first place, 7 decay 
to the ground level is long delayed; the excited level is a well-recognized 
isomeric level whose half-period is 2.6 min. This implies that there is an 
angular-momentum difference of several units between the ground level 
and the 0.66-Mev level in Ba 137 . Among the choices available for the 
0.66-Mev level we must elect 7 = TT- This decision is independently 
established by measurements of the internal-conversion coefficients for 



Stable 



Fig. 7.1 Decay scheme of Cs 137 , with 
angular-momentum and parity assign- 
ments for all levels. 



7] 



Nuclear Effects of Nuclear Moments 



233 



the transition to the ground level. For a 0.66-Mev transition at Z ~ 55, 
the K conversion coefficient is large (a K = N K /N y ~ 0.08) and also the 
K/L ratio is small (a K /a L = 4.6). Therefore, a large change in angular 
momentum is involved. All these data, when interpreted quantitatively 
(Wl, G27, G25), support the conclusion that this transition is a magnetic 
2 4 -pole or, more briefly, an M4 transition (Fig. 5.4), for which the selec- 
tion rule is A/ = 4, yes. This confirms the choice / = Vs odd, for the 
0.66-Mev excited level in Ba 137 . Because the parity is odd, I must be 
odd, and so I = T * = 5. Then the 81st. neutron is in an hy state, 
as shown in Fig. 7.1. 

Knowledge of / and parity for the 0.66-Mev level of Ba 137 permits us 
to estimate M and Q for this level, but there are as yet no measurements 



0- 0.31 Mev 
allowed shape 
log ft =7.5 

= const 




2.50 Mev 



- 1.1715 Mev 
a=1.72xl(T 4 
E2 



1.33 Mev 



/= 1.3316 Mev 
a=1.24xlO" 4 
E2 



Stable 



28 N' 
Fig. 7.2 Decay scheme of Co 80 -> Ni 80 . Sec also Figs. 8.6 and 8.8. 

of the magnetic and electric moments of this excited level with which to 
make comparisons. 

This experimental evidence on the decay and y decay of Cs 137 
* Ba 137 is seen to be sufficient to overdetermine the angular-momentum 
and parity assignments given in Fig. 7.1. In addition, experimental 
confirmation of the g\ character of the ground level of Cs 137 has been 
obtained by direct measurement of I = , M = +2.84, using the atomic- 
beam magnetic-resonance method (D9). 

c. /3-7 Decay of Co 60 Ni 60 . The decay scheme of the very impor- 
tant radionuclide Co 60 has been studied by substantially every available 
experimental method. We shall have many occasions to refer to its 
decay scheme. Figure 7.2 summarizes the presently known data on the 
isomers of Co 60 and on the excited levels in their decay product, Ni 60 . 



234 The Atomic Nadeus [CH. 6 

Under irradiation of Co 69 (/ = ^) with thermal neutrons (D32) the 
cross section for the formation of the 10.7-min isomeric level in Co 60 is 
nearly the same as the cross section for formation of the 5.2-yr ground 
level of Co 60 , both through the reaction Co b9 (71,7) Co 60 . The 10.7-min 
level has an excitation energy of only 59 kev and transforms almost 
entirely to the ground level of Co 60 by internal conversion. The half- 
period of 10.7 min is consistent with an M3 or E3 transition, and the 
K/L ratio of 4.55 implies that this is an M3 transition (magnetic octu- 
pole). The upper isomeric level transforms by j9 decay also; 0.3 per cent 
of the isomeric nuclei go by decay to the 1.33-Mev level in Ni 60 . Less 
than one m 10 6 isomeric nuclei transform by decay to the ground level 
of Ni 60 (D32). These data are consistent with assignments of 2 + and 5+ 
for the angular momentum and parity of the isomeric levels of Co 60 . 
Both transitions are then allowed. In Ni 60 , the 2.50-Mev "crossover" 
7 ray is not observed and has an abundance less than 2.5 X 10~ 7 of that 
of the 1.17 plus 1.33-Mev cascade (F55). 

Problems 

1. On a basis of the single-particle shell model, make scmiempirioal estimates 
of the magnetic dipole moment p and the electric quadrupole moment Q of the 
isomeric level at 0.66 Mev in Ba 137 , knowing thai this level is Jh,\ with odd parity. 

2. Look up the data on the nuclides (V 3b and Ba 13fc , especially the determina- 
tions of angular momentum and parity for the levels which are analogous to those 
of Cs 137 > Ba 137 shown in Fig. 7.1. Account qualitatively for the similarities 
and the differences. 

3. Compute the theoretical mean life of the 2.50-Mev excited level in J\ T i G0 . 

8. Angular Correlation of Successive Radiations 

The photons emitted by a sample in which a large number of nuclei 
are undergoing identical 7-ray transitions will be isotropic in the labora- 
tory coordinates. There is no preferred direction of emission for the 

7-ray photon from the individual transition I A * IB, because the atoms 
and nuclei are oriented at random. The same is true for a-ray, /3-ray, 
and conversion-election emission. If the transition I A > IB is followed 
by a second transition I B > Ic, the individual radiations from the second 
transition are likewise isotropic in the laboratory coordinates. 

However, in a two-step cascade transition, such as I A I B > /r, 
there is often an angular correlation between the directions of emission 
of two successive 7-ray photons, 71 and 72, which are emitted from the 
same nucleus. Often there are similar angular correlations for other pairs 
of successive radiations, such as a-y, 0-7, /3-e~ (where e~ means a conver- 
sion electron), 7-e~, .... Many of the details of the complicated 
theory of these angular correlations have been worked out. Experi- 
mental and theoretical developments have been summarized in a number 
of excellent review articles (D29, F66, B67). 

The existence of an angular correlation arises because the direction 
of the first radiation is related to the orientation of the angular momentum 



8] Nuclear Effects of Nuclear Moments 235 

IB of the intermediate level. This orientation can be expressed in terms 
of the magnetic-angular-momentum quantum number m B with respect 
to some laboratory direction such as that of the first radiation. If I s 
is not zero, and if the lifetime of the intermediate level is short enough 
so that the orientation of I B persists, then the direction of emission of the 
second radiation will be related to the direction of I B and hence to that 
of the first radiation. 

a. Dipole-Dipole Angular Correlation. Any pure 7-7 cascade can be 
represented in the obvious notation /A(/I)/A(^)/C, where /i and l z are the 
angular momenta of the two succes- 
sive 7 rays. As the simplest pos- / fl =l ^.-^"~ ? ^n 3 "* 

sible example, we shall consider two : \~ 






successive dipolc 7 rtiys (Zi = Z 2 = 1) 
in the cascade 0(1)1(1)0. 

At first, imagine that measure- 
ments are made only on the second 
7 ray, and that the source is in a 
magnetic field H, which serves here Zerof 'e' d Held H 



m a =-l 



, 

to give a fixed direction in the lab- f* 8 ; 1 M *F*^ ^ lcvel ? m * ! the 

oratory. The magnetic sublevels ^vcl / B = 1. Transitions /,- /c m- 

l , ., * ,1 -11 i volvc Uie emission of the dipoles I. rn 

m. = 0, 1 of the excited level with , _ ^ and m _ ^ _ mg _ +1> 

IB = 1 are shown schematically in o or 1 

Fig. 8.1. The relative populations 

of these sublevels will be proportional to the Boltzman factor 



where /i = magnetic dipole moment, nuclear magnetons 
IJLM = nuclear magneton 

k = Boltzman constant 

Equation (S.I) is substantially unity except at the strongest fields H and 
lowest temperatures T which are now available. Otherwise the mag- 
netic energy n(m B / 'In)puH is negligible compared with the thermal 
agitation energy k7\ The magnetic sublevels are therefore equally popu- 
lated under ordinary experimental conditions, and the transitions Am = 0, 
+ 1, and 1 will have equal abundance. 

The angular distribution of the intensity of electric multipole radia- 
tion is given by known analytic functions (p. 594 of B68, p. 251 of Sll) 
and is the same for electric 2 f -pole and magnetic 2 z -pole radiation. The 
electric and magnetic multipoles differ mathematically only in their par- 
ity and physically only in the orientation of their plane of polarization. 
For dipole radiation the angular distribution is 

w n (&) d$l = sin 2 rffi for m = (8.2a) 

oTT 

dtt = (1 + cos 2 0) d$l for m = +1 (8.26) 
16?r 

(in = -~(\+ cos 2 0) dfi for m = -1 (8.2c) 



236 



The Atomic Nucleus 



[CH. 6 



where & is the angle between H and the direction of emission of the 
photon and w(&) is the probability per unit solid angle that the photon 
will be emitted into the solid angle dfi at #. Equation (8.2a) is the 
common expression for the intensity of radiation from a classical linear 
dipole, when tf = denotes the direction of the axis of the dipole. 

If the states are equally populated, the total angular distribution 
is the sum of Eqs. (8.2), which is a constant. Hence, even in the pres- 
ence of the ordinary magnetic field, the total radiation is still isotropic. 
We see that the ability to observe anisotropy in the angular distribution 
depends on our ability to obtain a nonuniform population of magnetic 
sub states. This can be done most simply in the dipole case of Fig. 8.1 
if we can arrange experimentally to exclude observation of the m = 
transitions. The angular distribution of the m = 1 transitions would 
then have a (1 + cos 2 1>) distribution. 



ni A =0 



'A"-I 




m, 1 


L, 


> 

> 

> 


Ml!- 

' 


r+ 

t 


1 


i 

m,- 


-1 


| 


i 


i 


; 






1 
1 


> 1 


r, 




a" 1 


1 




\ 


F 



m fi + l 
m B =0 
mj,= -l 





m 2 --l 



. m c =0 



/C-0 * 

Fig. 8.2 Method of exciting the mag- 
netic Bublevels of IB by means of a pre- 
vious dipole transition from I A 0. 



Fig. 8.3 In the dipole-dipole, 71-79 
cascade, 0(1)1(1)0, the intermediate 
state has one unit of angular momen- 
tum, which must be annulled by the 
emission of the second quantum. 
This requirement imposes an angular 
dependence on the direction of emis- 
sion of the second quantum relative 
to the first. 



We can do this experimentally by forming the m B = sublevel in a 

preceding transition I A > IB, as shown in Fig. 8.2. No external mag- 
netic field is used. In its absence, the m B sublevels are degenerate, and 
the transition probabilities from I A = 0, m A = 0, to each of the m B levels 
are equal. Also the direction of tf = in the laboratory is arbitrary, 
and we will now take it as the direction of emission of the first quantum 
7i. The first transition cannot lead to the sublevel m B = in this 
particular coordinate system, because by Eq. (8.2a) its intensity in the 
& = direction is zero. All the photons in the # = direction therefore 
correspond to mi = 1 transitions. The nil = +1 transition has to be 
followed by m z = 1, and mi = 1 by m 2 = +1, in order to reach 
I c = 0. Both these second transitions have a (1 -\- cos 2 #) distribution, 
by Eqs. (8.2b, c). Thus if 71 is detected in a counter whose direction 



8] 



Nuclear Effects of Nuclear Moments 



237 



from the source is called # = 0, the probability that 72 will traverse a 
second counter set at an angle tf will vary as (1 + cos 2 #). 

These considerations can be reduced to pictorial terms as in Figs. 8.3 
and 8.4. In Fig. 8.3, if 71 is emitted in the z direction (tf = 0), its dipole 
character (l\ = 1) requires it to leave the residual system with one unit 
of angular momentum normal to the direction of propagation. (This 
corresponds in the previous argument to mi = 1.) Let the arbitrary 
direction of this unit of angular momentum be chosen as the x axis. 
The second radiation 72 must remove this unit of angular momentum, in 
order to reach I c = 0. The direction of 72 can therefore be any direction 
which is normal to the x axis, i.e., in the yz plane. If 7! and 72 are 
detected with instruments which 
are not sensitive to the plane of 
polarization of 71 and 72, then the 
observed 71-72 coincidence counts 
will include all possible orientations 
of the x axis. For unpolarized de- 
tection we must remove the arbi- 
trary selection of the x direction 
by rotating the xy plane about the 
z axis. This averaging operation 
must lead then to the angular cor- 
relation function 




TF(90)(1 + cos 2 



(8.3) 



Fig. 8.4 Spatial distribution of the angu- 
lar correlation for a dipole-dipole, -y-7 
cascade. The length of the radius vector 
W() gives the probability, per unit solid 
angle, that the angle between the direc- 
tions of the two successive photons will 
be tf. The directions of the two photons 
TI and 72 are, of course, interchangeable. 
Note the fore-and-aft symmetry. The 
angle # is actually in center-of-mass co- 
ordinates, but the nuclear recoil from 
T-ray emission is so slight that the dif- 
ference for laboratory coordinates is 
negligible. 



where W(d) is the probability, per 
unit solid angle, that the second 
quantum is directed into the ele- 
ment of solid angle dQ, at any an- 
gle & with the first quantum. This 
spatial distribution is indicated in 
Fig. 8.4, where the fore-and-aft 
symmetry which arises from even 
powers of cos # is evident. This 
fore-and-aft symmetry, in center-of-mass coordinates, is, in fact, charac- 
teristic of most angular correlation distributions. 

Equation (8.3) is applicable only to the 0(1)1(1)0 cascade. There 
are as yet no known nuclear examples of two successive dipole 7 rays. 
Most of the 7-7 cascades which have been measured thus far are 4(2)2(2)0, 
or quadrupole-quadrupole cascades, in even-Z even-JV nuclides such as 
Mg 24 , Ni 60 , etc. 

b. General Case for 7-7 Angular Correlation. The principles which 
we have just outlined can be applied to any 7-7 cascade involving arbi- 
trary multipole orders. The mathematical complications rapidly become 
insuperable unless more sophisticated methods are invoked. Yang (Yl) 
first applied group theory to obtain the form of the general angular- 
correlation function. For the generalized 7-7 cascade /A(ZI)/B(/S) Jc the 



238 The Atomic Nucleus fen. 6 

angular-correlation function W(&) for the angle d between the successive 
7 rays can be shown to be (Yl, F5, B67) 

i-Z, 

W(t) dQ = ^/Vros *) dfl (8.4) 

7=0 

where A zi are coefficients which depend on l\ and Z 2 , and Pa, (cos tf) a,re 
the even Legendrc polynomials. While this form is convenient for the 
theory, an equivalent and more common form is usually used for com- 
parison with experiments This is a power scries in even powers of cos tf, 
and normalized to W (90) = 1, as follows 

W(ff) dfi = (1 + a 2 cos 2 & + a* cos 4 tf + - - + a 2L cos 2L 0) dQ (8.5) 

where the coefficients a 2 , 04, ... are functions of the angular momenta 
I Aj I Pl I c , hj and 1 2 but not of the relative parity of the levels. 

There are rigorous restrictions on the number of terms in EqF. (8.4) 
and (8.5) ; the highest even power of cos $ is determined by /i, I B , or 1 2> 
whichever is smallest. Thus 2L is not larger than 2/,i or 2/ fl , or 2Z 2 , and 
will be one unit less than the smallest if the smallest is odd. For example, 
if IB = or -J-, W(ti) = 1, and the angular correlation distribution will be 
isotropic. 

A rough over-all index of the complexity of the angular distribution 
is given by the so-called anisotropy, defined as 



Anisotropy = ^7X0^: 1 = a a + a 4 + + ZZL (8.6) 

This is a convenient quantity experimentally as it involves measure- 
ments for only two values of #, corresponding to "back-to-back" and 
"normal" directions. However, this simple index may conceal some of 
the true complexity of the distribution, because the coefficients a 2 , a 4 , 
. . . can have negative as well as positive values. 

We shall see that Eqs. (8.4) and (8.5) are very general indeed and 
that with appropriate evaluation of a 2 , a 4 , . . . they apply to all two- 
step cascades, a-y, p-y, 7-7, 7-6', e~-e~ y etc., as well as to nuclear scatter- 
ing experiments and nuclear disintegrations. 

The conditions of validity of Eq. (8.4) or (8.5) entail all the assump- 
tions made in its derivation. These are: 

1. The magnetic sublevels m A of the initial level I A are equally popu- 
lated. This is generally true for ordinary radioactive sources at room 
temperature but can be altered deliberately in suitable cases by the 
influence of very low temperatures combined with very large magnetic 
fields, Eq. (8.1). 

2. Each nuclear level I A, IB, and Tr must be a single level with well- 
defined parity and angular momentum. Violations of this condition, 
caused by the occasional overlap of broad nuclear levels at high excitation 
energies, may give rise to interference effects and to the appearance of 
terms in odd powers of cos & in ir(tf). 

3. Each of the radiations li and l z must correspond to a pure multi- 



8] 



Nuclear Effects of Nuclear Moments 



239 



pole. Mixed radiations of opposite parity can give rise to interference 
effects and odd powers of cos #. 

4. Equation (8.5) restricts its attention to the relative directions of the 
two radiations, without cognizance of their states of polarization. There- 
fore Eq. (8.5) applies only to the usual experimental situations in which 
both detectors are insensitive to the plane of polarization of the radiations. 

5. The half-period of the intermediate level I B must be short enough 
to permit the orientation of IB to remain undisturbed. We have seen 
that IB > 1 if any anisotropy exists. The finite magnetic dipole moment 
fji of this intermediate level will therefore give rise to a Larmor precession 
of IB with Larmor frequency v in the field of the atomic electrons or in 



any applied strong external field. 
The half-period T of the interme- 
diate level must be short compared 
with the reciprocal of the angular 
velocity 2irv of the Larmor preces- 
sion if there is to be no influence 
on W(#). 

c. Magnetic Dipole Moment 
of an Excited Nuclear Level. Be- 
tween TI ~ 10~ 8 sec and a lower 
limit of ~ 10~ B sec (set experimen- 
tally by the accidental coincidence 
rate due to the resolving time of 
the coincidence circuits) it is pos- 
sible in a few selected cases to in- 
fluence the angular correlation by 
a known external field arid thus to 
determine the nuclear g factor for 
the excited level. This has been 
done in the case of the 243-kev 



In 111 

49 in 62 



2.8 day 



-'EC 



"* Jr-0.177 
i Ml 


Mev 




y-0.243 
E2 


Mev 



Fig. 8.5 The 7-7 cascade in Cd 111 , follow- 
ing the electron-capture transition In 111 
Cd 111 . The influence of an external mag- 
netic field on the 7-7 angular correlation 
leads to a value of /* ~ 0.7 nuclear mag- 
neton for the magnetic dipole moment of 
the rf s level at 0.243 Mev. Cd 111 also has 
another isomeric level (not shown here) 
lying 0.149 Mev above the d\ level. This 
is an hij. level, produced by Cd 110 (n,7)Cd 11J l 
and decaying to the d% level with a half- 
period of 48 min (G25). 



level in Cd 111 (Fig. 8.5), where the 
reduction in anisotropy with field 
strength (0 to 7,000 gauss), ap- 
plied perpendicular to the plane of 
the two 7 rays, leads to the value 
g = -(0.28 0.05) (F66). The 
ground level of Cd 111 has the directly measured values / = ^, p = 0.595 
and therefore is an ,<?j level in the shell model (Chap. 1 1, Sec. 2). The angu- 
lar momentum of the intermediate level is I B = I from the angular correla- 
tion measurements on W(&) without an external field. The negative 
sign for g makes the intermediate level d* (in agreement with the single- 
particle shell model) and gives for the excited level a't 243 kev in Cd 111 

/*(ds) = (0.7 + 0.1) nuclear magneton 

This is the first measurement of the magnetic dipole moment for an excited 
nuclear level. In magnitude, it is comparable with M of the ground level 
in this case. 



240 



The Atomic Nucleus 



[CH. 6 



d. 7-7 Angular-correlation Coefficients. Hamilton (H14), Falkoff 
(F4), and others have deduced the angular-correlation coefficients o 2 , a 4 , 
... for most of the multipoles expected in 7-7 cascades. As illustra- 
tions, we give in Table 8.1 the coefficients (H14) for some of the possible 
dipole and quadrupole 7-7 cascades for the important case I c = 0, as 
met in even-Z even-TV nuclei. 

TABLE 8.1. ANGULAR-CORRELATION COEFFICIENTS FOR SOME DIPOLE AND 

QUADBUPOLE 7-7 CASCADE TRANSITIONS 

[When the angular-momentum quantum number Ic for the final level, such 
as the ground level of even-Z even-TV nu elides (H14, B107).] 





W^(#) ^^ ~ (1 H~ *a cos 2 i? ~\" o>4 cos 4 i?) dn 


7-7 cascade 




WW/.(W/c 


2 


4 


0(1)1(1)0 


1 





1(1)10)0 


_ i 





1(2)1(1)0 


-i 





2(1)1(1)0 


+T5 





3(2)1(1)0 


-A 





0(2)2(2)0 


-3 


+4 


1(1)2(2)0 


i 





2(1)2(2)0 
2(2)2(2)0 


+ f 5 

TJ 





3(1)2(2)0 


_ 3 





4(2)2(2)0 


+ff 


+A 



Experimentally, the application of scintillation counters to the prob- 
lem of angular correlation of successive 7 rays, by Deutsch and coworkers 
(B107, M20), first gave the required combination of high sensitivity and 
good resolving time which facilitates routine coincidence counting. With 
these techniques W(d) was found to be anisotropic for the 7-7 cascades 
which follow the 0-ray transitions : Na 24 -> Mg 24 , Sc 46 -* Ti 46 , Co 60 -> Ni flo , 
Sr"' -* Y 88 , Rh 106 -* Pd 106 , and Cs 134 -> Ba 134 . 

Figure 8.6 shows the measured dependence of the coincidence count- 
ing rate on tf, for the 7-7 cascade in Ni 80 , and is in agreement with a 
quadrupole-quadrupole transition 4(2)2(2)0. This observation fixes the 
angular momenta of the excited levels at 1.33 and 2.50 Mev in Ni 80 as 
I B = 2 and I A = 4, as was shown in Fig. 7.2. The 7-7 cascades in 
Mg 24 , Ti 48 , Ba 134 , and Ce 140 have also been shown to be 4(2)2(2)0. In 
Hf 177 , the 7-7 cascade appears to be 1(1)1(2)4, while in Cd 111 (Fig. 8.5) 
the 7-7 cascade 0j(l)d 9 (2)sj seems well established (F66). 

e. Parity of Excited Levels. 7-7 Polarization-direction Correlation. 
The 7-7 angular-correlation coefficients of Table 8.1 depend only on 
multipole order and not on parity. This is because, for the same multi- 
pole order, magnetic and electric multipoles have the same angular 
distribution. They differ in their parities and in the corresponding orien- 



8] 



Nuclear Effects of Nuclear Moments 



241 



1,16 



1,12 



1.08 



1,04 



1.00, 




90 



135 



180 



Fig. 8.6 Coincidence counting rate, proportional to W(&), for the -y-y cascade in 
Ni 60 , following the /3 decay of Co ffl . The observations are in agreement with the 
unique angular correlation distribution for a 4(2)2(2)0 cascade (Table 8.1). These 
measurements determine the angular momentum of the 1.33-Mev and 2.50-Mev 
excited levels in Ni 60 , which were shown in Fig. 7.2. [Data from Brady and Deutsch 
(B107).] 

tation of the plane of polarization of the electromagnetic radiation. By 
measuring the orientation of the polarization vector (here defined as the 
vector of the electromagnetic radiation) relative to the plane of the two 
successive 7 rays, it can be determined whether the successive multipoles 
are electric or magnetic (H14). Hence the relative parity of the nuclear 
levels can be measured. 

Experimentally, Metzger and Deutsch (M44) developed a successful 
7-ray triple coincidence polarimeter, shown in Fig. 8.7, and measured the 
polarization-direction correlation of the 7-7 cascades in Ti 46 , Ni 60 , Pd 106 , 
and Ba 134 . 

Hamilton's (H14) theory of the polarization-direction correlation can 
be expressed qualitatively in terms of the measurable ratio ||/_i_, which 
is the ratio of the polarization of 71 parallel (||) to the d plane containing 
the two 7 rays, and perpendicular (_L) to the plane of the two 7 rays. 
In a quadrupole-quadrupole cascade, the polarization-correlation when 
= 90 is 

L > 1 f or E2-E2 (I A and I c have same parity) 

= 1 for E2-M2, or M2-E2 (I A and 7 C have opposite parity) 



^L < 1 for M2-M2 

x 



and I c have same parity) 



242 



The Atomic Nucleus 



[CH. 6 







Fig. 8.7 Tho 7-ray triple-coincidence polarimeter of Metssgrr and Deutsch (M441. 
Triple coincidences are registered between the throe scintillation counters A, B, and C. 

(1) The source <S sends a photon -> j into counter A, where > i projects a Compton 
electron in the scintillutor, thus producing a count in A. The Compton scattered 
photon -)r will be preferentially directed with its electric vector c parallel to the 
electric vector ] of the primary photon ) i [see the Klcin-Nishina formula, Chap. 23, 
Sec. 2, Eq. (2.3) and Fig. 2.2]. Thus if , lies normal to the plane of counters A 
and B, there is a maximum probability that ~yc will be directed toward counter B. 

(2) In counter B, yc must produce a countable secondary electron, either by photo- 
elect ne, absorption or another Compton collision. (3) In counter T, located at angle 
i? from the direction of A and B, the second photon 72 from the 7-7 cascade in the 
source S must also produce a countable secondary electron. Counters C and A form 
a coincidence pair which is insensitive to polarization, as in ordinary angular corre- 
lation experiments. 

Thus for two electric quadrupoles, E2-E2, the plane of the B vector tends to he 
parallel to the plane of the two y rays. For two magnetic quadrupoles 
M2-M2, the plane of the vector tends to lie perpendicular to the plane 
of the two y rays. When tf = 180, ||/fij_ = 1> independent of the 
electric or magnetic character of the successive quadrupoles. 

Figure 8.8 shows the polarization-directi _>n correlation of the cascade 
y rays in Ni 60 , following the ft decay of Co 60 . Unambiguously, Cy/Cj. > 1, 
showing that both y rays are electric quadrupoles. Now this cascade 
can be written more explicitly as 4(E2)2(E2)0. This proves that the 
intermediate level at 1.33 Mev has the same parity as the ground level, 
and HO does also the upper level at 2.50 Mev. Because Ni 60 is an even-Z 

TABLE 8.2. P-y ANGULAR CORRELATION (F5) 

Note that the angular distribution of 18-7 coincidences is isoiropic unless the ft 
spectrum has a forbidden shape. 



|3 Transition 


y Multipole 


WW/W(W) 


Allowed 


Any 


1 


Forbidden by selection rules, but having 
allowed spectrum shape 
First-forbidden . 


Any 
Any 


1 

1 + Oz COS 2 tf 


Second-forbidden . . . 


Dipole 


1 + a 2 cos 2 & 


Second-forbidden 


>Quadrupole 


1 H- O" COB 2 + Q,\ COB 4 # 









Nuclear Effects of Nuclear Moments 



243 



1.10 



1.05 - 



1.00 



0.95 - 



0,90 




90 



even-TV nucleus, it is assumed from the shell model that the ground level 
is I c = 0, even, or / = + . This is 
the experimental basis for the parity 
assignments given this nuclide in 
Fig. 7.2. 

f. 0-7 Angular -correlation Coef- 
ficients. Angular-correlation coeffi- 
cients for the successive emission of 
any two nuclear radiations, such as 
a-7, 0-7, 7-e~, . . . , have been de- 
veloped and tabulated by Falkoff 
and Uhlenbeck (F5) and others. In 
general, the coefficients a 2 , a 4 , . . . 
of Eq. (8.5) are found to be functions 
of the angular momenta of the three 
nuclear levels and the two emitted 
radiations, as well as the interaction 
between the emitted particles and 
the nucleus. 

For 0-7 angular correlations, 
irrespective of the character of the 
interaction, the generalizations 
shown in Table 8.2 apply to the com- 
plexity of the angular-distribution 
function W(d) of Eq. (8.5). Near 
the low-energy end of the spectrum 
there is no -7 angular correlation; 
the strongest correlation occurs for 
rays near the maximum energy of 
their spectrum, where the neutrino 
takes little energy and momentum. 
Anisotropic 0-7 angular correlations 



Fig. 8.8 Polarization-direction corre- 
lation of the, two quadruped r 7 rays in 
Ni 6l) , which follow the ft decay of Co 60 
(Fig. 7.2). The ordiiiates are the ex- 
perimental Iriple-coinridenrc counting 
ratios N^/N. For A'u the polarime- 
ter counters are parallel to the plane of 
the 7 rays (<P = in Fig. 8.7). Thus 
JV|| corresponds to j_. For N, the 
counters are at <? 00 to the plane of 
the 7. The data show JV|j/JVj_ < 1 at 
= 90, hence ||/fi-L > ] - The e l p - 
tric vector tends to lie parallel to the 
plane of the two 7 rays, which must 
both be electric quadru poles. The 
three curves show the different possi- 
ble parity assignments for the two suc- 
cessive transitions, which are known 
from the directional correlation alone 
to be two quadrupoles. [Adapted from 
Metzger and Deutsch (M44).] 



have been observed in K 42 , As 76 , Rb 96 , Sb 122 , Sb 124 , I 126 , Tin 170 , . . . 

Problems 

1. Show that the most probable plane angle, & to d + dtf, between two suc- 
cessive dipole 7-ray quanta, in the 0(1)1(1)0 cascade is about 55. 

2. In a 7-7 angular-correlation experiment show that the directional correla- 
tion will be disturbed if the half-period T of the intermediate level does not 
satisfy the inequality 



where AP is the hyperfine-structure separation for an atom with / = \ t Eq. (1.11) 
of Chap. 5. Determine the restriction on Tj in seconds for a middleweight 
nucleus, such as cesium. Ans.: ^10~ n sec. 

3. Under the same conditions as the previous problem, show that, if a strong 
external magnetic field H is applied, the angular correlation may be influenced if 



244 The Atomic Nucleus fen. 6 

the half-period Tj of the intermediate state is 

2 X 10~ 4 sec 



where M is the magnetic dipole moment in units of the nuclear magneton HM and 
7 is the angular-momentum quantum number of the intermediate level. Deter- 
mine a typical critical value for T if the applied field is 10 s gauss. Ans.: ~10~ B 
sec. 

9. Angular Distribution in Nuclear Reactions 

In any nuclear reaction, such as B^fopJC 1 *, the direction of the 
incident particle provides a reference axis for angular distribution studies. 
If we write out such a reaction in full, some obvious analogies appear 
with the case of the successive emission of two radiations. Thus, in 



5 B 10 + 2 He*^ ON 1 ') - 1 H 1 + 6 C 18 (9.1) 

the excited compound nucleus ( 7 N 14 ) plays the role of the intermediate 
level IB of the previous discussion. Generalizing, we can symbolize a 
large class of nuclear reactions as 

A +a->5->c + C (9.2) 

Elastic- and inelastic-resonance-scattering processes are included by 
noting that a and c may be identical particles. 

a* Channel Spin. In the dissociation of B, the products c and C 
have mutual orbital angular momentum l z . Analogously, in the forma- 
tion of By we may represent by l\ the mutual orbital angular momentum 
between A and a. Each of the four particles A, a, c, C may have a finite 
intrinsic nuclear angular momentum, denoted by the quantum numbers 
/i, *i, 12, /2, respectively. The vector sum of /i and i"i may have any 
value between |/i i\\ and |/i + i"i|, and the particular value which it 
does have is called the entrance channel spin s\. Similarly, the exit 
channel spin * 2 is the vector sum of / 2 and i z for the outgoing particles. 
These concepts and notation can be summarized mnemonically 

(A+a) - B -> (c + C) (9.2a) 

(Ii + ii) + li = Io = U + (ii + Ii) (9.25) 

Si + li = I = 1 2 + s 2 (9.2c) 

where the quantum number 7 denotes the angular momentum of the 
intermediate compound level. 7 is preserved throughout the reaction. 
The analogy is now complete. Our previous notation IA(II)IB(IZ)!C for 
the two-step process now becomes 



and is applicable to all nuclear reactions and scattering processes in 
which a compound intermediate level is formed and has definite parity 
and angular momentum. The angular distribution of reaction products 



9] Nuclear Effects of Nuclear Moments 245 

is measured in terms of cos tf, where # is the angle between the directions 
of a and c, in center-of-mass coordinates. 

b. Elastic Resonance Scattering. We may consider the case of elastic 
scattering between spinless particles without significant loss of generality 
(B67). The differential cross section da for scattering into the solid 
angle d!2, at angle tf, is developed in Appendix C [Eq. (107)] and is 



da = |/(i>)| 2 dfl (9.3) 

where the complex scattering amplitude /(tf ) is a summation of functions 
of the phase shifts d t) which are real quantities whose values depend on 
the nature of the central scattering force C/(r) and on the wave number 
k = I/ft of the colliding particles, as given by Eq. (118) of Appendix C. 
The summation over various partial waves I, and evaluation of 

I/WI 1 =/*(*)/(*) 
can be carried through rigorously and leads to the more convenient form 



da = X 2 t P t (cos tf) dtl (9.4) 

i = 

in which P, (cos tf) are the Legendre polynomials and B t is a real but 
complicated quantity which depends on the angular momenta and the 
phase shifts. There are^ rigorous restrictions which limit severely the 
number of terms in Eq. (9.4), as discussed below. 

c. Angular Distribution for Reactions in Which a Compound Nucleus 
Is Formed. Under conditions of validity which are the complete ana- 
logues of those given for Eqs. (8.4) and (8.5), Eq. (9.4) is applicable to 
all collision processes of the type A + a * B > c + C. As a conse- 
quence of conservation of parity, only the terms involving even powers 
of cos tf are finite if 7 is a pure level with single-valued parity and angular 
momentum. For convenience in comparisons with experiment, Eq. (9.4) 
can be put in the more common form 

OS ' * + ' " ' ] (9 ' 5) 

where the real coefficients A(E), B(E), . . . are complicated functions 
of the energy E of the incident particle, of the angular momenta Si, /o, 
s 2 , h, and Z 2 , and of the nature of the forces between the particles (Yl, 
B67). The highest power, cos 2L tf, is restricted by Zi, 7 0; or Z 2 , whichever 
is smallest, and 2L is not greater than 2/i, 2/o, or 21 2 . These restrictions 
are analogous to those which apply to Eq. (8.5). If /o = or , the 
distribution is isotropic in center-of-mass coordinates. Also, if li = 0, or 
Z 2 = (s waves), the distribution is isotropic. 

Thus, in all nuclear reactions, information on l\ or Jo or Z 2 is obtained 
directly by noting how many terms in cos 2 & are required to match the 
observed angular distribution. Interference terms in odd powers of cos tf 
may enter Eq. (9.5) if the compound state is a mixture of levels of opposite 



246 



The Atomic Nucleus 



[cu. 6 



s 

PQ 

. 
o 


^ 



Li 7 (p,ot)He 4 



A(E) 




parity, and if the incoming or outgoing particle waves contain mixtures 
of opposite parity. Examples of such mixtures have been observed in 
Li 8 (d,p)Li 7 , Li 6 (p,a)He a , and Li 7 (p,n)Be 7 , for example, (AlO). 

d. Angular Distribution for Li 7 (p,a)He 4 . The angular momentum 
and parity of the resonance level at 19.9 Mev above ground in Be 8 was 
given in Fig 1.1 as / = 2+. This is determined by the angular distribu- 
tion of the a rays from the reaction Li 7 (p,a)He 4 7 which has been the 
object of many careful experimental and theoretical studies, although 

several details require further work. 
Here we wish only to show how quali- 
tative interpretation of angular-dis- 
tribution data suffices to determine 
7 = 2+ for the 19.9-Mev level. 

From E = 0.5 to 3.5 Mev, the 
experimentally determined angular 
distribution follows 1 + A(E) cos 2 tf 
+ B(E) cos 4 tf, where A(E) and 
B(E) are empirical coefficients whose 
observed dependence on E is shown in 
Fig. 9.1. The shape of these curves 
can be matched by a detailed theory 
of the reaction (13, H50). Here we 
note only that a cos 4 # term is required, 
and that there is no positive experi- 
mental evidence for a cos 6 tf term. 
Then, from the conditions on the high- 
est power of cos 2 tf, li > 2, IQ> 2, 
and l z > 2. 

We have noted in Sec. 1 that the 
Bose statistics and spinless character 
of the two a particles from this reac- 
tion require Z 2 to be even. Because 
the intrinsic spin of the a particles is 
zero, the exit channel spin is zero, and 
therefore 7 = lz- Any level in Be 8 
which can dissociate into two a parti- 
cles is obliged by conservation of parity and of angular momentum to 
have even parity and even total angular momentum 7<j = 0, 2, 4, . . . . 
In the entrance channel, the ground level of Li 7 is 1 1 = -J, and it has 
odd intrinsic parity in every reasonable nuclear model, while the proton 
has i\ = \ and even intrinsic parity. Thus the entrance-channel spin 
has odd parity and is i = -|~ i+ = 1~ or 2~. This restricts the inci- 
dent orbital angular momentum to the odd values li = 1, 3, 5, . . . , of 
which only l\ = 3, 5, . . . are possible because of the cos 4 # term. The 
absence of a cos 6 & term is therefore not dictated by li but means 7 < 3. 
The only possible assignment for the resonance level is therefore 7 = 2, 
even. 

The yield of the Li 7 (p,a)He 4 reaction, as a function of bombarding 



01234 

(Mev) 

Fig. 9.1 The observed angular-dis- 
tribution coefficients 4C#) and B(E) 
in W(d)/W(9Q) = 1 4* A(K) cos z tf 
+ B(E) cos 4 #, for the reaction 
Li 7 (p,)Hc 4 . E is the kinetic energy 
of the incident protons in laboratory 
coordinates; d is the angle between p 
and <x in ccnter-of-mass coordinates. 
[Data are a composite of several authors 
in various energy ranges (H50, Tl, 
AlO).] 



9] Nuclear Effects of Nuclear Moments 247 

energy E, was shown in Fig. 1.1. The resonance peak in the yield at 
E ~ 3 Mev is attributed to the level with 7 = 2+, at 19.9 Mev above 
ground in Be 8 . Detailed analysis of the influence of E on the yield and 
on A(E) and B(E) indicates that this level, which has a width T at half 
maximum of ~ 1 Mev, is superimposed on a much broader level 7 = O 4 , 
which underlies the whole region. Both levels can be produced by both 
p-wave and /-wave protons (li = 1,3). 

e. Angular Distribution in Photodisintegration of the Deuteron. 
Two mechanisms for the disintegration of the deuteron by 7 rays are 
experimentally distinguishable, if the 7-ray energy is only slightly greater 
than the binding energy (2.22 Mev) of the deuteron. The proton and 
neutron have parallel spins ( 3 Si level) in the ground level of the deuteron. 
The antiparallel-spm state (V^o) ls an excited level which is unstable by 
about 65 kev against dissociation. There is a continuum corresponding 
to a wide 3 P state. The incident photon can be absorbed either as an 
electric dipole or as a magnetic dipole, and these two processes produce 
different angular distributions. 

In the photoelectric disintegration, the deuteron absorbs the incident 
photon as an electric dipole. This involves a change in parity and 
A/ = I (Table 4.2). Hence, in the struck deuteron, AN = 0, AA =- 1, 
yes, and the ('iS'i, even) level is converted to ( 3 P, odd), the dissociation of 
which is observed to have a (1 cos 2 tf) = sin 2 # angular distribution 
(F72). This corresponds classically to ejection of the proton by inter- 
action with the electric vector of the incident 7 ray and is peaked at 90 
from the direction of the incident Poynting vector. 

The photomagnetic disintegration results from absorption of the inci- 
dent photon as a magnetic dipole, A/ = 1, no. Then in the struck 
deuteron, A*S = 1, AL = 0, no, and the ( 3 *S T i, even) level is converted 
by the spin flip to O/So, even). This level is unstable and dissociates 
with an isotropic distribution in eenter-of-mass coordinates because the 
intermediate level has T (] = 0. The photomagnetic disintegration corre- 
sponds classically to an interaction between the magnetic vector of the 
incident photon and the spin magnetic', dipole moments of the proton and 
neutron. Being of opposite .sign, these magnetic moments are anti- 
parallel in the 3 #i ground level and parallel in the ^o excited level. 

The photomagnetic cross section is largest just above the 7-ray 
threshold at 2.22 Mev and falls as the 7-ray energy increases, as shown in 
Fig. 4.1 of Chap. 10. Above about 2.5 Mev, photoelectric disintegration 
becomes the dominant process (WTO). 

f. Deuteron Stripping Reactions. The outstanding peculiarities of 
the deuteron as a nuclear projectile arc its small internal binding energy 
and the large average separation (~ 4 X 10~ 13 cm) between the constitu- 
ent proton and neutron, which actually spend most of their time outside 
the "range" of their attractive mutual force (Chap. 10, Fig. 2.1). 

In nuclear reactions of the (d,a) type the incident deuteron joins the 
target nucleus to form a compound nucleus, in the manner of most other 
nuclear reactions. Much more commonly, the loosely joined deuteron 
structure dissociates in the external field of the target nucleus, ?nd only 



248 



The Atomic Nucleus 



[CH. 6 



one of its constituents is captured. These are the very common "strip- 
ping reactions" (d,p) and (d,n). 

Energetics of Stripping Reactions. In the (d,p) stripping reaction, the 
target nucleus accepts a neutron of orbital angular momentum l n directly 
into one of the levels of the final nucleus. The proton proceeds in a direc- 
tion determined by l n and with an energy determined by the reaction 
energy Q for the formation of the level into which its partner was cap- 
tured. Analogously, in (d,n) stripping reactions, the target nucleus 
accepts a proton of orbital angular momentum l p directly into one of the 

excited levels or the ground level of 
the final nucleus. Thus the ener- 
getics of the stripping reaction are 
indistinguishable from those in which 
a compound nucleus is formed and 
subsequently dissociates. 

Angular Distribution in Stripping 
Reactions. The angular distribu- 
tions of the product particles are en- 
tirely different in stripping reactions 
and in compound nucleus reactions. 
The direction of the uncaptured par- 
ticle in stripping reactions is deter- 
mined by the angular momentum l n 
or lp transferred to the final nucleus 
by the captured particle. The an- 
gular distribution docs not have fore- 
and-aft symmetry about tf = 90 but 
shows a pronounced forward maxi- 
mum. This maximum lies directly 
forward at # = 0, if l n or l p = 0, and 
moves out to progressively larger 
angles for larger values of l n or l p . 
There are also secondary maxima, for 
each l n or l p value, as shown in Fig. 
9.2. The theory of the angular dis- 
tribution in (d,p) and (d t n) strip- 
ping reactions has been developed by 

Butler, using approximations which are equivalent to the Born approxi- 
mation (B146, G18). Agreement with experiment is excellent over a 
wide range of target nuclei and deuteron energies, and the method is 
rapidly adding to our knowledge of the energy, angular momentum, and 
parity of excited levels in nuclei (A10, B128, B127, S31). 




Fig. 9.2 Angular distribution of the 
uncaptured particle in the stripping re- 
actions (d,p) and (d,n). The captured 
particle transfers orbital angular mo- 
mentum l n or lp directly into a level in 
the final nucleus. In general, the dif- 
ferential cross section is largest for / 
or lp = and decreases as the angular- 
momentum transfer increases. The 
illustrative angular distributions shown 
refer to any stripping reaction for which 
the incident deuteron energy is 14.9 
Mev and the uncaptured particle has 
19.4 Mev, both in center-of-mass cooi> 
dinates. [From Butler (B146).] 



Problems 

1. In the nuclear reaction A + a-+ B >c + C, show that, in center-of-mass 
coordinates, cos tf has the same absolute value, whether # is defined as the angle 
between the directions of the particles (a,c), or (a,C), or (A,c), or (A,C). 



9] Nuclear Effects of Nuclear Moments 249 

2. In the reaction Li 7 (p,a)He 4 , compare the height of the coulomb barrier 
with that of the centrifugal barrier for incident a-, p- f and /-wave protons, measur- 
ing each in Mev at the nuclear radius. Ana.: B^i = 1.5 Mev; B^t = 1(1 + 1) 
2.88 Mev. 

3. Explain qualitatively, in terms of angular momenta and parity of partial 
waves, why Rutherford scattering shows a characteristically forward distribu- 
tion, esc 4 (tf/2), instead of fore-and-aft symmetry in center-of-mass coordinates. 

4. When Li 6 (whose ground level is 7 = 1, even) is bombarded by deuterons 
whose kinetic energy in laboratory coordinates is Ed to 1 Mev, the yield of 
the reaction Li 6 (d,a) He 4 shows a moderate resonance peak at E d ^ 0.6 Mev. If 
i? is the center-of-mass angle between the direction of the incident deuterons and 
that of the observed a rays, the number of a rays per unit solid angle, at mean 
angle #, is found to be proportional to 1 + A cos 2 #, where the coefficient A is a 
smoothly varying function of Ed. If any cos 4 tf term is present, its coefficient in 
this energy domain is negligible compared with A. From this information alone, 
and the masses of the reacting constituents, determine systematically and clearly 
the energy, angular momentum, and parity of the excited resonance level in the 
compound nucleus. Which partial waves of deuterons (s, p, d, etc.) are effective 
in producing this excited level? 



CHAPTER 7 

Isotopic Abundance Ratios 



We have seen that many of the chemical elements consist of mixtures 
of isotopes. For nearly every element, the relative abundance of its 
several isotopes appears to be constant and completely independent of the 
ultimate geographical and geological origin of the specimen. Even in 
meteorites, the isotopic abundance ratios of all elements so far measured 
are the same as in the earth (E18). 

The principal exceptions to this generalization occur in specimens 
where radioactive disintegration processes result in the accumulation of 
stable isotopes as decay products. These include Sr 87 in rubidium micas 
(H6, M20), Ca 40 and A 40 in potassium minerals, He 4 in gas wells where the 
iatio of He 4 /He 3 is about ten times greater than in atmospheric helium 
(A26, A12, C39), and the well-known radiogenic leads (N14) which are 
found in uranium and thorium ores. 

Also, very slight variations in the normal isotopic ratios have been 
reported for some of the lightest elements (T13). The 18 /O 16 ratio is 
slightly greater in atmospheric; oxygen than in fresh water. Various 
carbon sources show maximum variations of 5 per cent in the C 12 /C 18 
ratio, limestone having a slightly higher ratio and plants a slightly lower 
ratio than the average of all sources (N20). Some minute differences in 
the lightest elements may well arise from natural evaporation and distilla- 
tion processes occurring geologically over long periods of time. In sev- 
eral cases the variations can be correlated with the equilibrium constant 
for isotopic exchange reactions (Sec. 8), such as that between water and 
carbonate ion. These physical-chemical equilibrium constants are tem- 
perature-dependent. Then precision measurements of isotopic abun- 
dance ratios, such as O 1H /O 16 and S 32 /S 34 , can be used to determine 
climatic conditions in the geological past (TJ3, TIG, NO). 

The relative abundance of isotopes in nature, by virtue of its almost 
universal constancy, seems to be closely related to the basic problems of 
nuclear stability and of the origin of the elements. Because of the wide 
range of abundance ratios, several experimental methods have been used 
in their determination. 

1. Ratios from Mass Spectroscopy 

By far the most accurate abundance ratios are obtained from mass 
spectroscopes especially designed for this purpose. Particular precau- 

250 



1] Isotopic Abundance Ratios 251 

tions must be taken with the ion source to assure proper representation 
in the ion beam of all the isotopes present in the source material. For 
this reason, sources depending on the evaporation of the element from a 
solid state are often unsatisfactory, because of the slightly greater vola- 
tility of the light isotopes. The presence of hydrides in the ion beam has 
to be particularly guarded against. The most reliable method is to 
secure the element in a suitable gaseous compound, to dissociate and 
ionize a portion of this gas by bombardment with a well-collimated beam 
of high-energy electrons, to withdraw electrostatically the ions so formed, 
and to send them through the usual energy and momentum filters. 




210 208 206 

Atomic mass units 

Fig. 1.1 The relative abundnnce of the isotopes (204, 206, 207, 208) of ordinary lead 
The numerical values of the relative abundances are 204 : 206 : 207 : 208 = 1.48:23.59: 
22.64:52.29. [From Nier (N12).] 

Best results usually are obtained using a single-focusing mass spec- 
trometer, such as that shown in Fig. 3.5 of Chap. 3. Ions of various 
ne-/M values may be brought into the collecting electrode through a 
fixed exit slit by varying the electrostatic field in the energy filter. Using 
modern vacuum-tube electrometer circuits, the relative ion currents can 
be determined with high accuracy, and the linearity of the electrical 
method is particularly suited to the study of very weak isotopes, such as 
K 40 , which has an abundance of only about 1 part in 8,000 parts of K 39 

Absolute values of relative abundance can now be obtained with an 
accuracy of 1 per cent, and abundances relative to a reference standard 



252 



The Atomic Nucleus 



[en. 7 



TABLE 1.1. RELATIVE ABUNDANCE OF THE ISOTOPES OF THE ELEMENTS 

FOUND IN NATURE 

[As compiled in 1950 by Bainbridge and Nier (B6). An asterisk denotes a 
naturally occurring radioactive nuelide. Parentheses denote questionable data.] 





Nuelide 




Relative 




Nuelide 




Relative 








i 








i j 


z 


Element 


A 


aounuance in 
atom per cent 


Z 


Element 


A 


iiuu.rLQ L iiicc in 
atom per cent 


1 

2 


H 
He 


1 
2 
3 


99 9851 
0.0149 
0.000 13 


19 


K 


39 
40* 


93.08 
0.0119 














41 


6 91 


3 


Li 


4 
6 


99.9999 
7.52 


20 


Ca 


40 

42 


96.97 
0.64 






7 


92.47 










4 


Be 


9 


100 






43 


0.145 


5 
6 

7 
8 


Bt 

ct 

N 

o 


10 
11 
12 
13 
14 
15 
16 
17 


18. 98 to 18 45 
81 02 to 81 55 
98.892 
1.108 
99.635 
365 
99.758 
0.0373 


21 
22 


Sc 
Ti 


44 
46 
48 
45 
46 
47 
48 
49 


2.06 
0033 
0.185 
100 
7.95 
7.75 
73.45 
5.51 


9 
10 

11 
12 

13 
14 

15 
16 


F 

Ne 

Na 
Mg 

Al 
Si 

P 

S 


18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 


0.2039 
100 
90.92 
0.257 
8.82 
100 
78.60 
10.11 
11.29 
100 
92.27 
4.68 
3.05 
100 
95.1 


23 
24 

25 
26 

27 

28 


V 
Cr 

Mn 
Fe 

Co 

Ni 


50 
50* 
51 
50 
52 
53 
54 
55 
54 
56 
57 
58 
59 
58 
60 


5.34 
0.24 
99.76 
4.31 
83.76 
9.55 
2.38 
100 
5.84 
91.68 
2.17 
0.31 
100 
67.76 
26.16 






33 


0.74 






61 
62 


1.25 
3.66 






34 


4.2 










17 


Cl 


36 
35 


0.016 
75.4 


29 


Cu 


64 
63 
65 


1.16 
69.1 
30.9 


18 


A 


37 
36 


24.6 
0.337 


30 


Zn 


64 


48.89 






38 


0.063 






66 


27.81 






40 


99.600 






67 


4 11 



t Abundances vary from different sources. 

t As found in limestone. 

As found in air. 

|| These abundances are recommended tentatively (B6). 



1] 



Isolopic Abundance Ratios 



253 



TABLE 1.1. RELATIVE ABUNDANCE OF THE ISOTOPES OF THE ELEMENTS 
FOUND IN NATURE (Continued") 



Nuclide 


Relative 
abundance in 
atom per cent 


I Nuclide 


Relative 
abundance in 
atom per cent 


Z 


Element 


A 


Z 


Element 


A 






68 


18.56 


44 


Ru|| 


96 


(5.68) 






70 


0.62 






98 


(2.22) 


31 


Ga 


69 


60.2 






99 


(12.81) 






71 


39.8 






100 


(12.70) 


32 


Ge 


70 


20.55 






101 


(16.98) 






72 


27 37 






102 


(31.34) 






73 


7.61 






104 


(18.27) 






74 


36.74 


45 


Rh 


103 


100 






76 


7.67 


46 


Pd 


102 


0.8 


33 


As 


75 


100 






104 


9.3 


34 


Se 


74 


0.87 






105 


22.6 






76 


9 02 






106 


27.2 






77 


7.58 






108 


26.8 






78 


23.52 






110 


13.5 






80 


49.82 


47 


Ag 


107 


51.35 






82 


9.19 






109 


48.65 


35 


Br 


79 


50.52 


48 


Cd 


106 


1.215 






81 


49.48 






108 


0.875 


36 


Kr 


78 


0.354 






110 


12.39 






80 


2.27 






111 


12.75 






82 


11.56 






112 


24.07 






83 


11.55 






113 


12.26 






84 


56.90 






114 


28.86 






86 


17.37 






116 


7.58 


37 


Rb 


85 


72.15 


49 


In 


113 


4.23 






87* 


27.85 






115* 


95.77 


38 


Sr 


84 


0.56 


50 


Sn 


112 


0.95 






86 


9.86 






114 


0.65 






87 


7.02 






115 


0.34 






88 


82.56 






116 


14.24 


39 


Y 


89 


100 






117 


7.57 


40 


Zr 


90 


51.46 






118 


24,01 






91 


11.23 






119 


8.58 






92 


17.11 






120 


32.97 






94 


17.40 






122 


4.71 






96 


2.80 






124 


5.98 


41 


Nb 


93 


100 


51 


Sb 


121 


57.25 


42 


Mo 


92 


15.86 






123 


42.75 






94 


9.12 


52 


Te 


120 


0.089 






95 


15.70 






122 


2.46 






96 


16.50 






123 


0.87 






97 


9.45 






124 


4.61 






98 


23.75 






125 


6.99 






100 


9.62 






126 


18.71 



254 



The Atomic Nucleus 



[CH. 7 



TABLE ].l. RELATIVE ABUNDANCE OF THE ISOTOPES OF THE ELEMENTS 
FOUND IN NATURE (Continued) 



Nuclide 


Relative 


Nuclide 


Relative 




11 




i | 


Z 


Element 


A 


atom per cent 


Z 


Element 


A 


atom per cent 






128 


31.79 


64 


Gd 


152 


0.20 


53 
54 

55 
56 

57 

58 


I 
Xe 

Cs 
Ha 

La 
Co 


130* 
127 
124 
126 
128 
129 
130 
131 
132 
134 
130 
133 
130 
132 
134 
135 
136 
137 
KJ8 
138* 
139 
130 
138 


34.49 
100 
0.096 
0.090 
1.919 
26.44 
4.08 
21.18 
26.89 
10.44 
8.87 
100 
101 
0.097 
2.42 
6.59 
7.81 
11.32 
71.66 
0.089 
99.911 
193 
250 


05 
66 

07 
08 

09 
70 


Tb 
Dy 

IIo 
Er 

Tm 
Yb 


154 
155 
150 
157 
158 
160 
159 
150 
158 
160 
161 
162 
163 
164 
165 
162 
164 
1Gb 
167 
108 
170 
109 
108 


2.15 
14.73 
20.47 
15.08 
24.87 
21.90 
100 
0.0524 
0.0902 
2.294 
18.88 
25.53 
24.97 
28 18 
100 
0.136 
1 50 
33.41 
22.94 
27.07 
14.88 
1(10 
140 






140 
142 


88.48 
11.07 






170 
171 


3.03 

14. rn 


59 
60 

02 
63 


Pr 
Nd 

Sm 
Eu 


141 
142 
143 
144* 
145 
140 
148 
150 
144 
147* 
148 
149 
150 
152 
154 
151 
153 


100 
27.13 
12.20 
23 87 
8.30 
17.18 
5.72 
5.60 
3.16 
15.07 
11 27 
13 84 
7.47 
26 63 
'22.53 
47.77 
52 23 


71 
7'2 

73 
74 


Lu 
Hf 

Ta 
W 


172 
173 
174 
170 
175 
170* 
174 
176 
177 
178 
179 
180 
181 
180 
182 
183 
184 
186 


21 82 
16 13 

:n.84 

12 73 
97 40 
2.00 
0.18 
5.15 
18.39 
27.08 
13.78 
35.44 
100 
135 
26.4 
14.4 
30.6 
28 4 



1] 



Isotopic Abundance Ratios 



255 



TABLE 1.1. RELATIVE ABUNDANCE or THE ISOTOPES OF THE ELEMENTS 
FOUND IN NATURE (Continued) 



z 


Nuclidc 


Relative 
abundance in 
atom per cent 


Nuclide 


Relative 
abundance in 
atom per cent . 


Element 


A 


Z 


Element 


A 


75 


Re 


185 


37.07 


80 


HR 


196 


0.146 






187* 


62.93 






198 


10.02 


76 


Os 


184 


0.018 






199 


16 84 






186 


1.59 






200 


23.13 






187 


1.64 






201 


13 22 






188 


13 3 






202 


29 80 






189 


16.1 






204 


6.85 






190 


20.4 


81 


Tl 


203 


29 50 






192 


41 






205 


70 50 


77 


Ir 


191 


38.5 


82 


Pbf 


204 


1 48 






193 


61 5 






206 


23.6 


78 


rt 


190 


012 






207 


22.6 






192 


0.78 






208 


52 3 






194 


32.8 


83 


Hi 


209 


100 






195 


33 7 


90 


Th 


232* 


100 






190 


25.4 


92 


V 


2:^* 


0.0058 






198 


7.23 






235* 


0.715 


79 


Au 


197 


100 






238* 


99 28 



t From iiuiiradiogi-iiic galona, (.Ireal Boar Laki\ 

can often ho relied on to about 0.1 per cent or better. Excellent reviews 
of the technical problems and of current results have been published by 
Thode and Shields (T17) and by Bain bridge (B4). 

A typical contemporary result of high quality is illustrated in Fift. 1.1, 
which is from NUT'S precision study of ordinary (nonradiogenic) lead 
(N12) pnd shows the existence of only four isotopes, ol mass numbers 
201, 206, 207, and 208. The additional masses 203, 205, 201), 210 
originally reported by Aston appear to have been spurious, some of them 
certainly due to the presence of hydrides. 

a. Tables of Relative Isotopic Abundance. Most of the currently 
accepted data on the relative abundances of isotopes in nature have been 
obtained by mass-spectroscopic methods. The measurements have now 
been extended to all the elements, although the results on some are still 
tentative. An excellent compilation and critical review of the results 
obtained up to 1950 by all methods was prepared by Bainbridge and Nier 
(BG). Table 1.1 summarizes the values which they adopted for each of 
the elements found in nature. Future work can be expected to produce 
some changes, but these will probably be minor. 

It is sometimes important to review the experimental evidence that a 
given nuclide, such as He 5 or Co 67 , does ttot occur in nature. The experi- 
mental upper limits for the relative abundance of the nonoccurring 
nuclides are given in the excellent isotope tables by Bainbridge (B4). 



256 The Atomic Nucleus [CH. 7 

Problem: 

Ordinary carbon monoxide is to be analyzed in a mass spectrometer. What 
fraction of the molecules will have atomic weights of 28, 29, 30, and 31 ? Ans.: 
(0.9865): (0.011 42): (0.002 02): (0.000 02). 

2. Isotope Shift in Line Spectra 

A number of rare but very important nuclides have been discovered 
by optical spectroscopic methods, after eluding early mass-spectroscopic 
searches. 

Isotope shifts in optical line spectra arise from two distinct causes: 
(a) an effect of reduced mass and (b) an effect of nuclear volume. 

a. In Light Elements. Bohr's theory of the atomic hydrogen spectra 
leads to an expression for the Rydberg constant, governing the frequency 
of the emission lines, which contains as a factor the reduced mass MQ of 
the electron and nuclear system. If the mass of the nucleus is M , and of 
the electron mo, then the moment of inertia about the center of mass of the 
system is Af a 2 , where a is the separation of the nucleus and electron and 
the reduced mass MQ is 



_ ( 

MQ - M + mo " 1 + (wo/If) (2 ' 1} 

It is well known that this expression, when combined with the experi- 
mentally determined Rydberg constants for hydrogen and for singly 
ionized helium, leads to an independent estimate of about 1,836 for the 
ratio of masses of the proton and the electron. It is also clear from Eq. 
(2.1) that the Balmer series lines for the deuterium atom will have a 
slightly shorter wavelength than the same lines in the light hydrogen 
spectrum. Deuterium owes its discovery (U5) to this slight difference in 
emission frequency in the Balmer series, which corresponds to 1.79 A for 
Hi - H* and 1.33 A for H - HJ, the H 2 satellites gaining in intensity 
as the heavy hydrogen was concentrated by fractional distillations. 

This spectroscopic method enjoyed dramatic success in the discovery 
of the very rare isotope H 2 . From the relative intensity of the lines, it 
further indicated that the atomic abundance of H 2 in ordinary hydrogen 
is only about 0.02 per cent. The observations cannot, however, be used 
for an accurate mass determination. 

The Bohr formula applies only to atoms or ions having a single 
electron. The theory of isotope shift in the line spectra of the two- and 
three-electron systems of Li+ and Li has been worked out and is in 
reasonable agreement with the separations of the Li 6 and Li 7 lines (H70). 
The separations are several times smaller than for the hydrogen isotopes. 
For heavier atoms the reduced mass correction will be very small and 
has not been deduced. 

b. In Heavy Elements. There is, however, an isotope shift in heavy 
elements which can be observed readily with the aid of the Fabry-Pe*rot 
interferometer. This isotope shift depends primarily on nuclear volume 



92] 



Isotopic Abundance Balios 



257 



(Chap. 2, Sec. 3) rather than nuclear mass. As we go from the lightest 
isotope of an element, an increase in mass causes a proportionate increase 
in nuclear volume such that the density remains approximately constant. 
The isotope shift, originating in departures from the coulomb distribution 
in the interior of the nucleus, is then proportional to the increment of 
nuclear volume, and each spectral line so affected will be represented by 
as many components as there are isotopes of the element,^ 

Each of these isotope-shifted lines may be further split by hyperfine 
structure, unless the line is due to an isotope for which the nuclear 
angular momentum 7 = (Chap. 5, Sec. 1). Even for odd- A nuclides 
some spectral lines have very small hyperfinc-structure splittings, arid 
these can be used when the isotope shift is of major experimental interest. 

Exact masses are not given by investigations of the isotope structure 
of line spectra, but under carefully controlled conditions relative abun- 
dances in some cases may be obtained with as much accuracy as intensity 
measurements on photographic plates will permit, say 1 to 5 per cent. 
Thus the routine assay of enriched samples of U 23B in U 238 is based on 
intensity measurements of the 4,244-A line where U 238 U 236 = 0.251 A 
(3145). 

Isotope Shift in Samarium. 8 
That the isotope shift in heavy jg-S 
elements is not due to nuclear mass J 
alone is well illustrated by com- 
parison of the mass-spectroscopic 
and atomic spectra observations on 
samarium. This element consists . 
of a mixture of seven isotopes, hav- | 
ing mass numbers of 144, 1 47, 148, - 
149, 150, 152, and 154, with rela- 
tive abundances as indicated in 
Fig. (2. la). Careful observations 
(SI 7) of the samarium atomic spec- 



144 150 155 A 

(a) Mass spectrum of samarium 



144 148 150 152 154 



A 

10" 3 



cm 



-l 



112 |]~ 103 " 

(6) Isotope shift in samarium X5321 
Fig. 2.1 Comparison of the mass spec- 
trum and isotope shift in the line spec- 
trum of samarium. Except for the hy- 
perfine splitting of the lines due to the 
isotopes of odd mass number, the two 
spectra parallel each other in intensity. 
The isotope shift between mass 150 and 
152 is, however, anomalously large, indi- 
cating a larger change in nuclear radius 
between these two isotopes. [Srhuler and 
Schmidt (SI 7).] 



tral line at X5,321 clearly .show the 

isotope shift in the lines from the 

isotopes of even atomic weight, 

while those due to the isotopes 147 

and 149 of odd atomic weight are 

split by hyperfine structure because 

of their finite nuclear moments, so 

that they spread over the region 

corresponding to masses between 146 and 149. The observed spectrum 

is indicated in an idealized fashion in Fig. (2.16) for comparison with the 

mass spectrum of Fig. (2. la). 

It will be seen at once that the isotope shift per atomic mass unit is 
nearly constant (0.03 cm" 1 ), except between the two isotopes having 
mass numbers 150 and 152 where the isotope shift is nearly twice as great. 
This anomaly precludes an explanation of isotope shift on a simple basis 
of changes in nuclear mass and lends additional support to the interpreta- 



258 The Atomic \ucleus [CH. 7 

tion in terms of a change in nuclear volume and hence alteration of the 
coulomb field near the origin. 

Anomalous Isotope Shift in Sm, Nd, and Eu. Samarium does have 
one isotope which is a-radioactive, and for a time this activity was 
thought to be associated with the isotope-shift anomaly and to be due to 
Sm 162 . However, further studies, using enriched isotopes, have now 
proved that Sm 147 is the a-active isotope (R7, W13). The implications 
of the anomalously large isotope shift between fl 2SmJg and fl 2SmJS 2 seem 
to be related to the neutron-shell configuration, because a similarly large 
isotope shift has been found between nnXdas 8 and 6 oNdJJ, and between 
egEuJS 1 and 6 3EuJo 3 (S48, K22). In every ease, the anomalous shift occurs 
between the isotopes whieh have 88 and 90 neutrons. In fact, for the 
same neutron numbers, all the relative shifts in Nd, Sm, and Eu agree 
within their experimental errors (F59). 

Direction of Isotope Shift. The nuclear-volume effect, as seen in the 
heaviest elements, is such that the largest isotope has the greatest wave- 
length. This is in the opposite direction to the mass effect, as seen in the 
lightest elements, where the lightest isotope has the longest wavelength 
in the isotope-shifted pattern. Some intermediate-weight elements 
exhibit combined effects; for example, in 3C Kr the heaviest isotopes have 
the shortest wavelength while in &4 Xe the heaviest isotopes have the 
longest wavelength (K27). The isotope shift in Xe is therefore pre- 
dominantly due to the nuclear-volume effect rather than to the nuclear- 
mass effect. 

Problem 

t, . 

Taking the Kydberg constant for an infinitely heavy nucleus as R M = 109, 
737 cm" 1 , compute the wavelength of H a and 110 for H 1 . Compute the isotope 
shift for tritium (H 3 ) as (H* HJ) and (II Hj}) in c in" ] and in angstroms. 
[H Q and H0 arc the Buhner series lines involving transitions between total quan- 
tum numbers 3 and 2, 4 and 2, respectively. The Bohr formula is 

cm" ~ l 

where R = (M /m )R^ and A/ is the reduced mass of the electron.] Compare 
with the measured tritium shift of 2.3o' + 0.05 A from H a (6,562.8 A) reported 
hy Pomcmiu-e and Terranova, Am. J. Phys., 18: 466L (1950). 

3. Isotope Shift in the Band Spectra of Diatomic Molecules 

The total energy W of any diatomic molecule is made up of a contri- 
bution E e from the electronic structure, together with energy E v due to 
states of vibration along the internuclear axis of the molecule, and energy 
E r due to states of rotation about an axis at right angles to the inter- 
nuclear axis. Each of these energy states is quantized, and the total 
energy hv emitted or absorbed when all three states change is 

hv = W - W = (E, + E v + E r ) - (E' e + E',+ E' r ) 

- (E. - E' m ) + (E 9 - E'J + (E r - E' r ) (3.1) 



3] hotopic Abundance Ratios 259 

a. Isotope Shift in Pure Rotational Band Spectra. Considering the 
pure rotational term (E r E' r ) first, we may write 



where 7 is the moment of inert in of the molecule about its center of mass 
and w is the angular velocity of rotation. According to the principles of 
wave mechanics, the angular momentum /w can have only certain dis- 
crete values 

7o> = h \/L(L +TJ (3.2) 

whore the integer L is the rotational-angular-momentum quantum num- 
ber. Then a quantum hv r in the pure rotational spectrum would have 
the energy 

hv T = E r - E' r = |J \L(L + 1) - L'(L' + 1)] (3.3) 

where L and L' are the rotational quantum numbers of the two states 
between which the transition takes place. 

The selection rule for the rotational quantum numbers requires that 
L - L' = 1 ; hence, there can be a scries of lines in the pure rotation 
spectrum in which no electronic or vibrational energy changes take place. 
Substitution in Eq. (3.3) of successive quantum numbers shows that 
these individual lines have an energy separation of h z /I. The rotational 
quanta hv T are very small, and hence the wavelength of these lines is 
very long. They occur in the far infrared, usually in the neighborhood 
of 50 to 100 M for molecules of small mass, and are, therefore, difficult to 
study by optical methods. 

The moment of inertia / about the center of mass of a diatomic mole- 
rule composed of atoms of mass M a and M & is 3/ofr 2 , where b is the separa- 
tion between the nuclei and M is the reduced mass M a ^h/(M a + Mb) of 
the system. It has been established experimentally that the internuclear 
distance b depends almost entirely on the electronic wave functions, rather 
than on the masses of the nuclei. Accordingly, the ratio of the moments 
of inertia of two isotopic molecules (M b ,Ma) and (M Cj M a ), in one of which 
the atom Mb is replaced by one of its isotopes M r , will be given by the 
ratio of the reduced masses of the two molecules, i.e., 



h = , = f = q *_* , 3 *, 

' h UW (M ) b M c + M a M b M a ( ' } 

The difference in the rotational energy hv r of Eq. (3.3), due to the same 
quantum transition L' > L in two such molecules, for example, H 1 C1 35 
and H 1 C1 37 , is therefore obtained from the difference between two expres- 
sions based on Eq. (3.3), and, with appropriate subscripts, is 



~ [L(L + 1) - L'(L' + 1)] (3.5) 

41 c 



260 



The Atomic Nucleus 



[CH. 7 



where Eq. (3.4) at once gives its dependence on the atomic masses 
Ma, Aft, and M e . When observed by pptical methods, the isotope effect 
is most readily studied in the vibration-rotation iands. The new tech- 
niques of microwave spectroscopy (Chap. 5, Sec. 3)'1iave made it jfossible 
to obtain precision mass data and relative abundance data from the 
rotational spectra of some molecules. 




Fig. 8.1 The isotope effect in the electronic bands of diatomic carbon. 
Birge (E18).] 



[King and 



b. Isotope Shift in the Vibration-Rotation Bands. Returning to Eq. 
(3.1), we examine the consequences of changes in yibrational energy. 
These vibrational quanta are, in general, considerably larger than the 
rotation quanta. Therefore, any given vibrational quantum change 
(E 9 E' v ) will be accompanied by a dozen or so smaller rotational 
quantum changes (E T - E' r ), and the rotational effects impose a sort of 
fine structure on the vibrational levels. Within this fine structure, the 
rotational isotope effects will appear as a further splitting of each of the 
levels. 



$3] Isotopic Abundance Ratios 261 

The vibration frequencies also depend on the masses of the nuclei 
composing the molecule, being inversely proportional to the square root 
of the reduced mass, i.e., 

-L + - (3.6) 



(-L 

\M a 



The vibration-rotation bands occur in the near infrared, in the neighbor- 
hood of 2.5- to 10-j* wavelengths, and the isotope effects in the vibra- 
tioual levels correspond to splittings of the order of 2 cm" 1 . 

c. Isotope r aift in Electronic Bands. Returning again to Eq. (3.1), 
we consider 1 ie consequences of changes in electronic energy states, 
(E e E' e ). These quanta are, in general, much larger than those due 
to the vibrational levels, and the electronic band spectra occur in the 
visible or ultraviolet regions. Each electronic transition, being a large 
energy change, is accompanied by several changes in vibrational energy, 
and in turn each of these vibrational transitions is accompanied by many 
rotational transitions. The result is a very complex band spectrum, the 
details of which may be reviewed in any of the standard treatises on band 
spectra (J15, H43). 

One of the great values of diatomic band spectra to nuclear physics 
has been in the discovery of rare isotopes which had escaped detection 
in the earlier mass-spect.roscopic studies, for example, C 13 . N 1B , O 17 , and 
O 1 *. Figure 3,1 illustrates the isotope effect in the electronic bands of 
C 12 C 12 , showing the presence of the C 13 C 12 components which accounted 
for the di&coverj' of C 13 . 

There are also important deductions from band spectra concerning 
nuclear angular momenta, nuclear statistics, and nuclear electric quad- 
rupole moments (Chap. 5, Sec. 3), 

Problems 

1. Compare the effectiveness of the nuclear mass of douieiium in producing 
isotope shift in a pure rotational band spectrum, say of IIF, and in producing iso- 
tope shift in the emission spectrum of atomic 1 hydrogen. Specificalty, calculate 
the fractional change (hv z hvi)/hv\ in the energy of the photons emitted as a 
result of (a) transitions between the same two rotational states (Z/ L) for H 2 F 
and for H ] F and (6) transitions in tlie Balmer series of atomic H 2 and of H 1 . 

2. Certain lines of the spectra of diatoirJc molecule,? arc due to tho vibrations 
of the molecules along their internuclear axis. In the quantum theory , these are 
due to transitions of the molecule from one state of vibrational energy to another. 
Assuming that the molecule is a harmonic oscillator (and this is only roughly 
true), the energy levels for pure vibration can be shown by wave mechanics 
(L. Pauling ard K B. Wilson, "Introduction to Quantum Mechanics," pp. 267- 
274, MoOaw-Hffl Book Company, Inc., New York, J935) to be 

W n = (n 
where n 0, 1 , 2, . . . 

i IT 



k = force constant 
o = reduced mass 



262 The Atomic Nucleus [CH. 7 

Thus PQ is the classical natural frequency. The selection rule is An = 1. Thus 
the energy levels are equally spaced and the frequency of the emission line is given 

' w^ w^ 

Emission frequency v ^-r = v 

h 

Actually, the observed vibrational levels show a convergence for increasing n. 
The fault with the above theory is the assumption of a parabolic potential func- 
tion (constant k) ; a better potential is the Morse function described in Pauling 
and Wilson. This leads to very accurate energy levels. 

Since HC1 molecules contain both Cl 35 and Cl 87 , there will be an isotopic split- 
ting of the vibrational levels. Assuming the simplified model (k = const), 
show that the separation in angstroms of the two components of the 17, 600- A line 
is about 14 A. This is close to the observed value. 

4. Isotope Ratios from Radioactive Decay Constants 

Among the naturally radioactive elements many isotopes exist in 
such minute amounts as to defy detection by mass-spectroscopic or 
spectroscopic methods. Their relative abundance can be obtained by 
computation from their decay constants. For example, uranium con- 
sists of a mixture of three isotopes, two of which are members of the 
uranium series of radioactive element, while the third is the independent 
parent of the actinium series. When radioactive equilibrium (see Chap. 
15) is present, the same number of atoms of each type decay in unit time. 
Hence their relative abundances are inversely proportional to their decay 
constants. Thus, taking the decay constants of U MB and U 284 as 4.8 
X 10~ 18 sec- 1 and 2 X 10~ 14 sec- 1 , the relative abundance of U 284 to 
U 238 in ordinary uranium would be about 1 to 4,000. Employing the 
same basic ideas, but in a somewhat more complicated manner (see 
Chap. 15), the computed value of the relative abundance of U 286 was 
first thought to be about 1 in 280. These abundances were the best 
available until Nier's successful mass-spectroscopic study (N13) showing 
that the ratios U 288 /U 284 = 17,000 2,000 and U 288 /U 28B = 139 1. 
These new data may now be used to improve the values assumed for the 
decay constants of U 284 and U 28fi , which were extremely difficult to 
observe directly until isotopically enriched samples became available. 

5. Chemical and Physical Scales of Atomic Weight 

The chemical-atomic-weight scale is based on the arbitrary selection 
of the atomic weight 16.000 for oxygen, and all other chemical atomic 
weights are obtained from measurements of the combining weights of 
the elements, using this oxygen standard. Band spectroscopic studies 
in 1929 (G22) first showed that ordinary oxygen is a mixture of three 
isotopes having mass numbers 16, 17, and 18, mass 16 being by far the 
most abundant. Moreover, there are variations of the order of 4 per 
cent in the relative abundance of O 1B /O lfl obtained from different sources. 
The O 18 /0 16 ratio is lowest in fresh water, intermediate in sea water, and 



5] Isolopic Abundance Ratios 263 

highest in limestone and in atmospheric oxygen (B6, T13). A more 
precise definition of the standard of atomic weight then became neces- 
sary. At present the chemical scale retains its traditional basis, O = 16, 
but the physical scale assigns a mass of exactly 16.000 to the most 
abundant oxygen isotope. Thus O 16 = 16 represents the physical- 
atomic-weight (or "isotopic mass") scale. 

The conversion factor between the chemical and physical scales of 
atomic weight depends upon the source of the oxygen used to define the 
chemical scale. As Bainbridge and Nier (B6) have said: "It becomes 
meaningless to give a conversion factor, or for that matter to make an 
atomic-weight determination, to more than five significant figures unless 
the isotopic composition of the oxygen used as a reference is clearly 
specified." 

The currently favored reference standard of oxygen is atmospheric 
oxygen, for which the relative isotopic abundances found by Nier (N18) 
are 



i = 4 89 - 2 O- 7 

OH (5-D 

55 = 2 ' 670 2 

Combining these data with the isotopic masses of O 17 and O 18 from Table 
5.1 of Chap. 3, we have for atmospheric oxygen 



Isotope 


Mass 


Atom per cent 


Q16 

0" 

Q1B 


16.000 000 
J7.004 533 
18.004 857 


99.758 
0.0373 
0.2039 



The arithmetic average mass of this mixture of atmospheric-oxygen 
isotopes, as computed from the relative abundances, is 16.004 452 ( 7) 
on the physical scale. Since this quantity is taken as exactly 16 in the 
chemical scale, we have for the ratio between the chemical- and physical- 
atomic-weight scales 

(Mass on physical scale) = 1.000 278 X (mass on chemical scale) (5.2) 

This ratio has fundamental consequences in many directions when- 
ever the small correction (A per cent) is numerically justified. Bearing 
in mind that the absolute mass of any single atom is independent of the 
scale on which it is measured, we note that one gram-equivalent weight 
of any element involves more atoms on the physical scale than on the 
chemical scale. For example, 1 mole (chemical scale) of oxygen is 
16.000 g of oxygen, but 1 mole (physical scale) of oxygen is 16.004 45 g 
of oxygen. This means that Avogadro's number and the faraday are 
both larger on the physical scale (Chap. 3, Sec. 4) than on the chemical 
scale. Thus, with e = 4.803 X 10~ 10 esu (D44), we obtain the values 
given in Table 5.1. 



264 The Atomic Nucleus 

TABLE 5.1. COMPARISON OF PHYSICAL, CHEMICAL, AND UNIFIED 
SCALES OF ATOMIC MASS 



[CH. 7 



Scale 


1 mole of 
atmospheric 
O 2 grams 


Avogadro's 
number in 
10" atoms 
per mole 


Faraday 
coulombs 


1 amu 
(or 1 u) 
Mev 


Physical (O" - 16 amu). . . 
Chemical (O mix - 16 amu). 
Unified (C 11 =* 12 u).. 


16.00445 
16.00000 
15 999 41 


6.0247 
6 0230 
6 0225 


96,520. 
96,493 
9(5, 4S7 


931.16 
931.42 
931.48 













Problems 

1. The density ratio between liquid H'CM'H 1 , HWH 1 , and HWH 1 is 
assumed to be equal to the ratio of the molecular weights, i.e., 18: 19:20. Water 
derived from atmospheric oxygen is about 6.6 parts per million more dense than 
fresh water. What is the mean atomic weight of fresh-water oxygen on the 
physical scale? If the density differences arise mainly from variations in the 
16 /O 18 abundance ratio, while 18 /O 17 ^ const., what is the O 1B /O 16 ratio in 
fresh water? 

2. Chlorine is a mixture of two isotopes whose percentage abundances and 
masses on the physical scale are 

Cl 36 : 75.4 atom per cent, 34.980 04 amu 
Cl: 24.6 atom per cent, 36.977 66 amu 

Calculate the chemical atomic weight of chlorine. (A slide rule combined with a 
little algebra will give a sufficiently accurate result for comparison with the 
chemists' gravimetric value of 35.457.) Ans.: 35.461. 

6. Mass-spectrographic Identification of Nuclides in Nuclear 
Reactions 

a. Direct Identification of Radionuclides. Dempster first used the 
mass spectrograph in 1938 for determining the mass number of a radio- 
active isotope by postponing the photographic development of the plate 
used for recording the ions of Sm until the radioactive isotope of Sm had 
produced latent a-ray tracks in the photographic emulsion. This com- 
bination of the mass spectrograph and autoradiographic techniques is 
especially useful for the identification of the mass number of long-lived 
0-ray-emitting isotopes produced in nuclear reactions (H25, L22). 

b. Identification of Nuclides with Large (71,7) Cross Sections. Sev- 
eral elements, such as Cd, Sm, Gd, have a number of stable isotopes, one 
of which has an unusually large capture cross section for slow neutrons. 
The particular isotope which accounts for the high nuclear reactivity of 
the element can be identified by comparing the relative abundance of the 
stable isotopes before and after intense irradiation with slow neutrons 
in a uranium reactor. For example, when normal Cd is exposed to 
thermal neutrons, Fig. 6.1 shows that there is an impoverishment in 
Cd 118 and a corresponding enhancement of Cd 114 , because of the very 
large cross section for the reaction Cd ll8 (n,7)Cd 114 (D23). In this way, 

the pvnontinnallv lorcro tharmol nonf i-rm-/ay\+iii>ci /iwAaa aAsi+;/*via fvf C^A Qm 



6] Isotopic Abundance Ratios 265 


and Gd have been shown to belong to 


Nuclide 


Cd 111 


Sm 1 " 


Gd lM 


Gd 7 












a(n.'v) in 10~* 4 cmVnucleus 


20,000 


65.000 


69.000 


240.000 



It is interesting to note that the usefulness of Cd in the control rods of a 
uranium reactor depends on the absorption of slow neutrons by the 
isotope Cd 118 , which comprises only 12.3 per cent of the atoms of normal 




Fig. 6.1 Mass spectrum of normal cadmium (above) and of cadmium after intense 
irradiation by thermal neutrons (below), showing the alteration produced in cadmium 
by its absorption of neutrons, predominantly through the reaction Cd n '(n y 7)Cd 114 . 
[Dempster (D23).] 



Normal Xe 




124 



136 



128 132 

Mass number 
Fig. 6.2 Mass spectrum of normal xenon gas. [Thode and Graham (T14).] 

Cd, while the seven other stable isotopes of Cd have relatively unimpor- 
tant (n,?) cross sections. 

c. Identification of End Products of Radioactive Series. By showing 
that Pb 207 , as well as Pb 206 , is an end product of the decay of uranium, 
Aston (A35) first proved the existence of U 286 . 

In an analogous way, the mass identification of several series of radio- 
nuclides, which result from the fission of U 28B by thermal neutrons, has 
been made by Thode and coworkers (T14). Figure 6.2 shows the mass 



266 



The Atomic Nucleui 



[CH.7 



spectrogram of normal xenon gas, which has nine stable isotopes. Figure 
6.3 is the mass spectrogram of the xenon gas which accumulates in 
uranium after irradiation with thermal neutrons. It is seen that the 
stable end products of four of the fission-product decay chains are Xe 
of mass number 131, 132, 134, and 136. If mass spectrograms are 
obtained directly after irradiation, Xe 183 is also found and can be shown 
with the mass spectrometer to decay with a half-period of 5.270 0.002 
days. 



Fission Xe 



I 



I 




124 



136 



128 132 

Mass number 

Fig. 6.8 Masb spectrum of xenon accumulated from the fission of U su . [Thode and 
Graham (T14)J 

From measurements of the absolute abundances of the Xe obtained 
in these five chains, the corresponding absolute fission yields can be 
determined to be (Ml) : 

Mass number, A 131 



132 



133 



134 



136 



Per cent of U" B fissions which give mass A 



2.8, 



4.2 



6.3 



7.4 



6.1 



7. The Separation of Isotopes by Direct Selection Methods 

The name isotope (from the Greek isos, equal, and topos, place) was 
selected by Soddy and connotes the chemical inseparability of various 
forms of the same element because they occupy the same place in the 
periodic table of the elements. In the ordinary sense, purely chem- 
ical methods will not successfully separate the isotopes of an element. 
Because the nuclear properties of the isotopes of any one element are 
usually highly dissimilar, the success of many nuclear studies depends on 
the availability of separated isotopes. 

Separation techniques must utilize the difference in mass, or in some 
physical or physicochemical property which, in its turn, depends on mass. 
These techniques divide into two groups, the "direct selection methods," 
and the "enrichment methods." The direct selection methods, by which 



71 Isotopic Abundance Ratios 267 

a single isotope is produced at substantially 100 per cent isotopic purity, 
are discussed in this section. Complete separation in chemical quantities 
was first achieved on a laboratory scale for the isotopes of H, Li, Ne, Cl. 
K, and Rb prior to 1940. Then the military significance of separated 
U 285 resulted in the extension of established laboratory methods to full 
industrial-plant scale. As an enormously useful by-product of this great 
technical development, the isotopes of any element can now be separated 
with the equipment at Oak Ridge (K10) whenever the operating expense 
is economically justified. Many separated or enriched isotopes are now 
catalogue items and are widely used both for studies of their nuclear 
properties and for the enormous field of tracer applications in chemistry, 
biology, and industry (H48, E3). 

a. The Mass Spectrograph. The mass spectrograph depends only on 
atomic mass and deals directly with the individual atoms. Therefore it 
effects complete separation regardless of the number of isotopes which 
an element may possess. Complete separation of isotopes in weighable 
quantities demands very intense ion sources, as wide slits as are possible 
without loss of complete resolution, and a method of " freezing" the 
atoms to a target which replaces the usual photographic plate or Faraday 
cup. Separated isotopes, in quantities of more than 1 mg, were first 
obtained for nuclear studies in the cases of lithium (R39), potassium 
(S53), and rubidium (H30) by high-intensity mass spectrographs. The 
electromagnetic mass separators, or "calutrons" (S52), at Oak Ridge 
are essentially large mass spectrographs combining high- in tensity ion 
sources (Kll) with filter systems of types b (accelerator energy filter) and 
c (180 magnetic momentum filter) of Fig. 3.1, Chap. 3. It has been 
pointed out (S50) that beam-current limitations imposed by the space 
charge within an intense beam of positive ions can be minimized by pro- 
viding an auxiliary supply of free electrons which can be attracted into 
the beam and trapped there by the strong potential gradients associated 
with the beam's positive space charge. 

The cyclotron acts as a mass spectrograph, since its resonant condi- 
tion is equivalent to a series of electrostatic accelerators and 1 80 momen- 
tum filters. The rare stable isotope He 3 was discovered by Alvarez and 
Cornog with the cyclotron (A26). 

b. Radioactive Recoil. Because of the conservation of momentum in 
each individual radioactive disintegration, the emission of an a ray 
imparts kinetic energy to the residual nucleus. Such recoil atoms, ionized 
by recoil or by collision, are positively charged and hence may be col- 
lected on a negatively charged plate. 

The recoils from a disintegration of the heavy elements have about 
2 per cent of the kinetic energy of the a ray and a range of about 0.1 to 
0.2 mm of air at atmospheric pressure. Separation of a radioactive 
decay product by recoil is applicable to any a emitter (for example, ThC" 
from the a disintegration of ThC) and has even been used successfully 
on a few ^-emitting elements, although the recoil atoms then have very 
small energy. The quantities separated are extremely minute and are 
unweighable by many orders of magnitude. They are, however, ade- 



268 The Atomic Nucleus [GH. 7 

quate for radioactive studies of the recoil products and may be thought 
of as "physical quantities," in contrast to weighable "chemical quanti- 
ties." It is conceivable that application of the recoil method to very 
intense artificially radioactive sources might result in the collection of 
sufficient quantities of the stable decay products to permit nuclear experi- 
ments to be conducted on them. 

Of course, the fission of uranium results in two fragments of compar- 
able mass MI and M 2 having equal momenta and therefore sharing the 
total kinetic energy (E\ + E z ) available to them such that M \Ei = M Z E^ 
Thus the radioactive fission products may be obtained by recoil from a 
uranium target experiencing neutron irradiation. 

c. Production of Isotopes by Radioactive Decay. Gold consists of 
only a single stable isotope, of mass number 197. When bombarded by 
slow neutrons, the gold nucleus captures a neutron, becoming radioactive 
79Au 198 , which has a half-period of 2.7 days and transforms into soHg 198 
by -ray decay. The complete decay of 1 curie of Au 198 produces only 
4.1 ng of Hg 198 , but this mercury isotope is unaccompanied by the six 
other stable isotopes of mercury. Substantial quantities of spectro- 
scopically pure Hg 198 have been produced by this method, and these 
have been used as spectroscopic sources of monochromatic radiation 
because of the absence of isotope shift and hyperfine structure. 

All the elements of odd-Z have one or, at most, two stable isotopes. 
The elements of even-Z often have many stable isotopes. Thus there 
are many cases in which one or two pure isotopes of elements having 
even-Z might be obtained through the decay of artificially radioactive 
isotopes of neighboring elements of oddrZ. 

In minerals containing rubidium, but no original strontium, pure Sr 8 '' 
accumulates in weighable amounts by the radioactive decay of Rb 87 , 
whose half-period is about 6 X 10 10 yr. Similarly, substantially pure 
Pb 208 is found in some thorium minerals as the end product of radioactive 
disintegration. 

d. Photochemical Excitation. Slight differences (isotope shift) exist 
between some of the optical levels of certain isotopes. By irradiating a 
photosensitive material, e.g,, mercury vapor in oxygen, with a resonance 
line of a particular isotope, this isotope alone may become excited, may 
undergo a chemical reaction, and subsequently may be removed by chem- 
ical methods. While sound in principle, this method is difficult to apply, 
the yields are discouragingly small, and the separations are incomplete (Z3) . 

e. Molecular-beam Method. A method of separating pure isotopes 
by a combination of an opposing magnetic field and the gravitational 
field or centrifugal force, which might have future implications, has been 
suggested by Stern (S72). 

Problems 

1. How many total milliamperes of singly charged iron ions would be needed 
in order to permit the collection of 1 g of Fe 54 in 24 hr of operation of a mass 
spectrometer, if the slits and focusing arrangements allow the collection ol 60 per 
cent of all ions emitted by the source? Ans.: 590 ma. 



8] Isotopic Abundance Ratios 269 

2. (a) Derive a general expression for the recoil kinetic energy T r of an atom 
of mass M as a result of its having emitted a 7 ray of energy E ( = hv). 

(b) Derive a similar expression for the case of a a emission. 

(c) Derive a similar expression for the maximum kinetic energy of recoil 
following 0-ray emission, if E is the maximum energy of the 0-ray spectrum. 

(d) Plot on a single graph three curves of (M T r ) vs. E (energy of emitted 
radiations) over the energy range < E < 5 Mev, for (1) a rays, (2) rays (plot 
maximum recoil energy), and (3) y rays. 

3. (a) What is the kinetic energy of recoil for a Br 80 atom recoiling after the 
emission of a 0.049-Mev y ray? 

(b) If the 0.049-Mev transition in Br 80 takes place by internal conversion in 
the K shell, what will be the kinetic energy of recoil of the residual Br 80 atom? 
The K edge of Br is 0.918 A, or 13.5 kev. Ans.: (a) 0.016 ev.; (6) 0.24 ev. 

8. The Separation of Isotopes by Enrichment Methods 

Partial separation of isotopes may be achieved sometimes by methods 
based on the statistical properties of a group of atoms. For example, in 
the gaseous state, the mean velocity of the lightest isotope of argon 
exceeds that of the heavier isotopes because of the equipartition of kinetic 
energy. The efficiency of some of these methods is greatly increased by 
operating at the lowest possible temperature, since then the fractional 
velocity differences become greater than at high temperatures. Enrich- 
ment methods are most suitable for the separation of isotopes when the 
element has only two abundant isotopes. Some enrichment methods 
depend upon slight differences in chemical equilibrium constants between 
isotopic ions. The existence of such chemical differences was first recog- 
nized after the discovery of deuterium and the comparison of the physico- 
chemical properties of heavy water and ordinary water. Isotope sub- 
stitution also exerts measurable effects on the rates of certain organic 
reactions (R27). 

a. Enrichment Factor and Separation Factor. The effectiveness of 
any enrichment process is characterized by the enrichment factor R, by 
the time necessary for the apparatus to come to equilibrium, and by the 
time required to produce unit quantity at this enrichment. If the mole 
fraction of the one isotope which we wish to separate is NQ in the original 
material, NI in one (e.g., the heavy) fraction, and A T 2 in the other fraction, 
then the enrichment factor RI for this isotope in the first fraction is defined 



A/i \\ '- 

N Q /(l - NQ) 

Accordingly RI = 1 represents no separation, while RI = <x> represents 
complete separation. If RI > 1, then there is a corresponding impover- 
ishment of the same isotope in the other fraction, where the enrichment. 
factor 



will be less than unity. 



270 



The Atomic Nucleus 



[CK.J 



In the theoretical analyses of various enrichment methods for separat- 
ing isotopes, the process fractionation factor, or separation factor a, where 



AV(1 - 



(8.3) 



is often a useful parameter. 




Fig. 8.1 Schematic diagram representing 
the general class of enrichment processes. 
An original amount Fo of iso topic, mate- 
rial having No mole fraction of the inter- 
esting isotope and (1 N ) of all other 
isotopes is separated into two fractions 
of amount V\ and Fz- The significance 
of the two "enrichment factors" and the 
"separation factor" is indicated on the 
diagram. Application of the principle of 
conservation of total material, NoVo = 
ATiF, + N Z V Z and F = Vi + V* allows 
one to deduce an analytical relationship 
between (Fi/V 2 ), N , Ri, and a, as de- 
nned by Eqs. (8.1) and (8.2). Note that 
if the feed is infinite, and only a small 
fraction Vi is drawn off, then V* C^ V , 
N 2 ^ N , and Ri ~ a. 



Comparison with Eqs. (8.1) and (8.2) shows 
that the process separation factor 
^ a is always greater than the Carre- 
^ sponding useful enrichment factor 
< Ri, since 



The relationships between enrich- 
ment and separation factors are 
shown in Fig. 8.1. 

When the enrichment process 
can be repeated n times, as by con- 
necting several units in series, the 
over-all enrichment becomes R n . 
Significant enrichment can thus be 
achieved even where R is small, since 
100 = 2 66 = 1.5 11 - J.I 48 = 1.01 463 , 
etc. 

Essentially complete separation 
of H 1 and H 2 , Ne 20 and Ne 22 , Cl 35 
and Cl 87 , as well as partial enrich- 
ment of the rare isotopes C 18 , N 15 , 
O 18 , S 34 , Kr s6 and minute changes in 
the mean atomic weight of K, Zn, 
Hg, and Pb, had been obtained by 
repeated applications of various en- 
richment processes on a laboratory 
scale prior to 1940. Using many 
enrichment stages in cascade, the 



[J 28B and U 238 isotopes were successfully separated in significant quantities 
at Oak Ridge (S52). 

b. Gaseous Diffusion. Continuous diffusion and recirculation through 
a series of porous tubes (H41, H18), Fig. 8.2, or through streaming 
mercury vapor (H42), was ably introduced in 1932 by Hertz and his 
coworkers for the essentially complete separation of Ne 20 and Ne 22 at a 
rate of 1 liter per 8 hr of operation. Over-all enrichments of the order 
of 10 to 20 were obtained for C 13 , N 16 , and O 18 on laboratory scale equip- 
ment prior to 1938 by diffusion of methane, ammonia, and water vapor, 
respectively. 

The lighter isotope, having a mean velocity inversely proportional 
to the square root of its molecular weight, diffuses slightly more rapidly 
than a heavier isotope. For a single diffusion stage, and if only a small 



8] Isotopic Abundance Ratios 271 

fraction of the feed material is permitted to diffuse, the enrichment factor 
K is given approximately by 

(8.5) 

where H and L are the molecular weights of the heavy and light gases. 
The process separation factor a for a single diffusion stage may therefore 
be somewhat larger than the square root of the ratio of the molecular 
weights; thus a = 1.1 was obtained in the case of methane by Sherr 
(S32). In multistage apparatus, approximately one-half the gas enter- 
ing each stage may be allowed to diffuse through the porous barrier, after 
which it is pumped back to the feed of the adjacent stage, Fig. 8.2. The 




p P p 

Fig. 8.2 Schematic presentation of the Hertz multiple-stage porous-tube apparatus 
for separating isotopes by diffusion. Tho progress of the main volume of gas is from 
right to left, as it becomes enriched in thr heavy fractions. The lighter fractions 
diffuse out through the porous tubes (rroHtdiatrhrd) and arc returned by the rccircu- 
lating pumps P to the previous stage, eventually collecting in the reservoir L, The 
heavier fractions progress from stage to stage, eventually collecting in the reservoir H. 
The over-all enrichment increases exponentially with the number of stages. Depend- 
ing on the isotopes to be separated, sonic 10 to 50 or more stages may be used. 

effective enrichment factor per stage is then less than the ideal value 
for a single stage with negligible throughput. As is well known, multi- 
stage gaseous diffusion methods have been applied successfully to obtain 
large-scale enrichments of U 236 from uranium hexafluoride vapor (S52, 
B32). 

c. Electrolysis. Electrolytic methods have thus far proved useful 
only in the case of the hydrogen isotopes. The commercial separation of 
heavy water D 2 O is carried out by the electrolysis of water. The hydro- 
gen liberated at the cathode is greatly enriched in H 1 , and by long con- 
tinued electrolysis D 2 O of any desired purity can be attained in the liquid 
residues. A number of isotope-discriminating processes appear to be 
involved in electrolysis, but the controlling process is thought to be a 
preferential adsorption of light hydrogen ions on the cathode and their 
subsequent combination to form neutral hydrogen molecules (117) . About 
1 cc of 99.9 per cent pure D 2 O can be obtained from 25 liters of ordinary 
water. 

d. Exchange Reactions and Free Evaporation. The isotopes of an 
element which is present in two phases in equilibrium usually have differ- 
ent concentrations in the two phases. For example, the equilibrium 
between gaseous ammonia NHi and aqueous ammonium ion NH| 

(8.6) 



272 The Atomic Nucleus [cfl. 7 

has an observed (T15) equilibrium constant 

[N"H 3 lfN"Hj] _ 

* ~ ~ L 



which leads to an enrichment of N 1B in the liquid phase and makes pos- 
sible separation of the nitrogen isotopes by repeated fractional distillation. 
Similarly, the equilibrium between gaseous SO 2 and aqueous HSOj ion, 
and gaseous CO 2 and aqueous HCOj ion, has equilibrium constants which 
favor slightly the concentration of the heavier isotope S 34 and C 13 in the 
solution (C25). In these particular cases the separation factor is the 
same as the equilibrium constant and is about 1.01 to 1.03 in the most 
favorable cases. The theory of such separation has been treated in 
detail by Urey and Greiff (U6) and by Cohen (C30). 

Similarly, the exchange may take place between the liquid and vapor 
phases of a single substance. For example, the vapor pressure of H 2 O is 
5 per cent greater than the vapor pressure of D 2 0; hence partial evapora- 
tion will result in enrichment of the liquid phase in D 2 0. Again, the 
vapor pressure of IT 2 exceeds that of D 2 , and deuterium was first discov- 
ered by concentrating it by evaporation of hydrogen near the triple point 
(U5) 

The principles of exchange equilibrium have led to the erection of 
large multiplate fractionating columns for the enrichment of certain iso- 
topes by substantially the same principles of fractional distillation which 
have long been applied in the petroleum and other chemical industries. 
If complete equilibrium between liquid and vapor were realized at each 
plate, then n plates, each giving a small enrichment R, would yield an 
over-all enrichment of R n . Actually, the number of plates required 
exceeds this theoretical number by some 20 to 100 per cent, because of 
lack of complete equilibrium. Straight columns, packed with glass 
helices, provide an inexpensive fractionating column which gives excellent 
results (U2). Fractional distillation has been applied to H, Li, C, N, O, 
Ne, S, and others with good enrichments of the heavier isotopes, while 
free evaporation methods have also been used on K, Cl, Hg, Zn, and Pb 
with slight changes in atomic weight. From the standpoint of the com- 
mercial production of enriched isotopes, Urey (U2) finds the chemical 
exchange methods the most economical. 

e. The Centrifuge. The ultracentrifuge offers separation factors of 
the order of 1. 1 to 1.7 at 300K for ideal gases having a mass difference 
of 1 to 4 amu (B21, B23, B22, H76). The enrichment increases with 
decreasing temperature; for example, at 200K these separation factors 
increase to 1.2 for unit mass difference and 2.2 for a mass difference of 
four units. 

The separation factor for the single-stage centrifuge in terms of the 
equilibrium mole fractions of the light isotope at the axis and at the 
periphery of the rotor is 

tt = e (Af 2 -Af 1 )(rV2fc7') (g g) 

where M 2 M\ is the difference in mass of the heavy and light isotopes, 



38] 



Isotopic Abundance Ratios 



273 



I Ught fraction^. Hot wjre or cy|jnder 
(~300C) 

^Cooled outer 
cylinder (~20C) 



Thermal 
convection 



Thermal diffusion 



v IB the peripheral velocity (~ 8 X 10 4 cm/sec), k is the gas constant, and 
T is the absolute temperature. In this case the enrichment depends on 
the absolute value of the difference in mass, not on the ratio of the masses 
as in the diffusion process. This characteristic gives the centrifuge a 
great advantage, particularly for heavy elements. If centrifuges could 
be operated in series at extremely low temperatures, very high over-al) 
enrichment factors could be realized. 

f. Thermal Diffusion. Self -diffusion and thermal diffusion in a mix- 
ture of two gases of different molecular weights, placed between a hot 
and a cold surface, result in a higher relative concentration of the heavier 
gas at the cold surface. Theory shows that this should be true for all 
molecules between which the interaction force varies more rapidly than 
the inverse fifth power of the separation of the molecules (J18, S41). 

The use of this thermal diffusion 
for the separation of isotopes was 
first suggested by Chapman, but it 
was not practical until ingeniously 
combined with noiiturbulent thermal 
convection by Clusius and Dickel 
(C2G) when it became one of the 
simplest and most effective methods 
available for isotope separation. 

Clusius and Dickel used a cooled 
vertical glass tube with an electri- 
cally heated wire along its axis. 
Optimum dimensions of this simple 
apparatus, shown schematically in 
Fig. 8.3, have been deduced subse- 
quently (K45) . The process separa- 
tion factor a, in terms of the equi- 
librium mole concentrations in the 
heavy and light reservoirs, is 

a = gUCAfr-JfO/df.+Jf!) (g 9) 

where Mi and M 2 are the masses of 
the light and heavy isotopic mole- 
cules being separated, I is the length 
of the column, and A is a function 
of the viscosity, self-diffusion, and 
density of the gas, the temperatures 
arid radii of the cylindrical walls, and 
the gravitational constant. The 
method owes some of its success to 
the fortunate fact that the separations achieved depend on the difference 
in mass, Eq. (8.9), of the substances being separated. A single column 
2 in. in diameter and 24 ft high gives an enrichment R = 4 for C 13 in 
methane (N15), and longer columns should easily yield 10 mg of C 13 per 
day, with about a 10-fold enrichment over the normal C"/C 12 ratio. 



4 



tl 
Heavy 
Light 



| Heavy fraction | 

Fig. 8.3 Schematic explanation of the 
Clusius-Dickcl thermal-diffusion iso- 
tope separator. By thermal diffusion, 
the heavier fraction tends to concen- 
trate at the cool outer wall while the 
lighter fraction concentrates at the hot 
inner cylinder or wire. The action of 
gravity then causes thermal convection 
which provides an effective downward 
transport for the heavy fraction at the 
cool outer wall and an upward transport 
for the light fraction along the axis. 
The pressure is maintained at a low 
enough value to avoid turbulence in the 
thermal oonvertive flow. 



274 The Atomic Nucleus fen. 7 

Rapid and substantially complete separations of the heavy isotopes of H, 
Ne, Cl, Kr have been obtained, and some extensions of the method to 
the separation of isotopes in liquid instead of gaseous phase have been 
undertaken with moderate success. 

Problems 

1. By an enrichment method, it is desired to produce chlorine which is at 
least 95 atom per cent Cl 37 . What is the over-all enrichment far-tor for such an 
apparatus? If the enrichment process selected has an enrichment factor of 1.5 
per stage, how many stages must be used in series? Ans.: Rc^ 58; 10 stages 
required. 

2. In a uranium separation proems, a sample of the ingoing material gives 10 4 a 
counts per second per gram with a certain experimental arrangement' of counter and 
sample, in which are U m (X = 1.527 X 1C)- 10 yi- 1 ), IT"* (X = 9.82 X 10 ln yr-'), 
l' 234 (X = 2.980 X lO-'yr- 1 ); the ratio Vaw/U" 6 - 139,andlTVU 2 " = 5 X 10~- r . 
A sample of the outgoing material, measured under exactly the same experimental 
conditions, gives 3 X 10* a counts per second per gram. Assuming that the 
apparatus is one which does not alter the U"VU a3B ratio, find the enrichment fac- 
tor R for U" 5 . .4775.: R ~ f>.1 . 

3. Assume that a, single-stage apparatus for the separation of isotopes by 
gaseous diffusion has M separation factor a equal to the ratio of the mean kinetic 
velocities of the molecules being separated. What is the minimum number of 
stages of such gaseous diffusion apparatus required theoretically to produce 
uranium having 20 atom per cent U Mi , if uranium hexafluoride is the diffusing 
gas? Ans.: 824 stages. 

4. It can be shown [e.g., from Eq. 1 of Humphreys, Phyx. Rev., 56 : 684 (1939) 1 
that, in a hollow cylindrical centrifuge, the density p t g/cm 3 of a light molecule, 
at any distance r from the axis, is 

p, = piof*'"*" 1 " 1 

where PIO is the density of the light molecules at the axis, M j is the mass of one 
molecule, u is the angular velocity, k is the gas constant, and T is the absolute 
temperature. A similar equation holds for the density p 2 of the heavy molecule, 
that is t pz Pactf Jlf2tt>arVur - If the separation factor at equilibrium is 



compute the enrichment factor R for the light isotopic molecule at the axis, in 
terms of the separation factor. Ans.: fl~ (M\/M*)a. 

5. A centrifuge is to be used at 20C for the enrichment of IT 235 in normal 
uranium hexafluoride vapor. If a hollow cylindrical rotor is used, having an 
inside radius of 5 cm and a speed of 1 ,000 rps, what separation factor a can be 
expected when equilibrium has been reached within the cylinder? How many 
times greater than the force of gravity is the radial force on a molecule at the 
periphery of this rotor? Ans.: a~ 1 .063; 2 X 10 5 . 

6. A certain thermal diffusion column, with a large reservoir of feed gas (so 
that 7?i ~ a) and having a height /,, is fed CH 4 containing the normal proportion 
of C 13 (1.1 atom per cent C 13 , 98,9 atom per cent C 12 ). The carbon contained in 
the heavy fraction of methane leaving the column contains 10 atom per cent C 13 , 
90 atom per cent C 12 . A completely similar column, except for its length, is to 
be built for the enrichment of radioactive C M . If the feed material for the new 
column is methane containing the normal isotopic ratio of C 12 and C 13 and also 
1 atom of C 14 per 10,000 stable atoms (this corresponds to a specific radioactiv- 



10] Isotopic Abundance Ratios 275 

ity of about 0.1 mc/g), how long a column (in units of h) must be built to give a 
100-fold enrichment of the C 14 ? Neglect the effects of deuterium in the system. 
Ans.: Z 2 ~ l.OS/i. 

7. Assume that an ion source which produces singly ionized Li 6 and Li 7 is 
available. The ions so produced are accelerated in vacuum through a potential 
difference of 2 X 10 6 volts and are then allowed to pass through a very thin 
metal foil. A fraction of the ions will be reflected (i.e., elastically scattered 
through 90 or more in the laboratory coordinates) by the foil. We shall collect 
these reflected ions. Compare the isotopic abundance of Li 6 in these reflected 
ions with that in the incident beam. What is the enrichment factor for Li 6 in 
this isotope separating process if (a) the reflecting foil is of beryllium ^Be 9 ) and 
(b) the reflecting foil is of gold ( 79 Au 197 )? 

(c) Comment briefly on the relative practicability of this method of separating 
isotopes, in comparison with existing production methods. 

9. Szilard-Chalmers Reaction for the Enrichment of Radioactive 
Isotopes 

When the nucleus of an atom which is present in an organic molecule, 
e.g., iodine in ethyl iodide, becomes radioactive by the capture of a slow 
neutron, the molecular bond is usually broken either by recoil from the 
neutron collision or by recoil from a 7 ray or other radiation emitted by 
the nucleus immediately after capturing the neutron. The radioactive 
atom thus set free from the molecule can then be made to combine with 
other ions present as "acceptors" in the solution, as was first shown by 
Szilard and Chalmers (S83). Thus, following the neutron irradiation of 
ethyl iodide, if water containing a trace of iodide ion be added and the 
two immiscible phases (water and ethyl iodide) be shaken together and 
then allowed to- separate, the hulk of the radioactive iodine will he found 
in the water layer. Thus it is possible to separate those iodine atoms 
(I 12K ) which have become radioactive from the overwhelmingly greater 
number of stable iodine atoms (I 127 ) in the target, because all the iodine 
atoms which have not been made radioactive remain bound in their 
original molecules while the activated atoms are liberated (S83, L29). 

This general method of enriching a radioactive isotope is widely used 
and often dictates the composition of the target material chosen for 
nuclear bombardment when the main objective is the production of 
radioactive material in a concentrated and useful form (M34). 

10. Separation of Radioactive I sowers 

Nuclear isomers have both the same mass number and atomic num- 
ber. Their separation offers special challenges to the radiochemist. It 
is often possible to separate the radioactive isomers by having the element 
combined in an organic molecule and then taking advantage of the dis- 
ruption of the molecular bonds which takes place during an isomcric 
transition to the ground level (SUG). The mechanism by which the trans- 
forming atom breaks its molecular bond probably is its acquisition of a 
large positive charge, owing to the emission of an internal-conversion 
electron and several Auger electrons (C41, E4). 



CHAPTER 8 

Systematics of Stable Nuclei 



Many of the basic properties of the subnuclcar const itucnts of matter 
emerge from a systematic catalogue of the nuclei found in nature. 

1. Constituents of Atomic Nuclei 

Before Chadwick's discovery of the neutron in 1932 it was assumed 
that nuclei were composed of protons and electrons. This incorrect 
notion arose from overinterpretation of the early studies of radioactivity. 
By 1932 there was enough evidence at hand to make nuclear electrons a 
distasteful concept. The neutron gained substantially immediate accept- 
ance as the proper subnuclear teammate for protons. The simplest and 
most compelling arguments concerning the subnuclear constituents of 
matter are reviewed in this section. 

We assume throughout that if a neutron, proton, electron, neutrino, 
or meson enters a nucleus, the particle retains its identity and extra- 
nuclear characteristics of spin, statistics, magnetic moment, and rest 
mass. 

a. Nonexistence of Nuclear Electrons. When a rays and ft rays were 
first identified as helium nuclei and electrons, the presumption was that 
both were contained in nuclei, because both were expelled from nuclei. 
This led to the incorrect notion that nuclei were composed of protons and 
electrons. Then the nucleus of yN 14 , for example, would contain A = 14 
protons and A Z = 7 electrons, or a total of (2/1 Z) elementary 
particles. Note that on the proton-electron model, all odd-Z nuclei 
would contain an odd total number of protons plus electrons. 

Both the proton and electron are known from direct evidence to be 
Ferrni-Dirac particles (fermions) and to have spin ^. Any nuclear aggre- 
gation of an odd number of fermions would have to obey Fermi- Dirac 
statistics and possess half-integer nuclear angular momentum, / (Chap. 
4). 

Nuclear Angular Momentum of N 14 . The earliest and soundest experi- 
mental contradiction to the proton-electron model came from observa- 
tions in 1928 of the band spectrum of N 14 N 14 , for which the intensity ratio 
of alternate lines, (7 + !)//, has the value 2. Then I = 1 for 7 N 14 , and 
this nucleus cannot be composed of an odd number (2 A Z) of fermions. 

276 



1] Systematic* of Stable Nuclei 277 

Statistics of N 14 . From band spectra and Raman spectra the statis- 
tics of 7 N 14 was found to be Einstein-Bose. Again there was disagree- 
ment with any model which involves an odd number of fermions. 

Magnetic Dipole Moment of N 14 . All nuclei have magnetic dipole 
moments p which are of the order of one nuclear magneton IL* = eh/kirMc 
(Chap. 4). For N 14 in particular, /* = 0.40 nuclear magneton. The 
magnetic dipole moment of a single electron is one Bohr magneton, or 
roughly 2,000 times larger. Therefore there cannot be an unpaired odd 
electron in any nucleus, for example, N 14 . These three arguments (7, 
statistics, and /i) apply equally to any other odd-Z nuclide, such as 
deuterium. 

ft Decay. The argument that electrons are contained in nuclei 
because electrons are emitted in ft decay lost its force when positron ft 
decay was found to be common. Indeed, dual ft decay is exhibited by 
many nuclides, such as Cu 64 , which can emit either positron ft rays or 
negatron ft rays. If both positive and negative electrons were in nuclei, 
they should annihilate each other. 

Electron-Neutrino Pairs. By differential measurements (B63, Ml 4), 
the neutrino is a fermion with spin ^, like the electron, proton, and 
neutron. Fermi's interpretation of ft decay in terms of the emission of an 
electron-neutrino pair (Chap. 17) gives a satisfactory over-all account 
of the emission of both positron and negatron ft rays. The mechanism 
by which the electron-neutrino pair arises during the nucleon transition 
is as yet obscure. It is clear, however, that the electron-neutrino pair 
originates during the transition and was not present initially in the 
nucleus. Cowan (C47a) has shown experimentally that if the neutrino 
has any magnetic dipole moment it is less than 10~ 7 Bohr magneton. 
Therefore electron -neutrino pairs, residing in a nucleus, would still possess 
a net magnetic dipole moment of one Bohr magneton, and u, for odd-Z 
nuclei, such as N 14 , would have to be in the neighborhood of one Bohr 
magneton, or about 2,000 nuclear magnetons, if there were electron- 
neutrino pairs in nuclei. 

De Broglie Wavelength. In order to be confined within a nucleus, a 
particle must have a rationalized dc Broglie wavelength X = h/p which 
is not greater than the nuclear dimensions. A 1-Mev electron has 
\ ~ 140 X 10~ 18 cm. This could not possibly be retained within nuclei 
whose radii are all smaller than 10 X 10" 13 cm (Chap. 2). To be confined 
within a nuclear volume, a nuclear electron would have to have a kinetic 
energy of ~ 30 Mev (fc ~ 7 X 10~ 1S cm). Such energies are too large 
to be admissible in any satisfactory model of mass defects and binding 
energies of nuclei (Chap. 9). 

b. Acceptability of Neutrons and Protons as Subnuclear Particles. 
We have seen previously that neutrons and protons are both Fermi- 
Dirac particles (fermions) with spin k and that their combination gives 
values of nuclear angular momentum, statistics, and of magnetic dipole 
moment (Chap. 4) which agree with observations. 

De Broglie Wavelength. In the neutron-proton model of nuclei, the 
binding energy is about 7 to 8 Mev/nucleon (Chap. 9). The kinetic 



278 The Atomic Nucleus [CH. 8 

energy of the nucleon is greater than this, because the binding energy is 
the difference between the potential and kinetic energy of the nucleons. 
A nucleon of only 8 Mev has a rationalized de Broglie wavelength of 
X ~ 1.7 X 10~ 13 cm and therefore easily can be localized within a nuclear 
volume. 

Nuclear Reactions. Many nuclear reactions involve only the addition 
or subtraction of one neutron or of one proton with respect to a target 
nucleus. For example, 



O 16 ( T .n)O 1B Mg(y,p)Na ?4 

The mere qualitative existence of such reactions does not of itself demon- 
strate the necessity of a neutron-proton model of nuclear constitution. 
Quantitatively, however, the detailed course of such reactions is in accord 
with theoretical deductions based on the neutron-proton model. 

Pions. The binding forces between nucleons are now thought, to be 
due to the exchange of ?r mesons, or "pions/' between protons and 
neutrons. At any given instant, pions may be "in transit-" between 
nucleons, thus producing a meson current in nuclei which may have 
effects on the nuclear multipole moments (Chap. 4, Sec. 4). The pion 
has 7 = and Einstein -Bone statistics (B63); therefore it docs not con- 
tribute to 7, /i, or the statistics of nuclei. 

c. Comparison of Possible Models. Actually, at least three forms 
of nuclear-electron model require consideration. These are: 

la. A protons + (A Z) electrons = (2.4 - Z) fermions. 
Ib. A neutrons + Z positrons = (A + Z) fermions. 
\c. A protons + (A Z) electron-neutrino pains = (3-4 2Z) ferm- 
ions. 

Models la and \b are excluded by considerations such as the statistics 
and angular momentum of N 14 . Model Jc, however, offers the statistics 
anvi angular momenta which are actually observed in nuclei. The 
neutrino could conceivably cancel the spin of its electron companion, but 
it cannot cancel the electron's large magnetic dipole moment. Therefore 
model Ic fails also. 

In disproving the existence of nuclear electrons, we establish at the 
same time that a neutron is riot a close combination of a proton and 
electron, nor is a proton a close combination of a neutron and a positron 
(with or without a companion neutrino). 

The neutron-proton model can also be visualized in several modifica- 
tions, chiefly: 

2a. Z protons + (A Z) neutrons. 

2b* Z protons + (A Z) neutrons + any number of TT mesons. 

Because the TT meson has 7 = arid Einstein-Bose statistics, either 
2a or 2b matches all known simple requirements. 

d. Summary of Physical Properties. We collect in Table 1.1 the 
known static properties of nucleons and other closely related particles. 



2] 



Sysiemalics of Stable Nuclei 



279 



e. Structure of Nucleons. Contemporary theoretical and experi- 
mental work is exploring the substructure of protons and neutrons. In 
the language of present theory (B63), the individual nucleon is composed 
of a core, or "nucleor," surrounded by a pion "cloud" made up of one 
or more v mesons. If the nucleor were found to obey the Dirac equation, 
then it would have a magnetic dipole moment of one nuclear magneton. 
The anomalous magnetic dipole moment of protons and neutrons would 
then be attributed to the contributions from the orbital moments of the 
circulating pion or pions. 

TABLE 1.1. STATIC PHYSICAL PROPERTIES OP SUBNUCLEAR PARTICLES AND 
THEIR CLOSE RELATIVES 











Magnetic 






Particle 


Charge 


Rest 

mass, 
amu 


Spin 


dipole 
moment, 
nuclear 


Statistics 


Intrinsic 
parity 










magnetons 






Proton . 


-fe 


1.008 14 - 7/io 


i 

Tj- 


+2.793 


Fermi 


Even 






= 1.007 59 


2, 








Neutron .... 





1.008 98 


\ 


-1.913 


Fermi 


Even 


Electron .... 


e 


TOO = 0.000 55 


1 
Y 


1,836 


Fermi 


Even 


Neutrino .... 








1 


<10~ 3 


Fermi 


Even 


ir meson . ... 


e 


273 m 








Bose 


Odd(B63) 


n meson . . . 





264 m 








Bose 


Odd 


ft meson 





207 m 


T 


g =2(1.00116) 


Fermi 


? 



In any case, these concepts are congenial with the experimental obser- 
vation, from the T~ + H 2 > 2n reaction, that the pion has 7 = 0, Bose 
statistics, and intrinsic negative parity. A pion would therefore have to 
circulate in a p orbit (I = 1) about a bare nucleor in order that the over- 
all parity of the nucleon could be even. Such circulation of a charged 
pion would contribute to the magnetic dipole moment of the nucleon. 
We have already noted in Chap. 4 some possible consequences of this 
model, in terms of the partial "quenching" of the anomalous nucleon 
magnetic moments when nucleons aggregate in nuclei. Quantitatively, 
the theory of the structure of nucleons is as yet very shaky, but progress 
is being made, and new experiments on pion-nucleon interactions will 
supply valuable new information. 

2. Relative Abundance of the Chemical Elements 

The observed abundance distribution of stable nuclides must be 
closely related to the mechanism by which the elements originated and 
also to the ultimate characteristics of nuclear forces. 

The experimental data consist mainly of measurements of the relative 
abundance of the elements in meteorites and in the earth's crust, hydro- 
sphere, and atmosphere. A.strophysical observations of solar and stellar 



280 The Atomic Nucleus [CH. 8 

spectra add some data and support the hypothesis that the earth is a 
reasonably typical cosmic sample. Among the individual elements, the 
relative isotopic abundances are found to be the same in terrestrial and 
meteoritic samples. As a working hypothesis, the isotopic constitution 
of each element is therefore taken as a constant of nature. 

The observed relative abundances of the elements show no systematic 
relationship with their chemical properties, but instead they are clearly 
related to the nuclear properties of their stable isotopes. 

Such obvious facts as the preponderance in the universe of even-Z 
even-JV nuclides, and of lightweight elements such as oxygen, must 
emerge as necessary consequences of any acceptable theory of nuclear 
forces and of the origin of the elements. 

a. Relative Abundance of Elements in the Earth's Crust. All avail- 
able relative abundance measurements on terrestrial materials were com- 
piled in 1932 by Hevesy (1147) and in 1938 by Goldschmidt (G28). One 
clear-cut generalization from these data is that by weight more than 85 
per cent of the sampled earth consists of cven-Z even-N nudides. Repre- 
sentative data for the eight most abundant elements in the earth's crust 
are given in Table 2.1. These alone account for 98 per cent of the earth's 
total mass. The hydrogen in the oceans makes up but a small part of the 
remaining 2 per cent (R40). 

We have noted previously that over 60 per cent of the known stable 
nuclides are even-Z even-N nuclides (Chap. 4, Table 4.1) and that of the 
remainder all but four have either even-Z or even-AT. 

TABLE 2.1. ABUNDANCE OF THE EIGHT ELEMENTS IN THE EAHTK'S CBUBTAL. 
ROCKS WHICH HAVE AN AVERAGE WEIGHT ABUNDANCE GREATER THAN 

1 PER CENT 
[These account for ~98 per cent of the earth's mass (G28).] 









Even-Z 




Odd-Z 


Element . . . 


uO 


i 4 Si 


26 Fe 20 Ca 


i 2 Mg 


W A1 n Na i-K 


Weight per cent abundance . . . 
Principal isotope 


48 
16 


26 

28 


5 3.5 
56 40 


2.0 
24 


8.5 2.8 2.5 
27 23 39 















b. Cosmic Abundance of the Elements. All available data on the 
relative abundance of the elements, from terrestrial, meteoritic, and 
stellar measurements, were compiled and summarized in 1949 by Harrison 
Brown (B131). A number of interpolations and judicious appraisals had 
to be made. Except for volatile constituents, such as the noble gases 
and the light elements which participate in thermonuclear reactions in the 
stars, stellar matter appears to be fairly well represented by average 
meteoritic matter, and meteoritic matter by terrestrial matter. Brown's 
tables have been reviewed and extended by Alpher and Herman (A20, 
A21), whose estimates of the mean cosmic abundance of nuclei are shown 
in Fig. 2.1. The abundance distribution is shown in terms of mass 
number A, which is the parameter used in most theories of the origin of 
the elements. Values which differ in detail but not in general trend have 
been compiled by Urey (TJ4). 



2] Systematic* of Stable Nuclei 281 

The general trend of these isobaric abundances clearly approximates 
an exponential decrease with increasing A, until A ~ 100, above which 
the relative abundance is roughly independent of A. There is no over- 
whelming distinction between even- A and odd- A. 




200 



250 



100 150 

Mass number A 

Fig. 2.1 Mean cosmic relative abundance of nuclides, of mass number A, normalized 
to 10,000 atoms of silicon (hence, 9,227 atoms of Si"). The nonequilibrium model. of 
the origin of the elements, by successive (n,y) capture processes, leads to predicted 
abundances which are in scmiquantitative agreement with this observed distribution. 
The solid curve shows a typical theoretical result, if the initial conditions at the 
starting time of the element-building process involved ~88 per cent free neutrons 
and ~12 per cent free protons, at a total nucleon concentration of ~1.07 X 10 17 cm~ 3 
(-1.8 X 10~ 7 g/cm j ) and at a temperature of ~1.28 X 10K (0.11 Mev). [Adapted 
from Alpher and Herman (A21).] 

c. The Origin of the Elements. Accompanying the gradual improve- 
ment of nuclear experimental data, the quantity and quality of theories 
of the origin of the elements have advanced markedly. 

Among the nuclear and astrophysical data which have to be matched 



282 The Atomic Nucleus [CH. 8 

by a proper theory arc the relative abundances of nuclides, the binding 
energy of nuclei, certain nuclear reaction cross sections, time scales which 
are compatible with the half-periods for ft decay of unstable nuclides, and 
the age and rate of expansion of the universe. 

The age of formation of the solid earth and of the solar system is 
about 3 X 10 9 yr, based on radioactivity studies of terrestrial and meteo- 
ritic samples (P2, A33). This age scale is confirmed by many other 
types of evidence and is also in agreement with the cosmic time scale 
derived from the Hubble red shift (L34). It has been shown experi- 
mentally (El 8) that the age of the atoms of potassium and of uranium 
found in the Pultusk meteorites is the same as the age of the atoms of 
terrestrial samples of these elements. 

These and other experimental facts point to a "great event" of 
creation which took place somewhat abruptly, about 3 X 10 9 yr ago. A 
number of very different theories have been developed , but none of these 
is yet free from serious difficulties. Details will be found in the interest- 
ing reviews by Alpher and Herman (A20, A21). Among the theories 
which invoke a great event rather than a continuum of creation, there 
are two broad classes: equilibrium and nonequilibrium theories, 

Equilibrium Hypothesis. Using as parameters the observed nuclear 
binding energies, and an assumed initial temperature and density, meth- 
ods of thermodynamics and statistical mechanics have been applied to the 
problem by Tolman and many others. Nuclear binding energies are of 
the order of 7 to 8 Mev/nucleon for most of the elements; hence in heavy 
elements the binding energies approach 2,000 Mev. Equilibrium condi- 
tions in a thermodynamic system would therefore require very high 
temperatures. The trend of the relative abundance data for A < 40 
can be matched by an initial temperature of ~ 8 X 10 9 K and an initial 
density of ~ 10 7 g/cm a . Because of the linear relationship between 
binding energy and mass number, the predicted abundance continues to 
fall exponentially with increasing A and for heavy nuclei is ~ 10 BI 
smaller than the observed abundances. No single combination of initial 
temperature and density can account for the observed abundances, and 
there are also important difficulties concerning the time scale and the 
mechanism for "freezing-in" a final mixture to give us the isotopically 
uniform cosmos in which we live. 

Nonequilibrium Hypothesis. A mechanism which does match the 
trend of the relative abundance data for all values of A was proposed by 
Gamow (G5). This is a nonequilibrium process, taking place during a 
very brief period of time. The subsequent quantitative development of 
this theory by Alpher, Herman, Gamow, Fermi, Turkevich, and others 
has been fruitful (A21). 

As initial conditions, the nonequilibrium theory contemplates a very 
small localized region of space containing mostly neutrons, at a concen- 
tration of ~ 10 17 cm~ 3 (^ 10~ 7 g/cm 3 ), a temperature of ^ 10 9 K 
(~0.1 Mev), and an initial rate of universal expansion corresponding 
to the present Hubble red shift. Within the first few minutes some 
neutrons have already undergone ft decay into protons, and these capture 







2] 



Syslernalics of Stable Nuclei 



283 



further neutrons to form douterons H(n,y)D. The heavier nuclides are 
built up by successive (n,y) reactions followed by decay to stable 
nuclides, and in competition with neutron decay. In the course of an 
hour or so the process is essentially terminated, due to ft decay of the 
uncaptured neutrons (half-period ~ 13 min), and to the reduction in 
mean density and reaction probability caused by universal expansion. 



0.3 




80 120 160 200 240 

Mass number A 

Fig. 2.2 Radiative-capture cross sections for 1-Me.v neutrons (?i ,7) as ji function of 
mass number A. The isolated points with extremely small (",7) cross sections are 
due, to nuclides containing closed shells of neutrons, N 50, 82, or 12G. [Data from 
Hughes, Garth, Eggler, andLwin (H68, II(>9).] 

Today, 3 X 10 9 yr later, we find the products of this nonequilibrium 
great event have a mean cosmic density of ~ 10~ 30 g/cm 3 and are still 
expanding. The isotopic constitution of the resulting mixture in this 
theory depends strongly on the fast neutron radiative-capture (n,-y) cross 
sections. These have been compiled by Hughes (H68) and coworkers 
and are shown in Fig. 2.2. A trend which correlates visually with the 
abundance data of Fig. 2.1 is evident. The fast (71,7) cross sections rise 
roughly exponentially with A up to A ~ 100, then level off to a substan- 
tially constant value of ~ 0.1 barn per nucleus. Using the (n,y) cross 
sections of Fig. 2.2, and an initial nucleon concentration of 1.07 X 10 17 



284 The Atomic Nucleus [CH. 8 

cm~ 3 at t = 0, the predicted relative abundance distribution is shown by 
the Holid line in Fig. 2.1. 

Many details remain to be clarified, but the nonequilibrium model 
for the origin of the elements appears to be an important step toward 
ultimate clarification of the origin of the elements. 

3. Empirical Rules of Nuclear Stability 

There are some 274 known stable nuclides. All these are found in 
natural terrestrial samples. As a result of extensive nuclear transmuta- 
tion and disintegration experiments, more than 800 new nuclides have 
been produced and studied. All these are radioactive. No previously 
unknown stable nuclides have been produced by nuclear reactions. The 
characteristics of the stable nuclides are basic input data for all theories 
of nuclear structure. 

a. The Naturally Occurring Nuclides- The creation of the elements 
must have involved the formation of all conceivable nuclear aggregates 
of neutrons and protons. Most of these were unstable and have under- 
gone radioactive decay into stable forms. There remain in nature today 
not only those nuclides which are truly stable but also the unstable 
nuclides whose radioactive half-periods are comparable with the age of 
the universe. 

Half-periods. The nuclides which occur in nature can therefore be 
defined in terms of their half-period T\ as 

~ 10 9 yr < T* < * (3.1) 

In addition, there are in nature some 40 shorter-lived radioactive nuclides, 
such as radium, which are members of the decay series of thorium and 
uranium (Chap. 16, Sec. 2). We exclude these from our present survey 
because their existence depends on the presence of their long-lived parent 
Th 232 , U 235 , or U 238 . 

With this limitation, the naturally occurring nuclides are those whose 
relative isotopic abundance is given in Table 1.1 of Chap. 7. 

Symmetry in Protons and Neutrons. Many types of visual arrange- 
ment of these data have been used. Of these we select, as the most phys- 
ical, the plot of neutron number N vs. proton number Z, Fig. 3.1. This 
arrangement emphasizes the important general symmetry in protonc and 
neutrons which is displayed by stable nuclides. The relative frequency 
distribution of isotopes (constant Z) is similar to the distribution of 
isotones (constant N). 

Radioactive Nuclides. Immediately after the discovery of radioactiv- 
ity in uranium and thorium, all the then available elements were sur- 
veyed for this new property of matter. Most of the elements were 
reported to emit radioactive radiations. Gradually it became evident 
that these radiations were usually due to the nearly universal contamina- 
tion of all materials by radium, in detectable but minute amounts 
(~ 10~ 13 g Ra/g). All reports were withdrawn or disproved except for 
the cases of potassium and rubidium, which remained for several decades 



3] Systematic* of Stable Nuclei 285 

as the only known radioactive substances outside the thorium and ura- 
nium series. 

In recent years several especially interesting nuclides, which seemed 
at first to form exceptions to the empirical rules of nuclear stability, have 
been restudied, using greatly improved chemical and physical techniques. 
By the end of 1954 most of these had been shown to be measurably 
unstable, although the half -periods, of some exceed 10 12 yr. In addition 
to the "stable" adjacent isobars Cd 113 , In 113 and Sb 128 , Te 128 , the only 
remaining exception is V B0 . Beta-ray transitions have yet to be found 
from V 60 , although this nuclide satisfies other criteria for instability. 
Mass data, obtained from mass-spectroscopic doublets (J17), show that 
both positron and negatron decay are to be expected: 

23 V BO -> 0+ + 2 2 Ti' + 2.4 Mev 
23 V BO f + 2 4Cr' + 1.2 Mev 



However, the nuclear angular momentum of V BO is I = 6, while both 
decay products are even-Z even-A r , hence probably 7 = 0. The ft transi- 
tions against A/ = 6 may well have such a long half-period that they 
will frustrate radiation detection techniques for some time to come. 
Because of the energetics of Eq. (3.2), and because stability or radio- 
activity at these levels becomes a matter of degree, we arbitrarily include 
V 60 in Table 3.1, which summarizes the naturally occurring parent radio- 
active nuclides. We shall exclude these nuclides from further considera- 
tion in connection with empirical rules of stability. 

b. Stability Rules Relating to Mass Number. Turning to Fig. 3.1, 
we note that the stable nuclides are confined to a narrow region of the 
N vs. Z diagram. Artificially radioactive nuclides have already been 
produced and studied which fill in most of the blank values of (N,Z) 
within this region and which also line the borders of the region for a 
distance of several neutron numbers above and below the region of 
stability. These nuclides transform by /3-ray emission, along lines of 
constant A, hence diagonally in Fig. 3.1, toward the center of the region 
of stability. 

Nuclear Energy Surface. When the N vs. Z diagram is viewed 
diagonally, along any isobaric line of constant A, it is noted that for 
odd-A nuclides there is generally only one stable nuclide. However, for 
even- A nuclides there are often two and occasionally three stable nuclides 
which have the same mass number. We can understand this consistent 
behavior most easily by adding atomic mass M , as a third coordinate, 
normal to the (N,Z) plane in Fig. 3.1. Then the region of stability 
becomes a valley, with the stable nuclides at the bottom of the mass- 
energy valley and the unstable nuclides lining the sides and rims of the 
valley. Such a mass-energy valley is called a nuclear energy surface. 

Relative Mass of Odd-A Isobars. Cross sections of the valley, in 
planes of constant A, have a characteristic appearance. For odd-A, the 
relationship between atomic mass M and nuclear charge Z is as shown in 
Fig. 3.2. The lowest isobar in the mast-energy valley is the stable 
nuclide for the particular odd mass number A. Isobars of larger Z decay 



286 



The Atomic Nucleus 



[CH. 8 



by positron rays or by electron capture. Isobars of smaller Z decay 
to the stable nuclide by successive negatron ft decay. Note that for 
odd- A each isobar is either even-Z odd-JV or odd-Z even-A r . 

The smoothed relationship between M and Z, for constant A } can be 
shown to be parabolic (Chap. 11), with a minimum lying at some value 
of nuclear charge Z which determines the "most stable isobar," and 

TABLE 3.1. PARENT RADIOACTIVE NUCLIDES WHICH ARB FOUND IN NATURE 
References to the original literature will be found in (H61) and (NO) 



Xuclide 


Atom 
per cent 

abundance 

i 


Half- 
period, yr 


Radiation 
observed 


Disinte- 
gration 
energy 
C. Mev 


Change in 
nuelear 
angular 
moment u in / 


Z 


A 


19 K 


40 0.0119 


1 2 X 10' 


0- EC\ -, 


1.4 


4 


23V 


50 


0.24 


-10" 


-y 


2 4 


U 


37 Uh 


S7 |' 27 85 


6 X 10 1U 


0- 


~0 3 3 




i 






49Tn 


115 95 77 


6 X 10 14 


0" 


00, 4 


52 To 


ISO 34 4< 


~10 21 | Growth of M Xe'' 

i 


<^l (i ' 


57 La 


138 


089 ~2 X K) 11 


0- F,O -:; ? 








1 


(HI Nd 


144 


23.9 


~1.5 X 10" 


(\V3) ; i <i o 


62 Sm 


147 


15.07 


1.4 X 10" 


a 21; ? 










I 


71 Lu 


176 


2.6 


7.5 X 10 10 j r, 7 


U >7 


75 Re 


187 


62.93 


4 X 10 12 


fl- 


04 


90 Th 


232 


100 


1.39 X 10 10 


at 


4.03 





92 U 


235 


0.715 


7.13 X 10 B 


a 


4.66 


? 


92 U 


238 


99.28 


4.49 X 10 B 


a 


4 . 25 






which usually is noninteger. The integer Z which lies nearest Z deter- 
mines the stable isobar of odd- 4. In the case shown in Fig. 3.2 the mass 
of BsUr lies below the smooth parabolic relationship which the other 
isobars follow. This is attributed to the closed shell of N = 82 neutrons 
in I 136 . 

Adjacent Stable Isobars. Figure 3.2 shows graphically why there is, in 
general, only one stable isobar- for any particular odd- A. If two odd- A 
isobars were to lie nearly symmetrically in the bottom of the energy 
valley, straddling Z , then the energy available for a transition between 



3] 



Syslemalics of Stable Nuclei 



287 




5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 9C 



A B Ne P Co Mn Zn Br Zr Rh Sn Cs Nd Tb Yb Re Hg Ai 
Be F Si K Cr Cu Se V Ru In Xe Pr Gd Tm W Au Po 
Li O AI A V Ni As Sr Tc Cd I Ce Eu Er To Pi Bi 
He N Mq Ci Ti Co Ge Rb Mo Ag Te La Sm HO Hf In Pb 
H C No 5 Sc Fe Go Kr Mb Pd Sb Ba Pm Dy Lu Os TI 

Fig. 3.1 The naturally occurring nuclides for Z < 83. Open circles show the radio 
active nuclides listed in Table 3.1. For Z > 80, all naturally occurring nuclides and 
artificial nuclides will be found in another NZ diagram which is given in Chap. 16, 
Fig. 3.1. The solid line shows the course of Z 0> which is the bottom of the mass- 
energy valley, or the "line of |9 stability 11 (Chap. 11), (S80, K33). The chemical 
elements are identified by their symbols along the Z axis. 



288 



The Atomic Nucleus 



[CH. 8 



them might be very small. If, in addition, there were a large difference 
ir nuclear angular momenta, then ft transitions could have such a long 
} Jf-period as to defy detection. This appears to be the situation for the 
two known cases of "stable" adjacent isobars, ( 4 8CdJ ia , 4Bln} 13 ) and 
UiSbj 28 , B jTeJ 23 ), where the subscripts denote the measured values of /. 
For several other pairs of naturally occurring adjacent isobars, one mem- 

ber of the pair has recently been 
shown to have a measurable half- 
period (Table 3.1). 

Relative Mass of Even- A Isobars. 
For even- A the mass-energy valley 
is more complicated. Successive 
isobars no longer fall on a single 
parabola. The isobars of even-Z 
even-JV fall on a lower parabola and 
therefore have more tightly bound 
nuclear structures than the alter- 
nating odd-Z odd- JV isobars. These 
relationships are shown in Fig. 3.3. 
The mass separation between 
these two parabolas is to be associ- 
ated with the even-even and odd- 
odd character of the proton-neutron 
configurations in these two sets of 
even- A isobars. An even number 
o f identical nucleons is seen to be 
relatiycly more tightly bound than 
^ odd number rf identical nude . 

Tr .,. {l . . ,, . 

^ , If <J "P"* energy" is 
called 5 > then the mass separation 




== 



55,7 



53 54 55 

* 2o= 

Fig. 3.2 Characteristic relationship be- 
tween atomic mass M and nuclear charge 
Z for odd- A isobars. For A = 135, the 
only stable isobar is 51 Ba, shown here as 
a solid circle. The bottom of the mass- 
energy valley for A = 135 occurs at 
Z - 55.7. 



between the two parabolas is 5 for 
the even-Z, plus another 6 for the 
even-#, or a total of 25. We shall give a generalized evaluation of the 
mass differences due to nucleon pairing energy in Chap. 11. 

Isobaric Pairs and Trios. Figure 3.3 also shows that there often can 
be two stable isobars for a particular even value of A. The isobars on the 
lower parabola can decay only by ft transitions to isobars on the upper 
parabola. Transitions between isobars on the lower parabola can take 
place only by way of two successive ft transitions, through the intermedi- 
ate odd-Z odd-N isobar on the upper parabola. When this is ener- 
getically impossible, both even-Z even-JV isobars are stable. Among the 
known stable nuclides there are 54 pairs of stable even-Z even- AT isobars 
and four cases in which three even-Z even-N isobars are stable (A = 96, 
124, 130, 136). 

Double ft Decay. The only alternative transitions between a pair of 
"stable" even-Z even-N isobars would be by the simultaneous emission 
of two ft rays, or double ft decay. The theoretical half -period for the 
emission of two electron-neutrino pairs, in a double ft transition, is ~ 10 24 



3] 



Systemalics of Stable Nbclei 



289 



yr for an allowed transition with 1.6 Mev of available energy. Presently 
available radiation-detection techniques, arranged for the observation of 
ft-ft coincidences, can at best explore down to a half-period of ~ 10 18 yr. 
Within this domain, no unequivocal cases of double ft decay have been 
found by radiation measurements. In the experimentally more favorable 
caae of B2 Te 1M -> ft~ft~ + 6 4Xe 180 (Q ~ 1.6 Mev), Inghram and Reynolds 
(12) have observed a measurable accumulation of Xe 180 in a Bi 2 Te 3 
mineral whose geological age is ~ 1.5 X 10 9 yr. The data lead to a half- 
period of ~ 1 X 10 21 yr for the double ft decay of Te 130 . 









Even^ 
A*"1Q 


4 
2 










1 


i 
\ 


















\ 
\ 

V -i 


^Odd- 


Z Odd-AT 






/ 




. 




V 


rEv 
, 


n-Z Even-W 






/ 
/ 


/ 


i 




\ 
\ 


\ 






^ 


/ / 

/ 








\ 


\^ 


V 




// 













\ 

V 


v\ x 


w^ 


r7 


/ 




* 






\ 


v\T 7 


^ 


^ 






1 








y T 


/ / 1 










h 


b M 


9 T 


1 i 

c Ru JF 
i 


h P 


d K 


K C 


d 




4 


1 4 

*g 


2 4 

* 


3 M? 


5 4i 

M 1 


5 4 


7 4 


B 





Wg. 1.8 Cross section of the energy valley for even-A isobars, showing the charac- 
teristic double-valued relationship between atomic mass M and nuclear charge Z. 
For A - 102, both 44 Ru and 4 Pd are stable (solid circles). The bottom of the mass- 
energy valley is at Z - 44.7 for A = 102. 

Aftiineutrinos in Double ft Decay. We may note here that the obser- 
vation of double ft decay with a half-period > 10 20 yr is presently the 
only positive experimental evidence which distinguishes between two 
formulations of 0-decay theory (K39) . If the neutrino is a particle which 
satisfies Dirac's equation for electrons (after setting the charge and rest 
mass equal to zero), then there would be two distinguishable solutions, a 
"neutrino," v, and an "antineutrino" v. The antineutrino may be 
regarded as a "hole," just as a positron is a "hole" in Dirac electron 



290 The Atomic Nuclew [CH. 8 

theory. Then positron decay involves the emission of a positron- 
neutrino pair, while in iicgatron /3 decay a ncgatron-antincutrino pair is 
emitted: 



p-+n 



Double 0~ decay requires the emission of two Dirae antineutrinos, and 
the calculated half-periods are generally > 10'- yr. 

In a modification due to Majorana, there is no distinction between a 
neutrino and an antineutrino. Then'm double $ decay, ihc neutrino 
emitted by one transforming nucleon can be absorbed by the other trans- 
forming nucleon. This type of double /?" decay would involve only the 
emission of two /3 rays but no neutrinos. This model leads to very much 
shorter predicted half-periods, ^ 10 1 - yr, for double decay (P34). 
Because these have not been found experimentally, the presumption is 
presently against Majorana neutrinos and in favor of distinguishable 
Dirac neutrinos and antineutrinos. 

Colloquially, the expression "electron-neutrino pair/' as used in dis- 
cussions of ordinary single (3 decay, can be underwood to include either 
form of neutrino, wherever the distinction between neutrino and anti- 
neutrino is significant. 

Frequency Distribution of Stable Isobars. Returning again to the NZ 
diagram of Fig. 3.1, study of the details of the distribution brings out 
several simple and important generali fictions concerning the stability of 
nuclei. With respect to mass number J, tl>e,se can be summarized for 
1 < A < 209 as follows: 

1. For even- A: There are always one, 1wo, or three stable rallies of Z, 
always with even-Z [exceptions: (a) H 2 , Li 6 , B JO , N n , which have odd-Z 
but are stable because N = Z; (b) no stable nuclide exists for A = 8]. 

2. For odd-^1 : There is only one stable, value of Z, and this valup of Z 
can be either odd or even [exceptions: (a) Cd 113 , In 113 and Sb m , Te ias ; (b) 
no stable nuclide exists for A = 5 and 147J. 

c. Stability Rules Relating to Proton Number and Neutron Number. 
Empirical rules for the distribution of stable isotopes and isotones also 
emerge from a study of Fig. 3.1. These rules are particularly significant 
for the theory of nuclear forces. 

Frequency of Stable Isotopes. Stable nuclides are found for all proton 
numbers in the range 1 < Z < 83, except for Z = 43 (technetium, Tc) 
and Z = 61 (promethium, Pm). The following generalizations can be 
made: 

1. For even-Z: There are always at least two values of N which give 
stable isotopes (exception: 4 Be 9 is simple). Usually two or more of these 
isotopes have even-JV (up to seven for Z = 50, Sn). There may also be 
stable odd-# isotopes, numbering 0, 1, 2 (for 15 elements), or 3 (for 
Z = 50, Sn, only). 

2. For odd-Z: There are never more than two stable isotopes. The 
element is usually simple, and if so its only stable isotope is invariably 
even-JV. In 10 cases there are two stable isotopes, both even-TV (01, K, 



3] 



Systematics of Stable Nuclei 



291 



Cu, Ga, Br, Ag, Sb, Eu, Ir, Tl). In two cases there are no stable isotopes 
(Tc, Pm). All odd-.V isotopes are unstable (exceptions: H 2 , Li 6 , B 10 , N 14 , 
for which N = Z). 

Frequency of Stable Isotones. Stable isotones are found for all neutron 
numbers in the range < N < 126, except for nine values, all of which 
are odd- AT. The following generalizations can be made: 

1. For even-JNT: There are always at least two values of Z which give 
stable isotones (exceptions: N = 2 and 4, where 2 HeJ and 3 LiJ are the only 
stable isotones). Usually two or more of these isotones have even-Z (up 
to five for N = 82). There may also be stable odd-Z isotones, numbering 
0, 1, or 2 (only for N = 20 and N = 82). 

2. For odd-iV: There are never more than two stable isotones. Usually 
there is only one stable isotone (example: aOj 7 ), and if so this is invariably 
even-Z (exception: 3 Li!j, where N = Z). In nine cases there are no stable 
isotones (N = 19, 21 , 35, 39, 45, 61, 89, 115, 123). All odd-Z isotones are 
unstable (exceptions: H 2 , Li 6 , B 10 , N 14 , for which N = Z). 

Correlation of Isobar, Isotope, and Isotone Distributions. The empiri- 
cal stability rules from the standpoint of mass number derive actually 
from the nuclear behavior of neutrons and protons, hence from the stabil- 
ity rules regarding isotopes and isotones. These can now be assembled 
in the symmetric form given in Table 3.2. 

TABLE 3.2. THE OBSERVED FREQUENCY DISTRIBUTION OF STABLE NUCLIDES 
(According to the odd and even character of the neutron number N, proton num- 
ber Z, and mass number A. The underscoring indicates the most abundant cases.) 



A 


Z 


N 


Total number 
of stable 
nu elides 


Number of stable 
isotopes for a 
particular value of Z 


Number of stable 
isotones for a 
particular value of N 


Odd 


Odd 


Even 


50 


0,1,2 


0, 1, 2 












(2 for N = 20, 82) 


Odd 


Even 


Odd 


55 


0, 1, 2, 3 


0,1,2 


Even 


Odd 


Odd 


4 


(3 for Z - 50) 
0, 1 


0,1 


Even 


Even 


Even 


105 


1. ?, 37 . . . , 7 


1,?,|, 4, 5 










(7 for Z - 50) 


(5 for N = 82) 



d. Conclusions from the Empirical Frequency Distributions of Iso- 
bars, Isotopes, and Isotones. The principal exceptions to the simplest 
generalizations about the occurrence of stable nuclei arise from two small 
groups of nudides. These deserve special mention. The four lowest 
odd-Z elements (H, Li, B, N) are able to form stable nuclides which con- 
tain equal odd numbers of protons and neutrons. For larger Z, and 
hence greater disruptive coulomb forces, nuclides containing equal num- 
bers of protons and neutrons are stable only if both are even. This 
implication that even numbers of identical nucleons are more tightly 



292 The Atomic Nucleus [CH. 8 

bound than odd numbers is confirmed by the nonoccurrence of stable 
odd-Z odd-N nuclides, when N > Z. The second unique group of 
nuclides comprises the stable adjacent isobars 48 Cd 113 , 4&In 113 and 
BiSb 123 , BzTe 123 , whose existence is attributable to their large difference 
in nuclear angular momentum. Note that these bracket the closed shell 
of protons at Z = 50. 

Detailed comparison of the stability rules for isotopes with those for 
isotones shows that neutrons exhibit a behavior in nuclei which is sub- 
stantially identical with the behavior of protons. The missing ele- 
ments Z = 43 and 61 are the analogues of the missing isotones N = 19, 




VO 80 120 160 200 240 

A 

Fig. 3.4 The excess-neutron number N Z as a function of mass number A for the 
stable nuclirles shown in Fig. 3.1. The smooth curve is AT Z - 0.0060 A a , as given 
by Eq. (3.27) of Chap. 11. 

21, ... , 123. The marked tendency of other odd-Z elements to be 
simple and evcn-JV is the analogue of the tendency for the odd-TV isotone,s 
to be simple and to have even-/?. The large number of stable isotopes 
found for evcn-Z elements is the analogue of the large number of stable 
even-jV isotones. These large frequencies for evcn--Z, and analogously 
but independently for even-AT, reach their maximum values for Z or 
N = 20, 28, 50, and 82. This observation was one of the earliest identi- 
fications of the "magic numbers/' or closed shells, in nuclei. 

We can draw at least three principal conclusions from the frequency 
distribution of stable nuclides. 



3] Systematics of Stable Nuclei 293 

1. Neutrons in nuclei behave in a manner which is similar, if not 
identical, to the behavior of protons in nuclei. There is every reason to 
regard neutrons and protons as two forms of a more fundamental particle, 
the nucleon. 

2. Even numbers of identical nucleons are more stable than odd 
numbers of the same uuclcons. 

3. Exceptional stability is associated with certain even numbers of 
identical nucleons, especially 20, 28, 50, and 82, and these magic numbers 
identify some of the closed-shell configurations of identical nucleons. 

e. Neutron Excess in Stable Nuclides. Only the lightest nuclei tend 
to have equal numbers of protons and neutrons. As Z increases, the 
disruptive forces due to coulomb repulsion between all the protons would 
prohibit the formation of stable nuclides if some extra attractive forces 
were not brought into the nuclear structure. These extra attractive 
forces are provided by neutrons, whose number N exceeds Z by a larger 
and larger amount as Z increases. In the NZ diagram of Fig. 3.1, the 
excess-neutron number, N Z, is seen as the vertical distance between 
the stable nuclides and the diagonal N = Z line. 

The empirical relationship between N Z and the mass number A 
becomes an important parameter in the liquid-drop model of nuclei 
(Chap. 11). From the data of Fig. 3.1 we can construct the graphical 
relationship between N Z and A, as shown in Fig. 3.4. Empirically, a 
good fit is obtained from the simple relationship 

N - Z = const A* (3.4) 

A slightly more sophisticated form emerges from the liquid-drop model 
and is given in Chap. 11. Equation (3.4) is of interest here because it, 
contains fundamental information about nuclear forces. The coulomb 
disruptive energy of a charge Ze, distributed throughout a volume of 
radius R, is proportional to (Ze) z /R. If nuclear matter has a constant 
density, then R is proportional to A*. To a first approximation, Fig. 3.1 
shows that A is proportional to Z. Then the coulomb energy should be 
approximately proportional to A 2 / A* = A*. It is a well-founded pre- 
sumption that the major role for the excess neutrons N Z is to neutral- 
ize the coulomb repulsion energy. 



CHAPTER 9 

Binding Energy of Nuclei 



The aggregate of protons and neutrons within nuclei is held together 
by strong forces of mutual attraction between the nucleons. There must 
also be short-range repulsive forces between nucleons within nuclei, such 
that the balance between attractive and repulsive forces causes nuclei to 
exhibit an approximately constant density and a radius which is pro- 
portional to ^4*. If short-range repulsion were absent, all nuclei should 
collapse into a small radius of the order of the range of the nucleon- 
nucleon force (~ 2 X 10~ 13 cm). 

Some of the characteristics of the net forces between nucleons are 
accessible to evaluation through an examination of the masses of nuclei, 
as compared with the masses of the constituent neutrons and protons. 

1. Packing Fraction 

Accurate values of the "isotopic weight," or ''neutral atomic mass" 
M, were first obtained by Aston in "prerieutron" days, when the nuclear 
constituents were thought to be protons and electrons. Aston expressed 
his results in terms of the quantity actually measured by his mass spectro- 
graph, the so-called packing fraction P, defined by 



. 

A nucleon 

where A is the mass number. By rearrangement of Eq. (1.1), the pack- 
ing fraction can be regarded physically as a small correction term (~ 10~ 8 
for many nuclides) which relates the isotopic mass M to the mass 
number A 

M E= A(l + P) amu (1.2) 

Aston correctly pointed out that the measured packing fraction P is 
connected in some way with the stability of nuclei, but the actual relation- 
ship to nuclear forces could not be inferred because the constituents of 
nuclei were unknown. 

The packing fraction is seen from Eq. (1.2) to be zero, by definition, 
for O lfl . Curves of P vs. A have been compiled by Aston (A36), Demp- 
ster (D22), Mattauch (M22), Collins, Johnson, and Nier (C34), and 
others to represent the accumulated mass-spectroscopic data on nuclear 

294 



1] 



10 



-10 



Binding Energy of Nuclei 



295 




i 



40 



80 120 160 200 240 

Mass number A 

Fig. 1.1 The pcncral trend of the variation of packing fraction P (in units of 
10~ 4 ainu/nurleon) with mass number A, from mass-spectrosropic data. 

masses. Figure 1.1 illustrates the general character of the packing- 
fraction curve. Note that P has a minimum value of about, 8 X 10~ 4 
in the vicinity of iron, cobalt, and nickel. 



2. Total Binding Energy 

Aston's early data (A3fi) were sufficiently accurate to establish what 
he justly called the "failure of the additive, law with regard to mass." 
Thus, the isotopic 1 mass of O 16 LS clearly not four times that of He 4 , and 
He 4 is not four times H 1 . These mass deficiencies were recognized as 
analogous to the heal of formation of a chemical compound and attribut- 
able to the energy liberated when the elementary nuclear constituents 
aggregate. What Aston called the mass defect, or loss of mass upon 
coalescence of the el mentary constituents, can be evaluated quanti- 
tatively only after the nuclear constituents have been identified or 
assumed. 

a. Binding Energy on the Proton-Neutron Model. When protons 
and neutrons are assumed to be the elementary constituents of all nuclei, 
the mass defect, f or binding energy B, of the nucleus is 



R -^ ZM P + NM n - 



(2.1) 



where M p , M tl , and AT are the masses of the proton, neutron, and bare 
nucleus. It is convenient to introduce the mass of Z atomic electrons 

t Ac-curding to its original definition (A3C) and some current usage (M22), "mass 
defect" is synonymous with binding energy. Some contemporary mass spectros- 
oopists (C35, C3ti) have used the name "mass defect" to mean (M A) and 
(.1 M). To minimize confusion, (M A), when it needs a name, can be called 
the mass excess (R18). 



296 



The Atomic Nucleus 



into the right-hand side of this equation, so that it becomes 
B = ZM* + NM n - M 



[CH. 9 



(2.2) 



where M H and M are the neutral atomic masses of hydrogen and of the 
nuclide in question. This conventional procedure allows binding ener- 
gies to be evaluated from tables of neutral atomic mass, such as those of 
Chap. 3, Sec. 5. Rigorously, the binding energy B f (Z) of the atomic 
electrons belongs in Eq. (2.2), both for M and for M. This refinement 
is customarily omitted because B r (Z) is at most about 3 kev/nucleon 
[Eq. (2.2) of Chap. 3], whereas B is of the order of 8 Mev/nucleon. 

Physically, we define the binding energy as the (positive) work 
necessary to disassemble a nucleus into neutrons and protons. Equiv- 
alently, it is the energy liberated when Z protons and N neutrons com- 
bine to form a nucleus. For visualization, the simplest examples are the 
photodisintegration of the deuteron Il z (y,n)H' 1 and the radiative capture 
of neutrons by hydrogen H 1 (H ,y)H 2 . The binding energy of the deuteron 
(2.22 Mev) is the Q value of the synthesis reaction H ! (n,7)H 2 , or Q for 
the dissociation reaction H^-^w)!! 1 , as evaluated in Chap. 3, Sec. 4. 

With respect to the interior of a nucleus, the binding energy is the 
difference between the mutual potential energy (taken as a positive 
quantity) and the total kinetic energy of the constituent nucleons. 

We emphasize that the definition of B is arbitrary and that the value 
of B depends on the model assumed. Even in the proton-neutron model, 
B is the energy liberated on coalescence only if the starting materials are 
exactly Z protons and N neutrons. If the nucleus ^X* is made, not by 
combining Z protons and N = (A Z) neutrons, but by combining 
(Z 1) protons and (A Z + 1) neutrons to form z-iX A , which then 
liberates an additional energy Q$ in a transition to Z X' 1 , the total energy 
liberated will not be equal to the binding energy B of ^X 1 . 

b. Binding Energy of the Lightest Nuclei. We can evaluate B from 
Eq. (2.2) for a number of nuclei, using the mass values M from Table 5.1 
of Chap. 3. For the lightest nuclides, the resulting values of B, and the 
average binding energy per micleou B/A, are given in Table 2.1. 

TABLE 2.1. BINDING ENERGY B, AND AVERAGE BINDING KNEJK.Y PER 

NUCLEON B/A, FUR THE LIGHTEST 



Nuclide 



.010 



B/A (Mcv/nuolpon). 



H 1 



i 



H 2 


IP 


JTo 3 


Ho 4 


Li c 


Li 7 


2 22 


8.48 


7.72 


28 3 


32.0 


39.2 


1.11 


2.83 


2 57 


7.07 


5 33 


5 ft) 



The Deuteron. We note especially the very small binding energy 
of the deuteron. This can be correlated with other evidence, to be dis- 
cussed in Chap. 10, which shows that the deuteron is a loosely joined 
structure in which the proton and neutron have an unusually large 
separation during a major portion of the time. In any nucleus, the 
rationalized de Broglie wavelength X of the constituent particles must 
not be greater than the nuclear dimensions. A large kinetic energy may 



3] Binding Energy of Nuclei 297 

be required in order to achieve a sufficiently small \. This is the case 
in the deuteron, where the kinetic energy is nearly as great as the poten- 
tial energy. Therefore the net binding energy (PE KE) is small. 

The a Particle. The number of possible attractive bonds between 
pairs of nucleons is one in H 2 , three in H 8 and He 8 , six in He 4 , and 15 in 
Li 6 . Clearly, the number of possible bonds for these lightest nuclides 
is not in proportion to the observed binding energies. He 4 stands out 
clearly as an exceptionally tightly bound configuration. In He 4 , the 
attractive forces have pulled the nucleons into a smaller and fully bound 
structure. He 4 contains the maximum possible number of Is nucleons, 
the four particles differing only with respect to their two possible spin 
orientations and their two possible values of charge, in accord with the 
Pauli principle. There is no orbital angular momentum in He 4 ; other- 
wise there would be a repulsion due to centrifugal force, and something 
other than He 4 would be the most stable simple configuration. He 4 then 
represents the smallest nuclear configuration of totally closed neutron 
and proton shells. 

c. Change of Binding Energy in Nuclear Transitions. Spontaneous 
nuclear transitions, such as ft decay and a decay, generally, but not 
necessarily, progress in the direction of increasing B. For example, H 3 
is 0-activc but has a greater binding energy than its decay product He 3 . 
It can be shown easily that if Qp- is the (ft + 7) energy released in ft~ decay 
between a parent and daughter nuclide identified by the subscripts 1 and 

2, then 

(V = M, - M, = (B t - B,) + (M n - A/ H ) (2.3) 

Because the neutron-hydrogen mass difference (M n Mn) is about 
0.78 Mov [Eq. (4A2), Chap. 3], all ft- transitions for which the total /3-ray 
and 7-ray energies is Q$ < 0.78 Mev will involve a decrease in binding 
energy. The criterion for (3 instability (either ft- or electron capture) is 

(.I/, - JI7 2 ) > not (B t - Bi) > (2.4) 

Problem 

Consider the energy released in the formation of any nuclide zX- 4 by two 
alternative processes: (a) combination o_ r Z protons and electrons with (A Z) 
neutrons, with release of binding energy B, and (b) combination of (Z -\- 1) protons 
and electrons with (A - Z [) neutrons, followed lv> positron decay. Show 
that the total energy released in processes (b) is B (M v M H ) = B 0.78 
Mev. Explain physically why the energy released in the formation of Z X^ by 
the processes (b) is not equal to the "binding energy" of Z X A . 

3. Average Binding Energy 

In middleweight and heavy nuclei (A > 40) the average binding 
energy per nucleon becomes an important empirical parameter in several 
theories of nuclear structure. 

a. Approximate Constancy of Average Binding Energy per Nucleon. 
From Eq. (2.2) the total binding energy, in terms of the mass number A 



298 The Atomic Nucleus [CH. 9 

and atomic number Z, is 

B = ZM H + (A - Z)M - M 

= AM n - Z(M n - M H ) - M (3.1) 

The average binding energy per nucleon can be expressed in several 
useful forms, including 




= (M n - 1) - 

- P (3.2) 

where P is the packing fraction of Eq. (1.1) and Fig. 1.1, in units of 
atomic mass units per nucleon. For the nuclides from Ca 40 to Sn 120 , 
Z/A varies only between 0.50 and 0.42; say its mean value is Z/A ~ 0.46. 
In the same region, P~ 6 X 10~ 4 amu/iuicleon. Then numerically 
Eq. (3.2) becomes 

^ ~ 0.008 982 - 0.46(0.000 840) + 0.0006 

= 0,0092 amu/nudeon 

= 8.5 Mev/nudeon (3,3) 

The mass excess (M n 1) of the neutron is clearly the predominant term 
in Eq. (3.3). The small observed values of the packing fraction P and 
the small neutron-hydrogen mass difference (M n Mn) act only as 
small correction terms. We see here the importance of accurate knowl- 
edge of the neutron mass M n , because to a first approximation (M n 1 ) 
is the average binding energy (B/ A) of the nucleons in all nuclei except 
the very lightest. 

Figure 3.1 shows the variation of B/A with A, for 1 < A < 238. 
For A < 28 there is a prominent cyclic recurrence of peaks, correspond- 
ing to maximum binding for nuclides in which A is a multiple of four. 
Each of these most tightly bound nuclides is even-Z even-JV, and N = Z. 
They correspond to a sequence of completed ''four-shells" and suggest 
an a model for light nuclei. The existence of these peaks is a compelling 
experimental demonstration of the applicability of the Pauli exclusion 
principle in nuclei, because each four-shell contains just two neutrons 
(with spin "up" and spin "down") and two protons (with spin "up" 
and spin "down"). 

Pairs of stable isobars first appear at leS 38 , i 8 A 86 and become frequent 
as A increases. Then B/A is no longer single-valued with respect to A, 
oven for stable nuclides. Also, for A > 30 (uSi'lJ), the B/A values 
begin to exhibit other effects, probably attributable to closed, shells in jy 
coupling (N or Z = 14, 20, 28, 40, 50, 82, 126), Above A ~ 60, accurate 
mass values are as yet available for only a small portion of the known 



3] 



Binding Energy of Nuclei 



299 



nuclides (M22, B4). Figure 3.1 therefore indicates only the smoothed 
general trend of B/A for 30 < A < 240, without explicit representa- 
tion of significant fine variations which undoubtedly will be quantified 
later. Even so, this region already shows several significant features. 
There is a broad maximum near A ~ 60 (Fe, Ni, Co) where B/A ~ 8.7 
Mev/nucleon. Above this region the mnan B/A values fall mono- 
tonically. Note also that B/A declines among the heavy emitters of a 
rays to a low of 7.3 Mev/nucleon for U r \ This small value of B/A 
approaches, but does not equal, the B/A = 7.07 Mev/nucleon exhibited 
in the a particle itself. Nuclides having A appreciably larger than 238, 



7 
g 

_O> 

I' 

^ 

.E 5 





4 B 12 16 20 24 



30 



150 



180 



210 240 



60 $0 120 

Mass number A 

Fig. 3.1 Average binding energy B/A in Mev per nucleon for the naturally occurring 
nuelides (and Be H ), as a function of mass number A. Note the change of magnifi- 
cation in the A scale at .4 = 30. The Pauli four-shells in the lightest nuclei are 
evident. For A > 16, B/A is roughly constant; hence, to a first approximation, 
B is proportional to A. 

and correspondingly smaller values of B/A, could be expected to be 
energetically unstable against total disruption into a particles. Thus 
there is a natural limitation on the maximum achievable value of A (and 
Z), even in the absence of the boundary set by spontaneous two-body 
fission, which is discussed in Chap. 1 1 . 

'b. Saturation of Nuclear Forces. If each nucleon exerted the same 
attractive force on all other nucleons in its nucleus, then there would be 
A(A l)/2 attractive bonds. For A 1, the binding energy would 
then increase at least as rapidly as A 2 , even assuming that in larger nuclei 
the nucleons are not drawn closer together, where they could experience 
still stronger forces. Experimentally, this square law is distinctly not 



300 The Atomic Nucleus [CH. 9 

realized, because B/A is not proportional to A. Instead B/A is sub- 
stantially constant. To a good approximation, the total binding energy 
B is proportional to the number of nucleons, or 

B ~ const X A (3.4) 

This is analogous to the chemical binding energy between the atoms in a 
liquid, which is known to be proportional to the total number of atoms 
present. We therefore take this analogy as a guide in our selection of the 
mathematical methods and terminology for the discussion of the funda- 
mental forces between nucleons. 

In a drop of liquid hydrogen we find a strong homopolar binding (H44) 
between individual pairs of hydrogen atoms, with the formation of H 2 
molecules. A third hydrogen atom is not nearly so strongly attracted, 
and the H 2 molecule is said to be saturated. The total binding energy 
of the drop is approximately equal to the combined energies of the 
individual pairs of hydrogen atoms, i.e., proportional to the total number 
of atoms present. The total energy is only .slightly increased by forces 
between the molecules. The successful mathematical representation of 
homopolar binding is that of exchange forces, which physically correspond 
to a continued process of exchanging the electrons of one atom with the 
other atom in the molecule. It is therefore assumed that the forces 
between nucleons may also be represented mathematically in terms of 
exchange operators, which perform the operation of exchanging the coordi- 
nates, between pairs of nucleons, in the potential energy term of the wave 
equation. This adoption of the concept of exchange forces in nuclear 
theory was made principally because such methods were known to give 
forces which show saturation. Its justification lies only in the success 
which the method has already had in dealing with the theory of light 
nuclei. 

Exchange Forces between Nucleons,, The particle which is ex changed f 
between two nucleons is assumed to be a w meson, or pi on. Symbolically, 
the exchange force between a proton and neutron can be described as 

p + n -> ri + w+ + n -> n' + p' (3.5) 

Here the initial proton becomes a neutron, by losing a positive pion, 
which then joins the original neutron and converts it into a proton. 
The original proton and neutron have now exchanged their coordinates. 
Negative pions and neutral pions can also be involved in the exchange 
force between nucleons, according to 

n + p > p' + TT- + p > p r + n 1 

n + n -> n' + ir + n - n' + n' (3.6) 

p + P-+P 1 +7r + p->p' + p' 

Dependence of Nuclear Forces on Spatial and Spin Coordinates. The 
nuclear unit which shows saturation does not contain two particles, as in 

f Falkoff's (F3) qualitative explanations of the concepts of exchange forces in bil- 
liard-ball and nucleon collisions should prove especially useful to those who have dealt 
previously with classical forces only. 



3] 



Binding Energy of Nuclei 



301 



the hydrogen molecule, but four. This is evident from the binding 
energies, which attain their first maximum for the a particle, an addi- 
tional neutron or proton being less tightly bound. Nucleons are seen to 
exert strong forces upon each other only if they are in the same quantum 
state with regard to their spatial coordinates. These internucleon forces 
are moderately indifferent to relative spin orientations. If, for example, 
the force between a neutron and proton were strong only when their 
spins were parallel, then the deuteron should be the saturated subunit, 
and an additional proton or neutron, as in He 3 and IP, should riot be 
strongly bound. Therefore the forces between nucleons can depend only 
moderately on the relative spin directions of the two nucleons. 




8.2 



150 



Fig. 3.2 Detail of a portion of the curve of binding energy per nurleon B/A in Mev 
per nudeon, against A, showing a discontinuity at BBOeJJ . The curve is drawn 
through the points for the family of even-Z even-JV nu elides. [From Duckworth et al. 
(D41).] 

c. Shell Structure and Binding Energy. The detailed behavior of 
the B/A vs. A curve of Fig. 3.1 has been explored over several restricted 
regions of A. For the heavier elements, the absolute values of M, and 
hence of B/A, obtained by different laboratories often disagree by more 
than the assigned errors of measurement. But by confining attention 
to any one self-consistent set of mass values, the systematic variations 
of binding energy with mass number may be revealed. In this way, 
Fig. 3.2 shows the discontinuity in B/A vs. A which has been reported 
by Duckworth et al. (D41) at 5 8 Cei2, and which apparently marks the 
closing of a shell of N = 82 neutrons. 

Similar discontinuities have been observed elsewhere, e.g., by Nier 
and coworkers at Z or N = 20 and 28 (C35, C36), and at Z = 50 (Hll), 



302 The Atomic Nucleus [CH. 9 

and by Dempster (D22) at the "doubly magic" Pb 208 (Z = 82, N = 126). 
The systematics of nuclear moments (Chap. 4, Sec. 4) has been fruit- 
ful in the development of the shell model of nuclei, but for all even-Z 
even-N nuclides these methods give only the information that 7 = 0. 
Mass values, and the binding energies derived from them, provide explicit 
quantitative dp.ta on families of even-Z even-A r mi elides and therefore 
supply complementary information on the location of closed shells in 
nuclei. 

Problems 

1. In Fig. 3.1, Li 6 has an average binding energy B/A which is less than that 
for the a particle. Why does Li 6 not undergo spontaneous a decay? 

2. Show that 

| = j (Af n - Af H ) + (Jlf H - 1) -P 

and that the uncertainty in B/A with respect to uncertainties SM n or 5A/ H , in 
M n or MU, is of the order of 5Af B /2 and fiAfn/2. 

3. From mass data, compute and plot the average binding energy per nucleon 
B/A against N in the vicinity of N = 20, 28 (C35 and C36, or Table 5.2, Chap. 3^ 
or B/A against Z in the neighborhood of Z = 50 (Hll). Comment on any 
closed-shell effects which may be displayed by your graph. 

4. (a) Show that for a nucleus A\ to be energetically unstable against a-ray 
emission, the slope of the B/A vs. A curve must be negative, and its absolute 
value must exceed 



where B/Ai refers to the decay product of the transition. 

(6) Show from a similar argument that the slope of the B *A vs. ,4 cm ve must 
be negative and that its absolute value must exceed 



At 

if the radioactive emission of a deuteron is to be possible energetically. 

(c) What conclusions can be drawn from (a) and (6) concerning the types of 
heavy-particle radioactivity which one can expect to observe? 

4. Separation Energy for One Nucleon 

A somewhat more detailed view of nuclear forces is given by the 
variations in the binding energy of the "last" proton or neutron in a 
group of nuclides. The energy required to remove one neutron from the 
nucleus (Z,N) is called the neutron separation energy 8 n , and can be 
written 

S n (Z,N) = ^ = M(Z, N-l)+M- M(Z,N) (4.1) 



where M n is the neutron mass. M(Z,N) is the atomic mass of the nuclide, 
and M (Z, N 1) is the atomic mass of the lighter isotope which results 



4] 



Binding Energy of Nuclei 



303 



when one neutron is removed from the nucleus (Z,N). In terms of 
binding energies, the neutron separation energy S n (Z,N) is the increment 
in total nuclear binding energy when one neutron is added to the lower 
isotope (Z, N 1), thus 



S n (Z,N) = B(Z,N) - B(Z, N -I) 



(4.2) 



For this reason S n is also called the "binding energy of the last neutron." 
In a completely analogous fashion, S p is the proton separation energy 
or the binding energy of the last proton, and is given by 



or 



S P (Z 7 N) = M(Z - 1, N) + M n - M(Z,N) 
S P (Z,N) = B(Z,N) - B(Z - \,N) 



(4.3) 
(4.4) 



Nucleon separation energies arc the nuclear analogues of the first 
ioriization potential of atoms. As is well known, the atomic ionization 
potentials exhibit a systematic cyclic behavior with increasing Z (p. 217 
of H44). The largest values of the first ionization potential occur for 
the atoms which have closed shells of electrons: He, Ne, A, Kr, Xe, Rn. 
In each case the next higher atomic number displays the smallest ioniza- 
tion potential, or "last electron binding energy/' in the sequence. 

The sequence of S n and ti p for successive nuclides shows a cyclic 
behavior and provides information on the nature of the forces between 
nucleons. 

a. Separation Energy in the Lightest Nuclides. Figure 4.1 shows 
the neutron and proton separation energies for the stable nudidos 1 < A 




4 8 12 16 20 24 048 12 16 

Mass number, A Mass number, A 

Fig. 4.1 The left-hand diagram shows the energy S n in Mev required to separate 
one neutron from the lightest stable nuclides and from H 3 and Be b . tiarh point is 
further identified by its neutron number N. The right-hand diagram shows the 
analogous separation energy S P in Mev required to remove the last proton from the 
same nuclides (except that He 3 replaces H 3 ) 7 each point being marked witli its proton 
number Z. In each diagram, note that the even-Z cven-JV nuclides are at or neai 
the top and the odd-Z or odd-# nuclides form the lower envelope. 

< 24, as computed from the self-consistent mass data of Table 5.1 of 
Chap. 3. Several principles emerge with dramatic clarity from these 
simple data. 

1. In odd- AT nuclides, the final neutron is lightly bound; for example. 
8 n = 5.4 Mev in 3 Li 6 3 . 



304 The Atomic Nucleus [CH. 9 

2. In even-AT nuclides, the last neutron is more tightly bound; for 
example, in 3 LiI, S n = 7.2 Mev. When, in addition, the nuclide has 
eveii-Z, then S n > 15 Mev. These are the even-2 even-AT nuclides which 
occupy the peaks of the B/A curve, Fig. 3.1, and also form the upper 
envelope of *S n in Fig. 4. 1 . The exceptionally tight binding of the second 
neutron, which completes an even-A r pair, is the origin of the pairing 
energy 5, whose qualitative presence in heavier nuclides we noted in Fig. 
3.3 of Chap. 8. The pairing energy for neutrons can be expressed as 



and is seen to be ~ 2 Mev for many of these lightest nuclides. The 
factor of one-half arises because each n represents the difference between 
an even-AT and an odd-JV nucleus. 

3. There is a complete parallelism between S n for odd-N and even-JV 
nuclides and the behavior of the proton separation energies S p in odd-Z 
and even-Z nuclides. The separation energies S n and S p are similar in 
absolute magnitude, and so are the neutron pairing energies 5 n and the 
proton pairing energies 8 P . 

4. The addition of a proton, as between K C\ 3 and yNy 4 , increases the 
separation energy *S n of the last neutron. Similarly, the presence of an 
additional neutron, as between 7 Ny 4 and 7NJ 5 , generally tightens the 
binding ti p of the last proton. 

5. These close similarities between 8 n and 8 P , 5 n and 5 P suggest that 
fundamentally the forces between any pair of nucleons are nearly inde- 
pendent of the charge character (n or p) of the nucleons. 

b. Models of the Lightest Nuclei. The l,s shells of neutrons and 
protons are filled at He 4 . Between He 4 and O 16 , the \p shells of six 
neutrons and protons (Chap. 4, Sec. 1) are being filled. In the p shell, 
neither pure jj coupling nor pure LK coupling agrees with the energy 
levels which are actually observed. Independent-particle, central- force 
(Hartree) models, whose wave functions do not correspond to preformed 
a particles but whose first-order energies may show a marked four-shell 
structure in LS coupling (F23), are favored over pure a models (112). 
For a closer match with experimental results, independent-particle models 
with intermediate coupling are required (14). 

As a simplification of this complexity, we may visualize the rough but 
illuminating model for the Is and }p shells which is shown in Fig. 4.2. 
The four-shell structure which was exhibited by B/A in Fig. 3.1, and by 
S n or S p in Fig. 4.1, is represented here as a sequence of levels, each 
capable of accepting at most two protons and two neutrons. In an 
L-coupling model, these four lowest levels correspond to the orbital 
quantum numbers I = 0, and I = 1, m z = 1,0, +1. 

Point-by-point comparisons between Figs. 4.1 and 4.2 .show in every 
case that nucleons are strongly bound only to other nucleons within the 
same subshell. There is very little net binding between nucleons which 
are not in the same quantum state Z, m z of orbital motion. We can say 
that the elementary forces between nucleons are nonadditive. This unique 



4] 



Binding Energy of Nuclei 



305 



and fundamental characteristic of the "specifically nuclear" forces is not 
exhibited by the other known basic types of interaction, i.e., by gravi- 
tational forces and by electromagnetic forces. 

The saturation character of the nuclear forces is emphasized by the 
nonexistence of stable nuclides with A = 5 and 8. The fully saturated 
He 4 structure declines to bind either an additional neutron or proton, so 
that He 5 and Li & have no bound levels. Even when two four-shells are 
offered, as in Be 8 , there is insufficient binding force between subshells to 
form a stable nucleus. Three four-shells are required, as in C 12 , before 
the forces between subshells are sufficient to form a stable even-Z eveii-AT 
nuclide. Again, in two nuclides such as B 11 and N JB , Fig. 4.2 suggests 

o = Neutron 
Proton 



Be 8 



Be 9 




C 12 C 13 N 14 N 15 

Fig. 4.2 Pictorial models of the IK shell and Ip subshells in some of the lightest 
stable nuclides and Be 8 . In accord with the Pauli principle, each "level " can accom- 
modate two neutrons (spin "up" and spin "down") and two protons (spin "up'" 
and spin "down") at the same "spatial position" (i", nil) in the configuration. The 
forces between nucleons in the same level are strong, while those between nueleona in 
different levels are weak, as is suggested schematically in the N 15 diagram. 

that if the forces between nucleons in different subshells were appreci- 
able, the last nucleons in N 15 should be more tightly bound than those in 
B 11 . However, Fig. 4.1 shows that S n and S p are the same, or possibly 
smaller, in N 16 than in B 11 . 

We should also visualize in the models of Fig. 4.2 all the generaliza- 
tions drawn earlier from the S n and S p diagrams of Fig. 4.1. For example, 
the last neutron in C 13 is very loosely bound, whereas the last proton in 
both C 12 and C 13 is tightly bound and gets only a little extra binding from 
the extra neutron in C 18 . 

c. Separation Energy in Heavy Nuclides. The nucleon separation 
energies S n and S p can be evaluated for many of the heavier nuclides, 
even where atomic-mass values are unknown, by using nuclear-reaction 
energetics. For example, the threshold energy for photoneutron produc- 



306 



The Atomic Nucleus 



[CH. 9 



tion is Q(y,ri), where (?(7,n) is the energy released in the (7,71) reaction 
on the target nuclide (Z,AT). Then 



It can be shown easily that 



S P (Z + \, N) - Sn(Z + }, N) = Q(p,n) 



(4.5) 



(4.6) 



where Q(p,n) is the energy released in the ground-to-ground levrl (p,n) 
reaction on the target nuclide (Z,N). Similarly, 



S P (Z + 1, N) - S n (Z 



- (M n - 



(4.7) 



where Qp is the total + y decay energy for a ft~ transition of the parent 
naclide (Z,N), and (Af n J/ H ) is the neutron-hydrogen mass difference. 

By utilizing all available reaction 
energetics, Feather (F13) was able by 
1953 to compile values of S n or S p in 
some 600 cases covering all values of 
Z from 1 to 98, except for the usual 
hiatus including 61 < Z < 72. A 
large proportion of the separation 
energies are in the domain of 8 2 
Mev for middleweight nur.lides, while 
for the heaviest uuclidcs K n and S p 
fall to the domain of 5 2 Mev. 

In Fig. 4.3 the neutron separation 
energies in the vicinity of N = 50 
(86 < A < 93) are plotted from 
Feather's tables. Qualitatively, the 
same physical phenomena which were 
so evident in the lightest nuclides, 
Fig. 4.1, are still present, but their 
magnitude is somewhat subdued. 
There is a clear maximum of S n at 
N = 50, marking the closing of a 
major neutron shell. For a given 
N, the neutron separation energy 
generally increases slightly with Z, 
hence with the total number of 
nucleons offering binding forces. S n 
is still clearly greater for even-AT 
than for odd-A", although the neu- 
tron pairing energy is now down to 
~ 1 Mev. 

We conclude that in nuclei of any A the forces between nucleons are 
nonadditive. Only nucleons which have the same "spatial position" 
are strongly bound to one another. These forces become saturated 
when, at most, two protons and two neutrons have the same spatial 




50 52 

Neutron number N 

Fig. 4.3 Neutron separation energy S n 
in Mev for the stable rind radioactive 
isotopes of aB Sr, 39 Y, and 40 Zr (86 < A 
< 93), which involve nuclides contain- 
ing about 50 neutrons. Isotones of 
odd N are shown as open circles. 
These have values which pass mono- 
tonically through W = 50, whereas the 
even-JV isotones exhibit maxima at 
N 50, which is identified as a major 
closed shell of neutrons. [From tables 
by Feather (F13).] 



4] Binding Energy of Nuclei 307 

coordinates. We therefore assume that the fundamental forces between 
nucleons are of an exchange character. 

Problems 

1. In the (d,p) reaction on a target nuclide (Z,N), show that the neutron 
separation energy S n (Z, N + 1) of the product nuclide is given by 

S(Z, N + ]) = Q(d,p) + J?(H Z ) 

where Q(d,p) is the energy released in the ground-to-ground level reaction and 
#(H 2 ) is the binding energy of the deuteron. 

2. When 79 Au 197 is bombarded with deuterons, a (d,p) reaction takes place, 
forming Au 198 which decays to stable Hg 198 with a half-period of 2.7 days. Au 198 
emits a simple negatron spectrum with a maximum energy of 0.963 Mev followed 
by one 0.412-Mev y ray. Take the neutral atomic masses to be 

7 Au 197 = 197.0394 amu Hg 19B = 198.0421 amu 

(a) Find the neutral atomic mass of Au 198 . 

(6) Find the Q value of the reaction Au J97 (d,p)Au 198 . 

(c) Find the separation energ3 r of the last neutron in Au 198 and compare it 
with the average binding energy per nurleon. 

(d) Explain briefly the basic physical reason for the large variation found 
in (c). 

3. Three of the five stable isotopes of nickel (Z = 28) have the following neu- 
tral atomic masses: Ni 60 = 59.949 01, Ni G1 = 60.949 07, Ni 62 = 61.946 81. 

(a) Determine the total nuclear binding energy, and the average binding energy 
per nucleon, in Ni 60 . 

(6) Determine the increase in the total nuclear binding energy when one 
neutron is added to Ni 60 to form Ni 61 . 

(c) Determine the increase in the total nuclear binding energy when one 
neutron is added to Ni 61 to form Ni 62 . 

(d) Explain the difference noted in the numerical answers to (b) and (V), in 
terms of the corresponding separation energies. 

(e) Ni" is stable, but Ni 59 transforms by electron capture to stable Go 59 . 
What type of nuclear force may be regarded as primarily responsible for the 
radioactivity of Ni 59 ? 

4. (a) Show clearly how you obtain the separation energy for one neutron 
from Pb 207 , knowing that the reaction Pb 206 (d,p)Pb 207 has a measured value of 
Q = +4.5 Mev for the protons which correspond to the formation of Pb 207 in its 
ground level. 

(6) What would be the quantum energy of the 7 rays emitted when thermal 
neutrons are captured by Pb 206 , in the reaction Pb 20r '(n,7)Pb 207 , for the Pb 207 
nuclei which are formed in the ground level? 

(c) Measured values of the separation energy for one neutron from each of 
several nuclides are given in the following table: 



Nuclide 


h ..Pb 207 


82Pb 20B 


82 Pb 209 


HvBi 210 












Seoaration enure v, Mev 


6.7 


7.4 


4.0 


4.2 



From general concepts regarding the shell structure of nuclei, explain wh}' these 
energies are not constant, and why each value deviates from the average in a 
way which is reasonable for the particular nuclide in question. 



308 The Atomic Nucleus [GH. 9 

' 6. The elements Z = 52 to 56 have the following stable isotopes: 

52 Te: 120, 122, 123, 124, 125, 126, 128, 130 

53 I: 127 

54 Xe: 124, 126, 128, 129, 130, 131, 132, 134, 136 

55 Cs: 133 

56 Ba: 130, 132, 134, 135, 136, 137, 138 

(a) Draw a schematic graph of mass vs. Z for the isobars of A = 130. and 
show why Te, Xe, and Ba can all be stable. 

(6) Show from the Pauli exclusion principle and other contemporary concepts 
of nuclear structure why elements having odd-Z have so few stable isotopes (and 
none of even- A unless A = 2Z), while neighboring elements of even-Z have many 
stable isotopes. 



CHAPTER 10 

Forces between JMucieons 



The ground levels and excited levels of all nuclei ca^n be explained by a 
quantitative theory only after we understand the simplest cases involving 
just the interaction between two nucleons. As Inglis (14) has pointed 
out, our progress toward a full understanding of nuclear spectroscopy 
involves three major steps. First, we have to see which of several pos- 
sible forms of interaction best fits the experimental data on two-body 
nucleon-nucleon interactions ("phenomenological theory of nucleon inter- 
actions "). Second, we must develop a theory of the structure of nucleons 
which will lead to the selected interaction in a natural way ("meson 
theory/' or other, of the nucleon-force field). Third, we must apply the 
nucleon-structure theory to the general problem of calculating nuclear 
energy levels. 

The main features of nucleon-nucleon interactions have become clear 
as a result of much experimental and theoretical work, but the two-body 
forces are still not completely understood. Thus, the first step is not 
finished, although the choices of interaction have been narrowed greatly 
in the two decades since the discovery of the neutron. The second step 
is in progress, but meson theory is still in an unsatisfactory state. Step 3 
has seen only exploratory sorties. 

In this chapter we make use of the experimental information previ- 
ously discussed, plus additional results on two-body interactions, in order 
to determine the principal characteristics of the interaction between 
individual nucleons. 

1 . General Characteristics of Specifically Nuclear Forces 

Prior to the discovery of the strange and intriguing character of intra- 
nuclear forces, substantially all types of material interactions could be 
described quantitatively in terms of either gravitational forces or electro- 
magnetic forces. Nuclear forces present a new, third major category of 
fundamental forces. 

a. Comparison of Atomic and Nuclear Forces. Atomic electrons are 
bound into atoms in a manner which is well understood in terms of 
coulomb forces and simple quantum-mechanical effects associated with 
spin. The atom possesses a predominant central particle which is the 
origin of a long-range coulomb field. The atomic electrons spend their 

309 



310 The Atomic Nucleus [CH. 10 

time at relatively large distances from this force center and have only a 
weak interaction with it. Thus the separation energy for valence elec- 
trons is only a few electron volts, while that of the innermost electrons 
in the heaviest elements does not exceed 0.1 Mev. 

In sharp contrast, the nucleus contains no predominant central particle. 
The forces which hold it together have to be mutual forces between the 
individual nucleons in the ensemble. These forces have a very short 
range of action, of the order of 10" 18 cm. Consequently, the nucleons find 
themselves closely packed together, with very smalt spacingsl In order 
to confine a nucleon to a region of this size, its rationalized de Broglic 
wavelength \ must be correspondingly small, and its kinetic energy must 
be of the order of p 2 /2M = h z /2M\ 2 ~ 20 Mev. This requires a very 
large average potential energy, of the order of 30 Mev if the residual average 
binding energy is to be ~8 Mev. / 

Clearly, the intranuclear forces cannot be dealt with as small pertur- 
bations, with consequent mathematical simplifications. The many-body 
problem here presented is prohibitive mathematically. What can be 
done is to deal with only the lightest nuclei and especially with the two- 
body problem represented by the deuteron. 

b. Inadequacy of Classical Forces. The force between nucleons can- 
not be a classical force which depends only on distance, because the total 
binding energy of nuclei is proportional to the number of nucleons A and 
not to A 2 . This qualitative conclusion is strengthened by simple quanti- 
tative considerations. 

The gravitational potential energy between a proton and neutron 
which are ~2 X 10~ 13 cm apart is smaller than 8 Mev by a factor of 
-^lO 36 . The electrostatic potential energy between the same two nucleons 
is identically zero, because the neutron is uncharged. The magnetic 
potential energy corresponding to the intrinsic magnetic moments M! and 
fi p of the neutron and proton is of the order of /iJ^J/r 8 and, at a separation 
of r ~ 2 X L0~ 13 cm, amounts to about 0.03 Mev. Whether the mag- 
netic force is attractive or disruptive depends on an average over the 
relative orientation of the neutron and proton but is clearly of opposite 
sign for parallel and antiparallel spin orientations. From our evaluation 
of the separation energies we have found that the force between a neutron 
and protron is attractive for both parallel and antiparallel spin orienta- 
tions. Hence the nuclear force cannot be of magnetic origin. 

We conclude that gravitational, electrostatic, and magnetic forces are 
quantitatively inadequate to act as anything more than very minor 
perturbations on the specifically nuclear forces. 

c. The Singlet and Triplet Two-body Forces between Nucleons. 
The number of important two-body forces which we must evaluate is 
fortunately limited. In nuclei, the proton and neutron separation ener- 
gies (Chap. 9) show that the important forces are those between nucleons 
which ajre in the stale spatial quantum state. Nucleons are strongly 
bound only to the small number of other nucleons which have the same I 
values. We are justified, therefore, in focusing' our attention on the 
forces between nucleons which have zero angular momentum relative to 



1] Forces between Nucleons 311 

each other, the so-called S states of even parity. Because the nucleons 
are fermions and obey the Pauli exclusion principle, there can be involved 
at most -two neutrons (spin "up" and spin "down") and two protons in 
such a group. The possible forces therefore include three tjrpes of singlet 
force (aiitiparallel spins), designated by the superscript 1: "" ~ 

w * 

1 (np) between a proton and neutron 
l (nn) between two neutrons 
(PP) between two protons 

The triplet forces (parallel spins) are restricted to one type for S states, 
namely, 

*(np) between a proton and neutron 

because the Pauli principle excludes a 3 (nn) and 3 (pp) force by providing 
that^no two identical particles can have identical quantum numbers. 
As long as we restrict our attention to even-parity, S-state interactions, 
we have only four forces to evaluate: 3 (np), l (np), *(nri), and l (pp). We 
may hereafter drop the singlet superscript from (nn) and (pp) when only 
S states are under consideration. 

In states of nonzero angular momentum the forces between identical 
particles (nn) and (pp) are restricted by the Pauli exclusion principle to 
singlet interactions for even-Z and to triplet interactions for odd-Z. 

The (pp) force represents the specifically nuclear attractive force 
between two protons and does not include their purely classical coulomb 
interaction. The attractive (pp) interaction greatly exceeds the coulomb 
interaction, in consonance with the observation that protons are not 
concentrated on the surface of nuclei but appear to be more or less uni- 
formly distributed throughout the nuclear volume (isotope shift, Chap. 
2) . That there exists also a strong attractive force (nn) between neutrons 
is shown by the fact that tlie neutron excess (N Z) in nuclei varies 
approximately as A* and appears to counterbalance the disruptive coulomb 
forces in heavy nuclei (Chap. 8). The finite strength and approximate 
equality of the (pp) and (nn) forces in mlclei are also shown qualitatively 
by the isotopic mass (Chap. 2) and excitation levels (B130) of mirror 
nuclei and by the presence of a proton-pairing energy and an approxi- 
mately equal neutron-pairing energy in nuclei of any A (Chap 9). 

d. Exchange Forces. The clear experimental evidence that nuclear 
forces show saturation directs our attention toward the purely quantum- 
mechanical concept of exchange forces (Chap. 9). > 

Three types of exchange force have been studied extensively, and these 
are commonly named for the investigators who first explored their 
characteristics. They are: " 

1. Heisenberg forces, in which there is exchange of both the position 
and spin coordinates of the two interacting nucleons. Heisenberg 'forces 
are attractive for triplet interactions and repulsive for singlet interactions 
(antiparallel spins). This would be acceptable if the deuteron were the., 
saturated subunit, but pure Heisenberg forces are ruled out by the clear ; 
experimental evidence that the a particle is the saturated subunit. . . j 



The Atomic Nucleus 



ICH. 1U 



2. Majorana forces, in which there is exchange of the position coordi- 
nates but not of spin. They can be visualized physically in terms of the 
exchange of T mesons and appear to have &n important place in nuclei. 
The Majorana force is attractive for two particles with even relative 
angular momentum (for example, S states) and repulsive for interactions 
involving odd relative angular momentum, 

3. Bartlett forces, in which there is exchange of the spin coordinates 
but not of the position coordinate*. 

The effect of the exchange operator on the sign of the force is summar- 
ized in Table 1.1, where for completeness we include also the entire class 
of short-range nonexchange forces, which are now generally known as 
Wigner forces. The Wigner exchange operator is unity and does nothing 
to the force. Taking the plus sign as representing an attractive force, the 
minus sign connotes a force of equal magnitude but repulsive. The two- 
nucleon system can usually be represented as a mixture of Majorana and 
Wigner forces. 

TABLE 1.1. EFFECT OF THE EXCHANGE OPERATORS ON THE SIGN OF THE 
NUCLEAR FORCE, IN THE TWO-BODY SYSTEM 









Sti 


ate 




Force 


Operator 


Ev< 


m-t 


Odd 


H 






Triplet 


Singlet 


Triplet 


Singlet 


Heisenberg 


PH 


1 


_1 


_1 


1 


Majorana 


PM 


1 


1 


I 


_1 


Bartlett 


PB 


1 


_1 


1 


_1 


Wigner 


Pw 


1 


1 


1 


1 















e. Tensor Forces. With central forces, the probability density of 
hucleons in S states must be spherically symmetric (Appendix C). 
The miain features of the measured interactions between two nucleons 
can be described in terms of central forces with or without exchange. 
However, there are a few small but absolutely definite effects whose 
existence cannot be explained in terms of central forces alone. Fore- 
most among these are the finite electric quadrupole moment of the deu- 
teron and the nonadditivity of the magnetic dipole moments of the 
neutron and proton in the deuteron (Chap. 4, Sec. 5). These and some 
other small effects are explicable if there is admixed with the dominant 
central force a small amount of a noncentral force. 

The strength of this noncentral force, or tensor force, depends not only 
on the separation between the interacting pair of particles but also on the 
angle between the spins of the particles and the line joining the particles, 
like the force between two bar magnets, Eq. (8.2}. The tensor force 
can be represented with or .without exchange, as in the case of central 
forces. 



(2] Forces between Nucleons 313 

f. Charge Independence of Singlet Forces between Nucleons. In 
the following sections we shall consider a variety of experimental evidence, 
principally that which concerns S-state interactions between all pairs of 
nucleons. The theoretical interpretation of these data depends some- 
what on the assumed character of the interaction. At low energies 
(< 10 Mev) many of the results are quite insensitive to the choice of 
interaction potential, so long as it is short-range. It is found that the 
singlet forces between all pairs of nucleons are substantially equal, i.e., 

'(TIP) - '(Tin) = '(PP) (1.1) 

This equality is spoken of as the "charge independence" of nuclear forces, 
and the extent and causes of small deviationsTrom Eq. (1.1) continue 
to be the object of many theoretical and experimental investigations. 
Equality between l (nri) and l (pp), without consideration of l (np), is 
spoken of aiTthe "charge symmetry" of nuclear forces!!"* 

The main features of the nucleon-nucleon interactions can be visual- 
ized from simple considerations outlined below, using a central-force, 
nonexchange (Wigner-force) approximation. The bound-state '(np) 
interaction is obtained most simply from the theory of the deuteron, while 
the continuum of unbound states is explored in n-p and p-p scattering 
experiments. 

2. Ground Level of the Deuteron 

a. Wave Function for the Rectangular-well Approximation. The 
wave function of the bound state of the deuteron is not markedly depend- 
ent on the exact shape of the potential U(r) between a proton and neu- 
tron, provided that a potential of short range is chosen. For simplicity, 
we may choose at first the rectangular potential well, of depth D and 
radius b, given by 

!7(r)--D r<b 

U(r) = r > b 

where r is the distance between the proton and neutron. 

The wave function t( r ,#,<p} which describes the relative motion of 
the proton and neutron can be separated into its radial ^i(r) and angular 
WtfjvO parts, because we have assumed a radially symmetric potential 
U(r). In our approximation we are interested only in the S state, I O f 
for which there is spherical symmetry and f 2 (#^>) is constant. Then if 
u(r) is the modified radial wave function, defined by 

u(r) = r,h(r) (2.2) 

the probability of finding the proton and neutron at a separation between 
r and r + dr is proportional to u*(r) dr. The boundary conditions on 
u(r) are 

u(r) = for r = and for r -> (2.3) 

so that ^i(r) will be noninfinite at r and will be zero at r - . 



314 The Atomic Nucleus [CH, 10 

The radial wave equation [Appendix C, Eq. (54)] for the relative motion 
becomes 



u . r , TT r^/\i f\ ff * *\ 

^ + ~^- [W - I (r)]* = (2.4) 

where M is the reduced mass of the proton and neutron (2M ~ M n ~ M p ) 
and W is their total energy in C coordinates. 

For the ground level of the deuteron, the total energy W is restricted 
to the single constant value 

W = -B (2.5) 

where B = 2.225 0.002 Mev is the observed binding energy of the 
deuteron. Then, for the regions inside and outside the rectangular 
potential well, the radial wave equation is 

g + ~ (D - B)u - r < 6 (2.6) 



The solutions which satisfy the boundary conditions of Eq. (2.3) on 

u(r) are 

u = Ai sin Kr r <b (2.8) 

n = A 2 c-< T/ * r > b (2.9) 

where A i and A z are arbitrary amplitudes, K is the effective wave number 
inside the potential well 



, 

X n 

and P = . . (2.11) 

V2MB 

Physically, p is equivalent to the rationalized de Broglie wavelength 
X of the relative motion of two particles having reduced mass M and 
sharing kinetic energy equal to the binding energy B of the deuteron. 
At r = b the usual boundary conditions [Appendix C, Eqs. (15), (16)] 
require that ^ and <ty/dr, and therefore u and du/dr, be continuous. 
Therefore 

/ '> (2.12) 



KA l cos Kb = - c-< 6/ "> (2.13) 

P 

Dividing, in order to eliminate AI and A 2 , we obtain 

K cot Kb = - - (2.14) 

P 

b. Relationship between Depth and Range of Potential. Equation 
(2.14) represents the relationship between the binding energy B of the 
deuteron and the depth D and radius b of a rectangular potential well. 



2] 



Forces between Nucleons 



315 



The explicit relationship is obtained by substituting Eqs. (2.10) and 
(2.11) intoEq. (2.14) and is 






This does not give explicit separate solutions for D and b, and it is help- 
ful to develop an approximation to Eq. (2.15). From many lines of 
evidence, the range of the nuclear forces is of the order of 2 X 10~ 13 cm. 
Substituting b ~ 2 X 10~ 13 cm and B = 2.22 Mev in Eq. (2.15) gives 
for the depth of the rectangular potential well D ~ 35 Mev. Then, in 
general, D B\ the right-hand side of Eq. (2.15) is small compared with 
unity, and the left-hand side can be roughly represented by cot (x/2). 
Then, approximately, 



Db 



AA 2 A 2 
~: ( I = 

\2/ 2M 



1.0 X 10- 24 Mev-cm 2 



(2.16) 



c. Shape of the Deuteron Ground-level Wave Function. The shape 
of the modified radial wave function u(r) is given by Eqs. (2.8) and (2.9) 
and is shown schematically in Fig. 2.1. Going out from r = 0, u(r) 
behaves like sin Kr for slightly more than one-quarter wavelength; then 
at and beyond the range of the force. u(r) becomes proportional to the 
exponentially decreasing function e~ (r/p) . The relaxation length p of the 
external part of u(r) is often referred to as ffie "radius of the deuteron. 19 



Ground level 3 S l ( bound) 




u(r) 



V=-D 



b 

- *T 

Fig. 2.1 The radial wave functions for the triplet and singlet slates of the deuteron, 
in the rectangular-potential-well approximation. The probability of finding the neu- 
tron and proton at a separation r is proportional to u 2 (r) dr. The external part of 
the triplet (ground level) wave function is nearly independent of the assumed potential 
and is proportional to fi~ (r/ ^, where the length p is called the radius of the deuteron. 



From Eq. (2.11), with 2M ~ M n , and B 
deuteron becomes 



2.22 Mev, the radius of the 



4.31 X 10" 13 cm 



(2.17) 



316 The Atomic Nucleus [CH. 10 

It is remarkable that the "radius" p of the deuteron is considerably 
larger than the range of the nuclear force. The neutron and proton 
actually spend the order of half the time at a separation greater than the 
range of the force which binds them together. This could not occur 
classically. It is a wave-mechanical phenomenon and is to be associated 
with the wave penetration, or tunneling, of potential barriers. The 
deuteron is seen to be a loosely bound, greatly extended structure, in 
which the average kinetic energy and the average potential energy of its 
constituents both greatly exceed the binding energy. 

d. Singlet State of the Deuteron. The ground level of the deuteron 
has nuclear angular momentum 7 = 1 and is therefore the *i state. 
Another 1 = state is formed when the proton and neutron have anti- 
parallel spins. Measurements of the cross section for the scattering of 
slow neutrons by hydrogen, to be discussed in Sec. 3, show that this ^o 
singlet state of the deuteron is definitely not a bound state but is unstable 
by the order of 70 kev. Its wave function may be represented sche- 
matically by the dotted line in Fig. 2.1, which does not quite reach a 
phase of v/2 at the edge of the well. 

e. Conventional Central-force Potential Wells. A number of poten- 
tials U(r) besides the rectangular well have been studied extensively. 
As long as these correspond to short-range forces the results are nearly 
independent of the exact potential U(r) assumed. In particular, it is 
always found that (a) the potential energy must be much greater than the 
binding energy of the deuteron and (b) the radial wave function of the deuteron 
decreases as e~ (r/p > outside the range of the nuclear force. 

The four most common potentials are 

Rectangular well: U(r) = E7 r <b /o IQ\ 

[/(r) =0 r > b l2 ' 18 ' 



Gaussian well: U(r) = - U e~W (2.19) 

Exponential well: U(r) = -Ur* (2.20) 



p- 

Yukawa well: U(r) = - U Q ' (2.21) 

(r/b) 

In each case, U represents the "depth" of the potential and b the 
"range" of the force. For precise studies, a comparative "well-depth 
parameter" and "intrinsic-range parameter" can be assigned to each 
form of potential (B68). 

Problem 

Recall that 6 ~ X/4 ~ v/2K for the ground level of the deuteron. From 
this, show that the radius of the deuteron, in the rectangular-well model, is 
given approximately by 

D 



3] 



Forces between Nucleons 



317 



3. Neutron-Proton Scattering at to 10 Mev 

The binding energy of the deuteron gives us a relationship between the 
depth and range of the 3 (np) force but does not suffice to determine either 
quantity uniquely, Eq. (2.16). To separate these parameters, and alsc 
to obtain information on the singlet interaction l (np), we turn to experi- 
ments on the scattering of neutrons by free protons. 

a. Energy Dependence of the n-p Scattering Cross Section. For 
incident neutron energies up to 10 Mev or somewhat greater, the angulai 
distribution of the observed scattering is isotropic in the center-of-mass , 
coordinates (A19). This means that only S-state interactions (I = 0) 
are involved in this energy range. On the reasonable assumption that 
some P-state interaction exists in nature, the failure to observe it at 
energies of ~10 Mev shows directly that the range of the (up) force is 
small. This is because a neutron and proton which share only 10 Mev of 
kinetic energy (laboratory coordinates) must be at a separation of at 
least / X 2.83 X 10~ 18 cm if they are to have a mutual angular momentum 
of Ih. As the I = 1 interaction is not observed, this minimum separation 
of 2.83 X 10~ 1J cm is greater than the range of the (np) force. 

A great many careful measurements have been made of the attenua- 
tion of monoenergetic neutrons by the hydrogen atoms in a variety of 
absorbers. From these, the total cross section cr t for neutron-proton 
interactions has been determined as a function of the laboratory kinetic 
energy of the incident neutrons. These values are shown in Fig. 3.1. 



JU 

10 

E 3 
I ' 
0,3 
tfO.1 

0.03 
0.01 




^^^ 





^ 


^^ 










H 














X, 


x 






















X 


\ 
























\ 
























\ 




\ 


















\ 


V 


f 


1 II Ml 





















0.001 0.003 0,01 0,03 0.1 0.3 1 3 10 30 100 300 
Neutron kinetic energy in laboratory coordinates. Mev 

Fig. 8.1 Observed total cross section for interaction of neutrons with protons, as a 
function of laboratory kinetic energy of the incident neutrons. [From Adair (A2).] 

The predominant contribution to the total cross section <r t is the n-p 
scattering cross section. The cross section for the competing radiative- 
capture reaction H 1 ^,?)!! 2 is only 0.05 barn per proton for 1-ev neutrons 
and decreases with 1/F, _where_ V IB the neutron velocity. 

The scattering of very slow neutrons by free protons is of special 
interest. The proton-containing absorber is commonly hydrogen gas, 



318 The Atomic Nucleus [CH. 10 

hydrocarbon gases and liquids, or liquid water. If thermal neutrons 
(~4/40 ev) are used, there are opportunities for exchanges of energy with 
molecular vibration levels, whose spacing is of the order of 0.1 ev. For 
neutrons whose energy is > 1 ev ("epithermal neutrons"), the proton 
may be considered to be unbound. From 1 ev to about 1,000 ev, the 
n-p scattering cross section, as measured with a slow neutron velocity 
spectrometer (M41), is nearly independent of neutron energy (Fig. 3.1). 
The hydrogen cross section in the energy range from 6.8 to 15 ev, after 
extrapolating out the small effects due to molecular binding, is variously 
identified as the scattering cross section <TQ for "zero-energy neutrons/ 1 
or for " epithermal neutrons' 1 by free protons, for which the measured 
value by Melkonian (M41) is 

<7 = 20.36 0.1 barns/proton " (3.1) 

b. Phase-shift Analysis of n-p Scattering. The theoretical descrip- 
tion of 5-wave, elastic n-p scattering emerges from the radial wave equa- 
tion (2.4) when the total energy W is taken as the (positive) mutual 
kinetic energy of the neutron and proton in C coordinates. The incident 
neutrons are represented as the plane wave e ikz , and the total disturbance 
^totai consists of the incident and scattered waves 

ikr 

(3.2) 



where the complex quantity f(d) is the scattering amplitude in the direc- 
tion tf and k is the wavfe number of relative motion, given by 

(3.3) 
n 

where M =- reduced mass ~ M n /2 ~ A/,,/2 

W = incident kinetic energy in C coordinates 
W ~ ^ (incident neutron energy E n in L coordinates) 
The corresponding solutions are developed in Appendix C, Sec. 6. 
We quote here only the pertinent results. For s-wave (I = 0) interac- 
tions, the scattering amplitude /(#) is isotropic and for any short-range 
central force has the value ' 

C 2i6, _ i e i** s i n fr x 

/o = -r = "IT- (3 - 4) 

where the purely real quantity 5 is the phase shift for s-wave scattering. 
The s-wave total-elastic-scattering cross section <r then becomes a func- 
tion of 6 , and of the incident neutron energy as represented by k, 

* = 47r|/c| 2 = ^sin 2 a (3.5) 

An experimental determination of a can therefore be expressed equally 
well as a measured phase shift 6 - It is then the task of theory to predict 



3] 



Forces between Nucleons 



319 



matching values of 5 , which will be expected to depend on k and on the 
shape, range, and depth of the assumed nuclear potential well. The 
measured phase shifts 5 have become the "common meeting ground" 
of experiment and theory. 

c. Scattering Length. In the limit of very small neutron energies 
E n * (and hence k 0), the scattering amplitude Eq. (3.4) takes on a 
particularly simple form. It can be seen from Eq (3,4) that as fc > 0, 
do must also approach zero, otherwise /o would become infinite. Then, in 
the limit, e* 6 * > 1 and Eq. (3.4) becomes 



as 



(3-0) 



where the length +a is called the scattering length in the convention of 
Fermi and Marshall (F40). Although / is, in general, a complex quan- 
tity, the scattering length a is to a very good approximation a real 
length. (Exceptions occur only in unusual cases near a resonance level 
where a large amount of absorption competes with the elastic scattering.) 




Positive scattering 
length, bound state 




Negative scattering 
length, unbound state 



Fig. 3.2 The "scattering lengtn a, or extrapolated Fermi intercept, is positive for 
scattering from a bound state and negative for scattering from an unbound state. 

A simple geometrical interpretation of the scattering length can be 
visualized. Outside the range of the nuclear force, U(r) = and the 
total wave function Eq. (3.2) has the value [Appendix C, Eq. (131)] 



e tfio 

= - sin (kr 



In the limit of fc * 0, this becomes 



(3.7) 



(3.8) 



which is the equation of a straight line crossing the r axis at r = a. 
Figure 3.2 shows that the scattering length can be interpreted physically 
as the intercept of r^o^i on the r axis, for zero-energy particles, when a 
linear extrapolation of r^ tot j is made from a distance just outside the 
range of the nuclear force. For this reason, the scattering length is some- 
times called the Fermi intercept of r 



320 The Atomic Nucleus [CH. 10 

From Eqs. (3.5) and (3.6) the zero-energy scattering cross section 
becomes 

ao = 4ira 2 (3.9) 

which is the same as the zero-energy scattering from an impenetrable 
sphere of radius a. We see from Eq. (3.9) that CTQ determines the magni- 
tude of (he scattering length a, but not its sign. 

d. n-p Scattering for a Rectangular Potential Well. For any poten- 
tial well, the phase shift 5 can be evaluated by joining the external wave 
function Eq. (3.7) to an internal wave function fan whose form is deter- 
mined by the parameters of the potential well. Inside a rectangular 
potential well of depth D and radius 6, the radial wave equation (2.4), 
for particles whose total energy has the positive value W, is 



T + Z>)u = (3.10) 

and has the solution 

u = rfcn = A! sin Kr (3.11) 



where K> = + (3.12) 

n* 

When this internal wave is joined to the external wave of Eq. (3.7) by 
requiring ^ and dty/dr to be continuous at the edge of the rectangular 
well r = 6, the result [Appendix C, Eq. (142)] is 

k cot (kb + ) = K cot Kb (3.13) 

Substituting for the internal and external wave numbers K and k their 
values from Eqs. (3.3) and (3.12) leads to 



. . t , /Q ,,. 

cot - b + 6o = - cot - - b (3.14) 



,\ 
bj 



as an implicit relationship which gives the phase shift 6 produced at a 
collision energy W by a rectangular well of depth D and radius b. This 
relationship is analogous in form to Eq. (2.15) which describes the binding 
energy B of the deuteron in terms of the same rectangular well D, b. 

By algebraically combining Eqs. (3.14) and (2.15) we can eliminate 
the well depth D. Then we can evaluate 6 and the n-p cross section a 
in terms of the remaining parameters. The result, after making use of 
the approximations D B, D T7, and for low-energy neutrons such 
that kb 1, is 

A_ is i / r. \ 

(3-15) 



h/V2MB = 4.31 X 10-" cm = deuteron radius, Eq. (2.17) 

M ~ M n /2 ~ M p /2 = reduced mass 

W ^ En/2 = kinetic energy in C coordinates 



3] Forces between Nucleons 321 

This relationship is found to be in satisfactory agreement with meas- 
ured n-p scattering cross sections for neutrons whose kinetic energy IB 
large enough so that the denominator (W + B) is not dominated by the 
binding energy B. Thus there is good agreement for E n ~ 5 to 10 Mev. 

e. Spin Dependence of Nuclear Forces. At low energies, however, 
the situation is very different. Numerical substitution in Eq. (3.15) of 
B = 2.22 Mev, M ~ M n /2, b ~ 2 X 10~ 18 cm gives a predicted value 
for "zero-energy neutrons," W = 0, 

<TO ^ 3.5 barns for B = 2.22 Mev (3.16) 

which is in violent disagreement with the measured value <TQ = 20.4 
barns. Wigner first pointed out^Ehat this disagreement is due to the 
tacit assumption that singlet and triplet s-wave interactions are equal. 
The binding energy B, which dominates W in the evaluation of <T O , applies 
only to the ground level of the deuteron, hence to a triplet interaction 
between the colliding neutron and proton. 

When unpolarized neutrons strike randomly oriented protons, their 
uncorrelated spins add up to unity in three-fourths of the collisions and 
to zero in one-fourth of the collisions. This is equivalent to saying that 
the triplet state (S = 1) has three times the statistical weight (2S + 1) 
of a singlet (S = 0) state. Accordingly, the average cross section cr for 
"zero-energy neutrons" should be written 

<ro = !('<ro) + TOoo) 

= T[3(a)> + Oa)>] (3.17) 

where 8 <r and 3 a refer to triplet collisions, while Vo and l a apply to 
singlet collisions. Then Eq. (3.16) becomes V ^3.5 barns, and it is 
clear that Vo )S> W ! 

To the extent that the simple central-force rectangular potential well 
is at all representative of the true character of nuclear forces, one would 
have to conclude that the 3 (np) and l (np) potentials are quite different. 
For orientation, one typical set of rectangular-well parameters, which is 
derived by making use of additional types of scattering experiments, is 
(C3) 

s b = 2.0 X 10~ 13 cm, to match neutron scattering by parahydrogen 
8 D = 36.6 Mev, combined with '& to match B ~ 2.22 Mev for the 
deuteron 

l b = 2.8 X 10~ 1B cm, to match p-p scattering 
1 D = 11.9 Mev, to match <T O ~ 20 barns for n-p scattering 

The "zero-energy neutron" cross section <T O shows that the singlet scatter- 
ing length l a must be large but, because of the squares in Eq. (3.17), 0-0 
does not tell the sign of 1 a. This can be done by neutron scattering in 
parahydrogen, as will be discussed later. It is found that l a is negative 
and that therefore the singlet state of the deuteron is not bound. 

f . Effective Range of Nuclear Forces. The various commonly used 
shapes of nuclear potential, Eqs. (2.18) to (2.21), have been blended by 
Schwinger, Bethe (B45), and others (B68) in an "effective-range theory 



322 The Atomic Nucleus [CH. 10 

of nuclear forces," or "shape-independent approximation/' by the intro- 
duction of a second parameter (in addition to the scattering length a) 
called the effective range r Q . 

Recall Eq. (3.13). It can be shown quite generally (B45, B68) that 
for any reasonable shape of potential well 



1 /"- 

k cot fi = h A- 2 / (i'i'o - MM<0 dr (3,18) 

a JQ 

where v = modified radial wave function r\f/, oulsidr range of nuclear 

potential, where U(r) = 
t'o = y, for zero incident energy W = 
u = modified radial wave function r\l/, inside range of nuclear 

potential U(r) 

u = u, for zero incident energy W = 

Equation (3.18) is exact. The significant contribution to the integral 
comes from inside the range of the nuclear force, < r < It. In this 
region, and for collision energies W which are not too large, U(r) W, 
and we can make the approximations v ~ z and u ~ w . Then the 
length TO, called the effective range, is defined by 

r =2/ o " (i>!-u!)dr (3.19) 

where the factor 2 is arbitrarily inserted so that the approximation will 
give an effective range TO which is near the outer "edge" of the potential 
well. Thus r is generally comparable in magnitude with b in Eqs. (2. L8) 
to (2.21). The phase shift 5 of Eq. (3.18) is given to good approximation 
(W < 10 Mev) by 

k cot 5 = - - + - fcVo (3.20) 

a 2 

where 5o = s-wave phase shift 

k = wave number of relative motion, Eq (3.3) 

a = scattering length of nuclear potential 

r = effective range of nuclear potential 

The effective range r depends upon the width and depth of the poten- 
tial well U(r), as Eq. (3.19) shows, but not upon the incident energy, 
which is given by fc 2 . The experimental values of 5 and k serve to 
determine the two parameters a and r . Any reasonable potential shape, 
such as those of Eqs. (2.18) to (2.21), can be made to give matching 
values of a and r by suitable choice of its depth f/ and range b. Hence 
the two experimentally determined lengths a and r do not determine the 
shape of the nuclear potential, but if the shape is chosen arbitrarily then a and 
r u fix the depth UQ and range b. For this reason, Eq. (3.20) is known as 
the "shape-independent approximation." 

g. Ground Level of the Deuteron in the Shape-independent Approxi- 
mation. By replacing the radial wave functions in Eq. (3.19) by those 
which are appropriate to the bound triplet state of the deuteron, for 



3] Forces between Nucleons 323 

which W = B, the shape-independent approximation leads to 



where p = H/V2MB = 4.31 X 10~ 13 cm is again the "radius" of the 
deuteron as defined by Eq. (2.17). 

h. n-p Scattering Cross Section in the Shape -independent Approxi- 
mation. For the s-wave, n-p scattering cross section a we can now write 
a more general expression than Eq. (3.15), which was derived for a rec- 
tangular potential well. In order to introduce the binding energy of the 
deuteron, we eliminate the scattering length a, between Eqs. (3.20) and 
(3.21), and substitute the resulting value of cot 6 into 

V = ~ sin 2 So = -^ - 2 (3.22) 

The result is 



\ + kVj [I - CTO/P) + (V /2p) 2 (l + 



i 

*V)J 



where p = h/V2AfB 

fc2p2 = W/B 

from Eqs. (2.17) and (3.3). Notice that Eq. (3.23) is essentially the same 
as Eq. (3.15), the difference being that the radius b of the rectangular well 
is replaced by the effective range r , and a second-order range correction 
term in (V /2p) 2 appears in the final bracket of Eq. (3.23). 

Equation (3.23) and also the rectangular-well approximation Eq. 
(3.15) apply rigorously only to the triplet scattering, which involves the 
binding energy B of the ground level of the deuteron. If there were also 
a bound singlet state, with binding energy VB T Eqs. (3.23) and (3.15) 
could be applied for V by using 1 B in place of B. In the absence of a 
bound singlet state, it is best to return to Eq. (3.20) and thus to express 
the singlet n-p scattering cross section in terms of the singlet effective 
range V and the singlet scattering length 1 a. Then 



, = _ _ = __ __ ( . 

fc 2 1 + cot 2 5 [1 - k*( l a)( v r*)/2Y + /^('a) 2 ' 

The total cross section is then given by 

<r = !(V)+l(V) (3.25) 

i. Coherent Scattering of Slow Neutrons. An experimental decision 
on the sign of the singlet n-p scattering length l a, and hence whether 
the singlet state of the deuteron is bound ( l a > 0) or virtual ( l a < 0), 
can be obtained by measurements on any phenomenon which has a 
strong dependence on the first power of 1 o. Such effects occur in several 
important types of experiments on the coherent scattering of neutrons. 

Scattering of Neutrons by Para- and Orthohydrogen. The internuclear 
distance in the hydrogen molecule is'0.78 X 10~ 8 cm. Coherent scatter- 



324 The Atomic Nucleus [cfl. 10 

ing, in which the amplitudes instead of the intensities add, may be 
obtained from the pair of protons in the hydrogen molecule provided 
that the de Broglie wavelength of the neutrons is much greater than the 
internuclear distance. Neutron velocity selector techniques can be used 
to conduct scattering experiments With "cold" neutrons, in the energy 
domain 0.0008 ev (~10K) to 0.0025 ev (~30K). Here the de Broglie 
wavelength is 10 to 5.7 X 10~ 8 cm, and only a small correction (~10 per 
cent) needs to be made for the relative phase introduced by the finite 
separation of the two protons in the hydrogen molecule. 

In the parahydrogen molecule, the nuclear spins of the two protons 
are antiparalleL Then an incident unpolarized neutron can have, so to 
s~peakj a triplet collision with one of the protons and a singlet collision 
with the other proton. The actual interference effects can be determined 
by using the Pauli spin operators of the neutron and protons. Then it 
can be shown that (S23, B43, B68) the coherent scattering cross section 
for cold neutrons by parahydrogen is N 



ap.) 2 ' (3.26) 

where the coherent scattering length a pm is 

= 2[f ('a) + *Ca)] (3.27) 



Here *a and l a are the usual triplet and singlet n-p scattering lengths 
Eq. (3.6) for free^protouB. The factor 2 in Eq. (3.27) represents the 
two protons in each hydrogen molecule. In Eq. (3.26), the factor 
^f- = (-J)* corrects for the reduced mass M in the neutron-hydrogen- 
molecule collision where M = |Af , as compared with the reduced mass 
in the neutron-free-proton collision where M = ?M . 

In the orthohydrogen molecule, the proton spins are parallel, and the 
neutron coherent scattering cross section is 



(3.28) 
where the coherent ortho scattering length a ortho can be shown to be 

* (3.29) 



Here the factor 2 at the second square brackets represents physically 
T(T + 1), where T is the total nuclear spin of the hydrogen molecule. 
This term is present only for orthohydrogen (T = 1). Equation (3.29) 
is physically the same as Eq. (3.27), except that, forjparahydrogen 7 = 0, 
the second square-bracket term disappears, and the general form 'which is 
implicit in Eq. (3.29) reduces to Eq. (3.27). 

The ratio of the cross sections given by Eqs. (3.26) and (3.28), for 
elastic coherent neutron scattering from the two forms of molecular 
hydrogen, then reduces to 



Several clear-cut predictions follow at once. 



3] Forces between Nucleons 325 

1. If the total (up) forces are spin-independent, i.e., if *(np) = l (np), 
then there should be no physical difference between ortho and para 
scattering 

5=S= - 1 if "a = 'a (3.31) 

<TP 

2. When the measured cross section of <T O ~ 20 barns for elastic 
scattering of epithermal neutrons by free protons and the measured bind- 
ing energy of the deuteron B = 2.22 Mev are combined with the theory 
of the (up) force in the shape-independent approximation Eqs. (3.23) and 
(3.24), one would conclude that for purely central forces 

'a ~ P = +4.3 X 10- 13 cm ,, ,v 

l a ~ 24 X 10- 18 cm l ; 

where the sign of l a cannot be determined because l a occurs only quad- 
ra tic ally in Eq. (3.24) for very slow neutrons k > 0. Making use of 
these orders of magnitude in Eq. (3.30), we find that the ratio 0- or thoA w 
is extremely sensitive to the sign of the singlet scattering length. Indeed 

~ 1.5 if 'a ~ +20 X 10- cm (3.33) 

~ 14 if l a ~ -20 X 10-" cm (3.34) 



The experimental values for the effective cross sections, at 0.002 ev, are 

(S81) 

<r ortho ~ 120 barns (3.35) 

(Tp. ~ 4 barns (3.36) 

Hence ^ 30 (3.37) 

<Tp.r. 

Unequivocally, these observations prove that: 

1. The total (np) forces are spin-dependent, '(np) ^ l (np). 

2. The singlet n-p scattering length l a is negative; therefore the singlet 
state of the deuteron is unbound. 

An additional consequence of the large observed value of <T or * /<rm 
relates to the spin of the neutron. The ground state of the deuteron 
could still have I = and 7 = 1 if the neutron spin were s n = 1 and 
were aligned antiparallel to the proton spin (which is known definitely 
from band spectra to be s p i). In such a model the relative statistical 
weights for n-p collisions with free protons would change from their values 
of I and T in Eq. (3.17) to values of and f. Analogous reweightings 
would occur in the ortho- and parahydrogen scattering cross sections. 
The over-all result of these reweightings is 



Tortho 



~ 2 if 8n = 1 (3.38) 



The observations therefore serve a third purpose, by showing that: 
3. The neutron has soin i. not 4. 



326 The Atomic Nucleus [CH. 10 

Experimentally, there are many complications in the ortho- and para- 
hydrogen experiment. Among others, these include: (1) transitions 
between orthohydrogen (ground-state molecular-rotation quantum num- 
ber L = 1) and parahydrogcn (ground-state molecular-rotation quantum 
number L = 0) induced by inelastic collisions with neutrons; (2) Doppler 
corrections for the thermal motion of hydrogen molecules, even at the 
usual operating temperatures of ~20K; (3) possible intermolecular forces 
between adjacent hydrogen molecules; and (4) radiative-capture reactions 
H l (n,y)ll 2 in which a neutron is removed from the incident beam and a 
deuteron is formed. 

Scattering of Slow Neutrons by Crystals. The_de Broglie wavelength 
of a 0.1-ev neutron is 0.9 A, which is comparable with the atomic separa- 
tions in solids and liquids. Therefore diffraction studies can be carried 
out using slow neutrons (H(i8, Bl) under physical principles which are 
entirely analogous to those involved in the Bragg coherent scattering of 
X rays. 

The neutron scattering amplitudes, for various atoms, depend upon 
nuclear properties, whereas the corresponding X-ray scattering amph- 
fu3es" depend upon the number of electrons in the atom. There are 
therefore important differences in th~e relative intensity of neutron and 
of X-ray scattering by different types of atoms in a crystal. For example, 
in crystalline sodium hydride NaH the X-ray diffraction patterns are 
dominated by the scattering from Na and give no information on the loca- 
tion or behavior of H. On the other hand, the neutron scattering ampli- 
tude of Na is small enough that the neutron diffraction pattern is clearly 
influenced by scattering from H and serves to determine both the position 
of H in the crystal structure and the neutron scattering amplitude ol 
H (S36). 

Neutron-diffraction studies on crystalline powders can now be carried 
out routinely by utilizing the strong neutron flux available from uranium 
reactors. Monoenergetic beams of neutrons, in the angstrom region oi 
wavelengths, are obtainable by Bragg reflection from crystals in the same 
manner as X rays are monochromatized. 

The incoherent (diffuse) scattering cross section for a mixture contain- 
ing several nuclides is 

<r = **(pial + p&\ + - - -) (3.39) 

where pi, p 2 , . - . are the relative abundances of nuclides whose bound 
scattering lengths are oi, a 2 , .... These bound scattering lengths cor- 
respond to the scattering which would be observed if the struck nucleus 
were infinitely heavy, so that the reduced mass for a neutron and a bound 
atom is equal to the mass of the neutron. Therefore the bound scattering 
lengths differ from the scattering lengths for a free atom, and 

(3-40) 



where A is the mass number of the atom. 



3] Forces between Nueleons 327 

The coherent (Bragg) scattering is observed only at the appropriate 
Bragg angles and has the cross section 

tfcou = 47r(pitti + pza 2 + -) 2 (3.41) 

where the p's are the relative abundances and the a's are the bound 
scattering lengths. 

For nuclei with nonzero nuclear angular momenta /, each bound 
scattering length is the statistically weighted sum for the (/ + -J-) and 
(/ jt) interaction with the incident neutron. Thus the bound coherent 
scattering length for hydrogen a becomes 

an = 2[( 3 a) + i(i a )] (3.42) 

where 3 a and l a are the usual triplet and singlet scattering lengths for 
free protons and the factor 2 arises from the reduced mass correction of 
Eq. (3.40). Recall Eq. (3.27) and note that a H is identical with a^, 
the coherent scattering length for both atoms in the parahydrogen 
molecule. 

Coherent neutron scattering from crystalline powders such as NaH 
can therefore be used to evaluate a H = fl p r tt by experimental methods 
which are free from many of the difficulties of the low-temperature para- 
hydrogen scattering experiment. In this way, Shull and coworkers 
(S36) obtained in 1948 the value 

OH = *[3( 3 a) + l a] = (-3.96 0.2) X 10~ 13 cm (3.43) 

which is an improvement on the parahydrogen result, Eq. (3.36), but, 
in the light of later evidence, Eq. (3.43) appears to contain a small but 
significant systematic error. 

Reflection of ftlow Neutrons from Liquid Mirrors. Jt can be shown 
that, if absorption is small compared with scattering, the, index- of refrac- 
tion n for neutron waves on a homogeneous material is 



1 /Q^n 

n = 1 ---- , (3.44) 

! 27T 

I 

where a = average bound coherent scattering length 

X = de Broglie wavelength of incident neutrons ! 
N = scattering nuclei per cm 3 

If a is positive, so that n is less than one, there will be a critical angle tf c 

given by 

cos tf c = n (3.45) 

at which neutrons impinging on the material will experience total specular 
reflection back into the air. A material from which neutrons incident 
at a glancing angle arc totally reflected is spoken of as a neutron "mirror." 
The angle # r is always small, e.g., in the case of beryllium a = 7.7 X 10" 13 
cm, and for neutrons of 1-A wavelength, 1 n = 1.5 X 10~ 6 , and 

# c ~o.r. 

The critical angle for total neutron reflection can be measured in a 
straightforward way. Various incoherent effects which might seriously 



328 The Atomic Nucleus [GH. 10 

disturb other types of scattering experiments are less troublesome because 
they only reduce the intensity of the reflected beam, without changing 
& c which is the quantity measured. Observations on substances whose 
bound coherent scattering length is negative can be accomplished by 
mixing them with substances which have sufficiently positive scattering 
lengths to give a net positive value, and hence observable total reflection. 
In this way, Hughes, Burgy, and Ringo (H68) have measured the bound 
coherent scattering length for hydrogen a n , by using various liquid 
"mirrors' 1 of the hydrocarbons d 2 Hi 8 , C 6 Hio, C 6 Hi2, and taking advan- 
tage of the accurately known positive bound coherent scattering length 
of carbon a c = (+6.63 0.03) X 10~ 18 cm. These experiments gave 
a c /a H = (-1.753 0.005) and therefore 

a H = *L3( 3 a) + 'a] = (-3.78 0.02) X 1Q- 18 cm (3.46) 

which is currently regarded as the best available measurement of the 
coherent scattering lengths for the (up) interaction. 

This important experimental parameter of the (np) interaction has 
recently been confirmed by very careful ortho- and parahydrogen neutron 
scattering experiments at the Cavendish Laboratory (S75), which give 
for the coherent scattering length 

a pm = (-3.80 0.05) X 10~ 13 cm 

in good agreement with Eq. (3.4G). 

j. Nuclear-force Parameters in the Shape -independent Central-force 
Approximation. We now have enough experimental data and intercon- 
necting theory to permit evaluation of the four parameters which enter 
the "shape-independent approximation" or "effective-range theory" of 
nuclear forces. We summarize the theoretical and experimental rela- 
tionships, with their original equation numbers for ease of reference. 

1. n-p scattering cross section for free protons, extrapolated to "zero 
energy neutrons" 

cro = 7r[3( 3 a) 2 + ('a) 2 ] (3.17) 

a = 20.36 X 10~ 24 cm 2 (3.1) 

2. Bound coherent n-p scattering length 

<W = a = ^[3( 3 a) + ']* (3.27), (3.42) 

o H = (-3.78 + 0.02) X 10-" cm (3.4G) 

3. Binding energy of the deuteron 

B = H> + 71 - IP 

= 2.225 0.002 Mev Chap. 3, Eq. (4.46) 

4. Radius of the deuteron 

P 7 _L^ = 4.31 X 10- 11 cm (2.17) 

V2MB 



3] Forces between Nuckons 

5. Effective range 8 r of triplet (up) force 

1 ,1,1 ( a r ) 



329 



(3.21) 



P ( 3 a) 2 p* 

6. Variation of total n-p scattering cross section with neutron energy 

cr = f (V) + |(V) (3.25) 

TTtV)] (3 ' 23) 



= 47r( 1 a) 2 



(3.24) 



Simultaneous solution of items 1 and 2 gives the two triplet and singlet 
scattering lengths 3 a and 1 a. With 3 a determined, item 5 gives the triplet 
effective range 3 r . The experimental variation of (7, for neutron energies 
from 0.8 to 5 Mcv (L3), gives the singlet effective range V . The "1952 
best values" for the central-force, shape-independent approximation to 
the (np) nuclear force are (S3, B114) 



For \np) 
and for l (np) 

Problems 



3 a = +5.378(1 0.0038) X 10~ 13 cm 
"TO = +1.70(1 0.018) X 10~ 13 cm 
l a = -23.69(1 0.0022) X 10~ 13 cm 
V = +2.7(1 0.19) X 10- 18 cm 



(3.47) 
(3.48) 



1. In the collision of a neutron with a proton, show that the classical impact 
parameter x must exceed (9J/\/^) X 10^ 18 cm, if I is the angular-momentum 
quantum number for the collision and E n is the laboratory kinetic energy of the 
incident neutron in Mev. 

2. When neutrons of the order of 1 Mev are scattered by hydrogen, it is found 
that the angular distribution of recoil protons is isotropic in the center-of-mass 
coordinates. Thus, in the C coordinates, the differential cross section per unit 
solid angle is independent of the angle of scattering. This could be written 
da = cdtl = c2v sin d, where c is a constant whose value is <r/47r, if the total 
scattering cross section is a. If E n is the kinetic energy of an incident neutron, 
show that the differential cross section for the production of a recoil proton with 
kinetic energy between E p and E p + dE p in the laboratory coordinates is inde- 
pendent of E p and is equal to 

dE p 
da = a 




330 The Atomic Nucleus [CH. 10 

3. Carry through the derivation of the total n-p scattering cross section <r as 
given by Eq. (3.15), using the phase-shift relationship of Eq. (3.14) as starting 
point. 

, 4, Using the parameters of the shape-independent approximation, determine 
the absolute value of the (negative) binding energy \ 1 B\ of the singlet state of the 
deuteron. Arts.: 66 kev. 

4. Electromagnetic Transitions in the n-p System 

The 3 (np) and l (np) forces also govern two other ba.sic processes in the 
neutron-proton system : 

n + H 1 > H 2 + hv radiative capture (4.1) 

and the inverse process 

H 2 + hv > n + H 1 photodisintegration (4.2) 

From a study of these two interactions we can: 

1. Confirm that 3 (np) j l (np) 

2. Show that the singlet state of the deuteron is unbound 

3. Obtain a rough measurement of (V Vo) 

a. Selection Rules for Transitions below 10 Mev. The cross section 
0-o.p for the radiative-capture process, Eq. (4.1), is only significant for 
very-low-energy neutrons. Even then, it is very small compared 
with the elastic n-p scattering cross section <r . For thermal neutrons 
(E n = 0.025 ev) cr Mp = 0.3 barn, whereas <T O is over 60 times as large. 
Physically, the radiative-capture cross section is small because the cou- 
pling between matter and electromagnetic radiation is always weak, 
being represented in general by the appearance of the fine-strurture 
constant 2ire 2 /hc = r^r in the formulas for all types of radiative 
processes. 

Magnetic Dipole Radiative Capture. The capture of a neutron by a 
proton is a radiative transition from the continuum of unbound n-p states 
to the 8 Si ground state of the deuteron. In principle, the initial state 
may have angular momentum L = 0, 1, 2, ... and either parallel or 
antiparallel spins. However, for L > 1, and incident neutron energies 
below about 10 Mev, the n-p separation exceeds the range of the (np) 
force. Consequently, capture is only important from S states (L = 0) 
in the continuum. Then the initial and final states both have even 
parity, and the 7-ray transition between them has no change in parity. 
This restricts the important radiative-capture transitions to the magnetic 
dipole, or Ml, transition, for which AL = 0, AS = 1, A/ = 1, no (Chap. 
6, Sec. 4). This Ml transition involves only a spin flip and takes place 
between a singlet in the continuum and the triplet deuteron ground 
state, ^So -> 'Si. 

Photomagnetic Disintegration of the Deuteron. In the inverse process, 
Eq. (4.2), the " photomagnetic disintegration" consists of Ml transitions 
from the deuteron ground state *Si to ^o states in the continuum. 



4] Forces between Nucleons 331 

Photoelectric Disintegration of the Deuteron. An additional process 
can be effective if the incident photon energy hv is clearly greater than 
the binding energy of the deuteron. Then an El, or electric dipole, 
transition (AL = 1, AS = 0, A/ = 1, yes) can carry the n-p system from 
the 3 Si ground state to a Z P state in the continuum. 

Quadrupole transitions, M2 and E2, are generally much less probable 
than dipole transitions (Chap. 6, Sec. 4); hence at moderate photon 
energies (< 10 Mev) only the Ml and El transitions can be expected 
to be significant. 

b. Cross Section for Photomagnetic Capture. The matrix element 
which governs the probability of the " spin-flip" magnetic dipole transi- 
tion can be shown to be proportional to the net magnetic dipole moment 
in the singlet state (B68). For the central-force model the net magnetic 
dipole moment is 

- - M - } (4 - 3) 



where \L P and \L U are the magnetic dipole moments of the proton and 
neutron in units of the nuclear magneton eh/4irMpC. It is interesting 
to note that the 1 S > VJ transition probability would be zero if the 
neutron and proton had equal magnetic dipole moments. 

The matrix element for the photomagnetic-capture transition is also 
proportional to the integral over all space of the product of the singlet 
and triplet wave functions (^XV)- This integral has a nonzero value 
because the singlet potential differs from the triplet potential. The 
1 H > 3 jS Y transition probability and the corresponding radiative-cap- 
ture cross section are finite because the nuclear force is spin-dependent, 
that is, *(np) 7* l (np)j as well as because fj. p ^ fi n . The principal con- 
tribution to the integral over all space comes from the region outside the 
range of the nuclear forces. Then a good approximation can be obtained 
simply by assuming that the external wave functions are valid all the 
way from r = eo in to the origin, r = 0. This procedure is equivalent to 
neglecting the effective ranges V and V of the nuclear force. In this 
central-force zero-range approximation, the cross section o- oap for photo- 
magnetic capture of a neutron by a proton is (R4, BC8) 



- ~ 137 * n \2Alc kp* [1 + ( W] 

where the definition of each symbol remains the same as in Sec. 3. 

The physical behavior of a mv is more apparent if we replace the wave 
number k of the colliding neutron and proton, and the radius p of the 
deuteron, by their energy equivalents 

, 2 2MW , 2 ft 2 tA ,. 

V = and p> = _ (4.5) 

where W = incident kinetic energy in C coordinates 
B = binding energy of deuteron 
M = reduced mass of neutron and proton 



332 The Atomic Nucleus [CH. 10 

Alter some algebra, Eq. (4.4) becomes 



- (46) 

137 ' Mc. \2Mc> \ [TTOa/cW l } 

Several important results emerge from the comparison of Eq. (4.6) with 
the experimentally determined value (W43, H20) 

fftMp = 0.332(1 0.02) barn for 0.025-ev neutrons (4.7) 

Singlet State of the Deuteron. First, consider the factor [1 ( l a/p)] 2 
in Eq. (4.6). Using the numerical values of the singlet scattering length 
l a and the deuteron radius p as given by Eqs. (3.48) and (2.17), we find 
(*a/p) = i5.50. Then the expression in the square brackets has the 
value [6.50] 2 = 42.25, and Eq. (4.6) gives a^ = 0.30 barn for 0.025-ev 
neutrons, in fair agreement with the experimental values. If we were 
to use the same absolute value of l a, but with a positive sign (corre- 
sponding physically to a bound singlet state of the deuteron), then the 
square brackets become [4.50] 2 = 20.25, and o- cap = 0.14 barn. This is 
clearly excluded. 

Historically, the first experimental proof that the *S state of the 
deuteron is virtual was obtained by Fermi (F35) who showed in 1935 that 
(1) the lifetime of slow neutrons in paraffin is only about 10~ 4 sec, (2) 
their disappearance is due to photomagnetic capture by protons, (3) the 
cross section for capture is ~0.3 barn, and (4) therefore the singlet state 
of the deuteron is unbound. 

The l/V Law. For low-energy neutrons, W B and also ('a/.*) 2 <C 1 
in Eq. (4.G). Then the velocity V of the neutron is proportional to II' 5 , 
and the cross section for neutron capture takes on the particularly simple 
form 

for W B (4.8) 



Physically, this says that the probability of neutron capture is propor- 
tional to 1 /velocity, i.e., to the length of time which the neutron spends 
in the vicinity of the target proton. The l/V law is applicable not only 
to the simple H ! (n,7)H 2 capture reaction but to neutron-capture reactions 
generally. All such reactions (n,y), (n,a), etc.. exhibit a l/V cross sec- 
tion for slow neutrons. In addition, there may be peaks, superimposed 
on the l/V cross section, at particular neutron energies which correspond 
to the resonance capture of the neutron into a level of the compound 
nucleus [Chap. 14, Eq. (1.1)]. 

Effective Range of Nuclear Forces. Approximate expressions for the 
correction to Eq. (4.6) for the finite effective ranges V and 8 r of the 
nuclear force were first obtained by Bethe and Longmire (B50). The 
first-order correction does not depend on the absolute value of Vo or 8 r 
but upon their difference ( T r V ). From the presently available data 
(B50, S3) ( X r - 3 r ) = (0.8 + 0.4) X 1Q- 18 cm, which is in acceptable 
agreement with the values of effective range obtained from scattering 
experiments, Eqs. (3.47) and (3.48), as interpreted on the central-force 
shape-independent model. 



4] Forces betuven Nuckons 333 

Additional corrections for the effects of tensor forces, and for the 
additional (exchange) magnetic moment, which can be associated with 
the exchange of charge between the nucleons (F48), are small and are 
presently comparable with the ~6 per cent uncertainty in the experi- 
mental value of cr^, Eq. (4.7). 

c. Reciprocity Relationship between Cross Sections for Inverse Proc- 
esses. The reactions Eqs. (4.1) and (4.2) are illustrative of inverse 
nuclear processes, and their cross sections obey a universal reciprocity 
relationship. Symbolically, consider any two inverse reactions 

a + A-+B + b + Q (4.9) 

b + B-> A+a- Q (4.10) 

where Q is the energy release in the "direct" reaction A B. Then 
if the particles a, A , B, b are all spinless, it can be shown that (F41, B43, 
B52, BG8) the cross section a(B > A) for the inverse reaction Eq. (4.10) 
is 

<r(B -> A) 

<r(A -> B) = 

where a(A * B) is the cross section for the direct reaction Eq. (4.9) and 
p a is the momentum of relative motion of the incident particle a in the 
entrance channel which produces the momentum p h of b in the exit 
channel of the direct reaction. More explicitly, the momentum p a is 
related to the channel wavelength X a = h/p a and is given by 

pi = 2M m W m (4.12) 

where M a = reduced mass of a + A 

W a = kinetic energy of a + A , in C coordinates 
Analogously, the exit channel wavelength is X b = h/p b and 

pi = 2M b W b (4.13) 

where Mb = reduced mass of b + B 

Wb = kinetic energy of b + B, in C coordinates 

From conservation of energy, the kinetic energies follow the usual 
relationship 

W a + Q = W b (4.14) 

If the particles have intrinsic; angular momenta 7, but the reaction cross 
sections are independent of the relative orientations of the 7's, then the 
cross sections are each to be multiplied by the statistical weight of all 
the states in their respective channel. Thus, for particles with angular 
momenta 7 tt , I A, IB, 7 6 , the general relationship which replaces Eq. (4.11) 
is 

(27 b + 1)(27 B + l)pl<r(B-> A) 

= (2I a + 1)(27 A + l)p! *(A - B) (4.15H 

t An interesting experimental example of Eq. (4.15) is the detailed correspondence 
between the reaction Al"(p,)Mg" (Q _ +1.61 Mev; 7 = ; I A = |) and its 
inverse reaction Mg a4 (a,p)Al 27 , as reported by Kaufmann and coworkers (K8). See 
Fig. 1.4 of Chap. 14. 



334 The Atomic Nucleus [CH. 10 

If a or b is an unpolarized photon, its statistical weight (2/ a + 1) or 
(2/ b + 1) is to be taken as 2, which corresponds physically to the two 
possible directions of polarization which remain after the direction of 
propagation has been fixed. 

The angular distribution, in C coordinates, will be the same for both 
the direct and inverse reactions, and so Eqs. (4.11) and (4.15) apply 
equally well to differential cross sections or to total cross sections. 

d. Photomagnetic Disintegration of the Deuteron. Fermi first 
pointed out that the disintegration of the deuteron by photons must 
include a photomagnetic disintegration process 3 & > 1 S which is the 
inverse of the 1 S > *S photomagnetic-capture reaction. From Eq. 
(4.15), the cross section er dlB (Ml) for the magnetic; dipole absorption 
3 5 > 1 S must be given by a(B > A), 

where p a = kh p* = 

c 

(2I a + 1) = 2 (proton) (2/ 6 + 1) = 2 (photon) 

(21 A + 1) = 2 (neutron) (21 B + 1) =3 (deuteron) 

Hence, by Eq. (4.15) 

**.(M1) = 2/MV 
3 \hv/cj 

Substituting k 2 = 2MW/W, and W + B = hv [because B is the Q value 
of the H l (n,y)H 2 reaction], we find 

* , . 

^ ' } 



3 (W + R)* 
in which the reduced mass M is given to good approximation by 

2M = i(Jf + MJ 

and the corresponding rest energy is 2Mc 2 c^ 939 Mev. Recall Eq. (4.8), 
ffcmp * I/W*, for slow neutrons. Then for the photomagnetic 'disintegra- 
tion, Eq. (4.17) shows that o\i 1B (Ml) is zero at the threshold hv = B, and 
for small values of W = hv B, o- dlH rises linearly with 10, that is, with 
the velocity V of the disintegration particles. 

The general expression for the photomugnetic cross section is obtained 
by substituting <7 cap from Eq. (4.4) into Eq. (4.17). This gives 



or, expressing k and p in terms of hv and B, 



(4-Y I >'" 

\2Mc/ { 





3 137 
These cross sections, like o- c . p , correspond to the central-force zero-range 



Forces between Nucleons 



335 



model and are subject to the same small corrections for the finite range 
of nuclear forces, for exchange magnetic moments, and for tensor forces. 
The photomagnetic disintegration cross section increases as (hv B)* 
near the threshold at hv = B. In contrast with <r np , the term in [1 + 
( J a/r) 2 ] therefore becomes important. The cross section o- dli (Ml) reaches 
a maximum when l ak ~ 1, i.e., when hv ~ B + I 1 /?!, where 



is the absolute value of the " binding energy," 1 B, of the virtual 1 S level 
of the ti-p system, in the zero-range, central-force approximation. The 
variation of o- dl .(Ml) with hv is shown graphically in Fig. 4.1. 




B- 2.225 



4 6 8 10 

Photon energy in Mev 



12 14 



Fig. 4.1 Photodisintegration of the dcuteron. Solid curves: theory, Eq. (4.21) for 
electric dipole (El); Eq. (4.19) for magnetic dipole (Ml), Points: representative 
experimental data (B58, P21, F48). 

e. Photoelectric Disintegration of the Deuteron. We have noted that 
the electric dipole capture reaction 3 P 8 S has a negligible cross section 
because of the small interaction energy of the P state. The inverse 
reaction is the photoelectric disintegration of the deuteron by the absorp- 
tion of El radiation. The ratios of the cross sections for El disintegra- 
tion and for El capture are the same as the corresponding ratios for Ml 
disintegration and capture. Then Eq. (4,17) shows that for W ~ B, 
kdi.(El)]/[<r cap (El)] ~ 150. The photoelectric disintegration therefore 
becomes significant. 

Derivation of the cross section o- dii (El) is completely analogous to the 
procedure we have followed for <r dil (Ml). In the matrix element for the 
8 P 3 S capture transition, the interaction energy between the neutron 
and proton in the P state is set equal to zero. The correction for the 
effective range 8 r of the 8 S interaction can be made easily. Including 



336 The Atomic Nucleus [CH. 10 

this, the cross section for disintegration of the deuteron by electric dipole 
absorption is (B68) 



The variation with photon energy hv is seen more clearly by expressing 
jfc and p in terms of W ( = hv - B) and B. From Eq. (4.5), fc/r = VW/B; 
hence Eq. (4.20) becomes 



- B) 



&r_p^r 
= 3 137 L 

The range correction term has a constant value, as 
r (1.70 X 10- 13 cm) 



- WP) 



(4.21) 



= 
p ~ (4.31 X 10- 18 cm) 



= 0.394 



Near the threshold at hv = 5, <r dlB (El) increases with fc 8 , or (hv 
or F 3 , where V is the velocity of the photoneutrons produced in the 
reaction. The photoelectric cross section reaches a maximum of 0.0023 
barn per deuteron at hv = 2B ~ 4.45 Mev and then decreases. The 
order of magnitude of <rdi.(El) is seen to be the "geometrical area" of the 
deuteron irp 2 (where p = h/\^2MB = 4.31 X 10~ 13 cm is the deuteron 
radius) multiplied by the fine-structure constant TST = e z /hc which 
represents the strength of coupling between radiation and matter. The 
other terms in Eq. (4.21) are of the order of magnitude unity. Figure 
4.1 shows the variation of <r diB (Ml) and cr diB (El) with hv, according to the 
central-force shape-independent theory. The experimental data so far 
available are in agreement with this theory, within the present experi- 
mental and theoretical accuracy. The experimental results of Bishop 
et al. (B58), at photon energies just above the threshold, are summarized 
in Table 4.1. 

TABLE 4.1. EXPERIMENTAL AND THEORETICAL TOTAL CROSS SECTIONS 
<r = <Tdi B (Ml) + <r dill (El) FOR THE PHOTODISINTEGHATION OF DEUTERIUM (B58) 
These theoretical values were based on B = 2.231 0.005 Mev. This will depress 

the theoretical values somewhat as compared with the more recent value of B = 2.225 

Mev [Chap. 3, Sec. 4; Eq. (4.46)]. 



Source of 


hv, 


a (observed), 


ff (theory), 


7 rays 


Mev 


millibarns 


millibarns 


Ga 


2.504 


1.19 0.08 


1.01 0.03 


ThC" 


2.618 


1.39 0.06 


1.25 0.03 


Na" 


2.757 


1.59 0.06 


1.51 0.03 



f. Angular Distribution of Photoneutrons. The ratio of the photo- 
magnetic to photoelectric cross sections is determined from measure- 



4] 



Forces between Nucleons 



337 



ments of the angular distribution of the photoneutrons (G39). Recall 
from Chap. 6, Sec. 9, that the angular distribution in C coordinates is 
isotropic for Ml and varies as sin 2 i> for the El transition. Then the 
observed differential cross section da (tf) is 



da (*) = cfa(Ml) + 



3 

Sir 
<r(Ml) 
tr(El) 



<r BUI 



3 . t 
sm 2 



siu s tf 



dtt 



(4.22) 



Much of the experimental work on angular distribution has been summa- 
rized by Bishop and coworkers (B59), who add the value 



o-(Ml) 
cr(El) 



0.61 0.04 



for hv = 2.504 Mev (Ga 72 ), based on measurements of the relative photo- 
neutron intensities at # = 45, 90, and 135. Figure 4.2 shows that the 
measurements are well fitted by the central-force shape-independent 
model, within the present accuracy of both the experimental and theo- 
retical values. 



0.8 



0.6 



0.4 



0.2 



-II 

BQ 



2.2 




Ga 72 



ThC" 



Na 24 



2.8 



3.0 



2.4 26 

Photon energy, in Mev 

Fig. 4.2 Ratio of magnetic to electric photodisintegration of the deuteron, as a 
function of hv. Points: measured values determined from the angular distribution of 
photoneutrong (B59). Curve: the shape-independent (effective-range) theory (B50) 
calculated with B = 2.231 Mev by Bishop et al. (B59). 



338 The Atomic Nucleus [CH. 10 

Problems 

1. From the cross section for the photoelectric disintegration of the deuteron 
trdn(El) f derive an expression for the photoelectric-capture cross section o- oap (E]) 
and evaluate o- p (El) in barns per proton for thermal neutrons. Compare with 
the photomagnetic-capture cross section er C a p (Ml) = 0.30 barn per proton for 
thermal neutrons. Ans.: <r mv (ll) ^ 5 X 10~ 9 barn per proton for 0.025-ev 
neutrons. 

2. Numerically evaluate o- oap for the photomagnetic capture of thermal neutrons 
(0.025 ev in L coordinates) by protons, using the central-force zero-range model, 
Eq. (4.6). Ans.: 0.30 barn per proton. 

5. The Proton-Proton Force at to 10 Mev 

The energy domain up to ~10 Mev corresponds roughly to the separa- 
tion energies for a proton from a heavy nucleus and should involve most 
of the main characteristics of the force between two protons. 

Because of the saturation character ani finite range of the (pp) force, 
as seen qualitatively in the constancy of the average binding energy 
B/A per nucleon (Chap. 9), we can anticipate that the (pp) force for 
S-state (I = 0) interactions will be overwhelmingly stronger than the 
(pp) force for P-state and higher (I = 1 , 2, . . .) interactions so long as 
the energy is small. 

In the S state, the Pauli exclusion principle confines the possible 
interactions to the l (pp), or singlet, interaction, because the two protons 
are identical particles. In general (,T2), singlet (pp) interactions can 
occur only in states of even-/ (*S, Z), . . .), while triplet interactions 
*(pp) can occur only in states of odd-l (P, F, . . .). 

With primary interest thus centered in only one mode of interaction, 
the singlet S state, detailed information on the (pp) two-body forces can 
be obtained best from experiments on the scattering of protons by pro- 
tons. These experiments involve the production and detection of 
charged particles only and hence can be conducted with greater accuracy 
than the analogous n-p scattering experiments. However, the theoretical 
interpretation of the low-energy p-p scattering is much more complicated 
than for n-p scattering. This is due in part to wave interference effects 
produced by the joint action of the short-range (pp) force and the 
"infinite-range" coulomb force. 

Studies of p-p scattering entail a tremendous field of experimental and 
theoretical activity. Excellent comprehensive reviews appear with rea- 
sonable frequency. The details of the present situation will be found in 
such reviews as those of Breit (B113), Jackson and Blatt (J2), Breit 
and Gluckstern (B114), Breit and Hull (B115), and Squires (S64). We 
shall have to be content here with a qualitative description of the main 
phenomena. 

a. Theory of p-p Scattering at to 10 Mev. It can be shown from 
general arguments similar to those developed in Appendix C, Sec. 6, that 
the differential cross section da (tf) for the projection of a proton into the 



5] Forces between Nucleons 339 

solid angle dfl, at mean angle # (in C coordinates), is (M69, B117, J2) 

J cosh In tan 2 (fl/2)] 

cos 2 (tf/2) 




Rutherford 
scattering " 



Rutherford 

classical^ 
term 



_Wave -mechanical 
interference term 



Corrections for two identical^ 
particles 



-Mott scattering- 



2 siafio jcoslgo + i/lnsin 2 ^/^)] cos[a + v In COSW2)]] 
n \ sin z (>/2) cos 2 (#/2) j 



J/Vave interference cross terms between coulomb and I 

nuclear potential scattering I 



(5.1) 

I Pure nuclear I 

u . J 

| potential scattering | 



where M = reduced mass (2M = M p ) 

V = incident relative velocity 

# = scattering angle in C coordinates 
n = e z /hV = 1/1370 

d = nuclear phase shift f or I collision 

The phase shift 5 which we have used heretofore (Chap. 10, Sec. 3; 
Appendix C, Sec. 6; etc.) represented the departure of the total wave 
function from the wave function of the incident particles. However, in 
Eq. (5.1) 5 represents only the effect of turning on the specifica!!}*- 
nuclear (pp) force. The entire phase shift between the incident wave 
and the total wave function includes both 6 and a phase shift 5 C due to 
the effect of the coulomb force. The coulomb phase shift does not appear 
explicitly in Eq. (5.1), having been eliminated algebraically by use of the 
known form of the coulomb potential. The predicted value of 5 , for a 
given proton energy, depends upon the shape, depth, and range of the 
nuclear potential. 

The term in 17 In tan 2 (tf/2) is small for proton energies above 1 Mev 
and for angles tf not too close to or 180. When each of the three 
logarithmic terms is assumed to be zero, a commonly seen approximate 
form emerges. 

Equation (5.1) is applicable when only the s-wave phase shift is 
appreciable, and when the nuclear forces are central forces. The physical 
origin of each term is indicated explicitly in Eq. (5.1). These can be 



340 



The Atomic Nucleus 



[CH. 10 



visualized with the help of Fig. 5.1, which corresponds to Eq. (5.1) when 
the incident proton energy is 2.4 Mev in L coordinates, hence 1.2 Mev in 
C coordinates. 

For small incident energies (large 77), or for small scattering angles, 
the scattering is essentially classical. The first term in Eq. (5.1) is the 
standard Rutherford scattering [Chap. 1, Eq. (3.10)]. The second term 
arises because the projectile and target proton are indistinguishable. 
Scattering of the incident proton at tf or at v tf results in the partner 
proton emerging from the collision at angle IT # or at #. The first two 
terms are entirely classical. However the third term is completely non- 
classical. It is the result of wave interference between two identical 
particles with spin , such as two protons or two electrons. This term 
was originally developed by M ott (M65) ; hence the first three terms are 



!0.3 









- o 



90" 180 90 

& (C coordinates) i* (C coordinates) 

Fig. 5.1 Theoretical differential cross section for scattering of 2.4-Mev protons (labo- 
ratory energy) by protons vs. scattering angle tf in C coordinates. Near tf = 90 
(laboratory angle 45) the scattering is mainly nuclear. Near & = and 180 (labo- 
ratory, and 90) the scattering is substantially coulomb. The dips at intermediate 
angles, ~ 45 and 135 , are due to destructive interference between nuclear and 
Mott scattering. [The curves are from Jackson and Blatt (J2).] 

commonly known as Mott scattering. Note that the Mott scattering has 
fore-and-aft symmetry about 90, because of the identity of the two 
particles, even though the interaction forces are purely coulomb. 

At higher incident energies (small 17), or for nearly head-on collisions, 
the specifically nuclear forces become effective. There are two results. 
The scattering amplitude is now the sum of coulomb and nuclear (pp) 
effects. As shown in Appendix C, Sec. 6, a cross term appears, represent- 
ing the intensity due to interference between the coulomb and nuclear 
scattering amplitudes. Secondly, a term due purely to nuclear potential 
scattering also appears in Eq. (5.1). At moderate energies, as in Fig. 5.1, 
the observed total scattering differs from Mott scattering predominantly 
near tf = 90 and is therefore due mainly to the cross term in Eq. (5.1). 
This cross term makes the nuclear effect important by coupling it to the 
coulomb effect. The purely nuclear term (4/V) sin 2 6 is smaller than 
the cross term at moderate energies and is independent of scattering 
angle, because of the spherical symmetry of s-wave interactions. 



5] Forces between Nucleons 341 

In comparisons with experimental values, the sign of the cross term 
gives immediate information about the nature of the (pp) nuclear force. 
Positive values of 6 correspond to an attractive interaction; negative 
values of 60 signify a repulsive force (Appendix C, Sec. 6, Fig. 11). 

b. Experimental Results on p-p Scattering from to 10 Mev. The 
existence of a short-range attraction between protons was demonstrated 
in 1936 when White (W40) and Tuve, Heydenburg, and Hafstad (T32) 
first observed the anomalous (non-Mott) scattering of protons by hydro- 
gen, beginning at about 0.7 0.1 Mev. These findings were confirmed 
and extended in experimental studies by Herb, Kerst, Parkinson, and 
Plain (H37), and in 1939 all available data in the energy range from 0.8. 
to 2.4 Mev were compared with the predictions of a variety of assumed 
nuclear potentials by Breit, Thaxton, and Eisenbud (B117), and Hoising- 
ton, Share, and Breit (H60). The essential results of this analysis remain 
unaltered and have been confirmed by much further work. It was found 
that: 

1. Cross sections having an accuracy of a few per cent can be fitted 
by Eq. (5.1) and therefore correspond to s- wave (I = 0) interactions, 
with no significant contribution from p waves, 

2. The observed phase shifts 6 could be accounted for by any of 
several shapes of central-force potential wells, with suitable adjustment 
of the depth and range Eqs. (2.18) to (2.21), e.g., by a rectangular well of 
11 Mev depth and 2.8 X 10~ 18 cm radius. 

3. The 1 (pp) interaction was found to be nearly equal to the l (np) 
interaction but about 2 per cent weaker, the difference being independent 
of the assumed shape of the potential well. 

Figure 5.2 shows the differential p-p cross section, measured with an 
accuracy of 0.3 per rent, at incident proton energies from about 1.8 to 
4.2 Mev (W71). The general physical effects noted in Fig. 5.1 are evi- 
dent. The curves are fitted from Eq. (5.1) for s-wave nuclear interactions 
only, with the " nuclear phase shift" 5 as the only adjustable parameter. 
The resulting nuclear phase shifts (H7) 5 are plotted in Fig. 5.3 as a 
function of the incident proton energy. 

c. Scattering Length and Effective Range of the (pp) Singlet Force. 
In the interpretation of the p-p scattering results, the phase shifts 5 are, 
as usual, the common meeting ground of experiment and theory. As is 
shown in detail by Jackson and Blatt (J2), the observed variation of 5 
with incident proton energy can be matched by any of the conventional 
shapes of potential well, with suitable choices of depth and range. A 
shape-independent theory therefore has a natural attractiveness. For 
the (pp) interaction, such a theory contains as parameters the effective 
range r and a constant a which is the (pp) analogue of the singlet scatter- 
ing length in the (up) interaction. The best fit to the data of Fig. 5.3 is 
obtained with 

r = 2.65(1 0.03) X 1Q- 13 cm (5.2) 

a = -7.7(1 0.07) X 1Q- 13 cm (5.3) 

Within the present accuracy of measurement, the singlet effective range 



342 



T7ie Atomic Nucleus 



[CH. 10 



0.18 r-r 




0.10 



30 40 50 60 



70 80 90 
* + (C coordinates) 

Fig. 5.2 Angular distribution of p-p scattering, at angles tf (in C coordinates', for 
the incident proton energies (in L coordinates) marked on each curve. [Adapted from 
Worthington, McGruer, and Find ley (W71).] 



Nuclear phase shift, tg in degrees 
o 5 8 8 S S S 














, 


1 


B 











* 


. 
















e 
o 















































































012345 

Proton energy (Mev)(laboratory coordinates) 

Fig. 5.8 Experimental p-p differential scattering cross sections as represented by the 
nuclear phase shift 5 , Eq. (5.1), as a function of the incident proton energy. Open 
circles represent the data compiled by Jackson and Blatt (J2) ; solid circles are from 
data of Worthington et al. (W71) as interpreted by Hall and Powell (H7). 



5] Forces between Nacleons 343 

r for the (pp) interaction is seen to be the same as for the l (np) interac- 
tion, Eq. (3.48). 

The p-rp scattering length a is of special interest. It turns out to be 
negative in sign. This tells us at once (compare Fig. 3.2) that the 
singlet p-p system is unbound. Therefore the di-proton, or He 2 , has no 
stable bound level. The absolute value of a requires further interpreta- 
tion because it includes coulomb as well as specifically nuclear effects. 
The specifically nuclear effects can be isolated by treating the coulomb 
force, within the range of the nuclear force, as a small perturbation. 
When this is done (J2, B68), the specifically nuclear scattering length a 1 
for the l (pp) interaction is increased to the value 

a'~ -17 X 10- 13 cm (5.4) 

d. Equivalence of the (pp] and (np) Singlet Forces. Quantitative 
comparison of a' with the singlet l (np) scattering length 

] a = -23.69 X 1Q- 13 rm 

Eq. (3.48), shows that both interactions have virtual singlet states whose 
(negative) binding energy is close to zero (~50 kev). Also, if the 
potential wells for the \pp) and l (np) interactions are assumed to have 
the same shape and range, then the potential well for 1 (np) is only slightly 
deeper (1.5 to 3 per cent) than for the l (pp) intersection. 

Finally, Schwinger (S22) has drawn attention to the difference in the 
magnetic forces between l (np) and l (pp), which arises because the intrin- 
sic magnetic dipole moments of the neutron and proton are of opposite 
sign. By a variational method, Schwinger has evaluated a net attractive- 
magnetic interaction in the observed l (np) force and a net repulsive mag- 
netic interaction in the observed l (pp) force. If the Yukawa shape, 
Eq. (2.21), is chosen for the nuclear potential, then the small observed 
difference between l (pp) and 1 (np) is exactly ( + per cent) accounted for. 

This equality of the parameters of the (pp) and (np) singlet interac- 
tions is the central evidence on which rests the hypothesis of the charge 
independence of nuclear forces. The evidence is valid within present 
experimental and theoretical accuracy. There is supportive experi- 
mental evidence, of a less accurate kind, that the (nri) singlet force equals 
the (pp) singlet force. 

The hypothesis of charge independence implies that the forces between 
two protons, or two neutrons, or a neutron and proton, are all equal in 
1 S states and are all equal in 3 P states, etc. While there is as yet no 
contradictory evidence, the present experimental evidence must be 
regarded as inadequate to prove so sweeping a generalization. In the 
meantime, the hypothesis of charge independence has proved fruitful in 
some theories of the heavier nuclei. 

e. P -state p-p Repulsion. In Fig. 5.2, the observed p-p scattering 
cross sections have been well matched by a 1 S phase shift 5 appropriate 
to each bombarding energy. However, there are slight discrepancies, of 
the order of 1 per cent, especially for scattering angles near $ ~ 30 15. 
Figure 5.4 illustrates these small deviations from a pure s-wave effect. 



344 



The Atomic Nucleus 



[CH. 10 



Upon analysis (H7), these effects are interpreted as evidence for a small 
negative p-wave nuclear phase shift whose value is about 5i ~ 0.1 at 
4 Mev. The negative sign means that the 1 P interaction is repulsive. 
These inferences are not inconsistent with the interpretation of very-high- 
energy proton-proton scattering experiments to be discussed in Sec. 9. 




2.425 Mev 



* * 



I- 



M t 



Cross se 


f* 
H 



l>*^ 3.899 Mev 


* ^*n i i 




t i 




1 T t 4 
i i r 


' 4 




T 




0" 20 40" 60 


80 


100 


120 14 



& * (C coordinates) 

Fig. 5.4 Experimental evidence for a small nuclear repulsive force between two 
protons in a S P state. The points represent the difference between the observed and 
calculated p-p differential scattering cross sections which are not apparent on the 
scale of Fig. 5.2. [Adapted from Worthington, McGrueT, and Findley (W71).] 



Problem 

Show under what conditions the ps-p differential cross section at # = 90 (C 
coordinates) should vary as I/ (incident proton energy) and compare with the 
experimental findings of Cork, Phys. Rev., 80: 321 (1950) (da/dil oc \/E p 
between 19 and 32 Mev), and with Figs. 5.3 and 9.4. 

6. Equivalence of (nn) and (pp) Forces 

Direct two-body evidence on the force between two neutrons is 
extremely limited. Scattering experiments, analogous to the n-p and 
p-p scattering studies, seem permanently excluded. This is because a 
thermal neutron flux of 10 13 neutrons/ (cm 2 ) (sec), such as is attainable 
in present reactors, corresponds in neutron density only to a monatomic 
gas at 10~ 12 -atm pressure. 

a. The Di-neutron. There is one piece of negative evidence, the 
nonexistence of a stable di-neutron (C29). This might be expected by 
analogy with the nonexistence of the di-proton He 2 , because, even with 
the coulomb field turned off, the p-p scattering length a' is negative, 
Eq. (5.4). The scattering length for the (nri) interaction appears at 
least to have the same (negative) sign as that for the (pp) interaction. 

b. Binding Energy of H 3 and He 3 . The lightest pair of adjacent 
mirror nuclei, H 3 and He 3 , provide the most direct evidence on the 
equivalence of l (nri) and 1 (pp). Counting up the number of possible 
two-body interactions, we have in H 3 , 8 (wp) + 1 (np) + 1 (nri) J and in 
He 3 , 8 (np) + l (np) + l (pp). These are all S-state interactions because 
neither nucleus has any orbital angular momentum. The binding energy 



7] Forces between Nucleons 345 

of the two nuclides should therefore differ only by the coulomb energy 
and by any difference which may exist between l (nri) and l (pp). The 
observed binding-energy difference is 0.76 Mev (Chap. 9). If this is 
attributed to the coulomb energy of two discrete protons, then, by Chap. 
2, Eq. (2.3), 

JFcoui = !! 2 Z(Z- 1) (6.1) 

and R ~ 2.3 X 10~ 13 cm. This is a reasonable radius for H 3 or He 3 
and suggests that l (nri) = l (pp). When tensor forces are included, and 
charge independence is assumed, the detailed theory of H 3 by Pease and 
Feshbach (Pll) shows that the binding-energy difference between H 8 
and He 3 can be attributed entirely to the coulomb interaction between 
the two protons in He 3 . 

c. Coulomb -energy Difference of Heavier Mirror Nuclides. The 
heavier mirror nuclides, such as uSU and uPii, differ from each other only 
by the interchange of one proton for a neutron. This interchange alters 
the number of p-p and n-n bonds, but in a manner whose details depend 
on the coupling scheme. In any case, we have found in Chap. 2, Eq. (2. 13) 
and Fig. 2.1, that the difference in total binding due to specifically nuclear 
forces is negligible between mirror nuclides, and this supports the hypoth- 
esis of charge symmetry, i.e., the presumed equality of l (nri) and 1 (pp). 

d. Neutron -Deuteron Interactions. Neutron scattering by deuterons 
provides a means of exploring the (nri) force. At present, clear-cut quanti- 
tative conclusions have yet to emerge from the theoretical and experi- 
mental work on elastic n-d scattering (A6, A19, C20, T29). Inelastic 
scattering, i.e., the disruption of H 2 according to the reaction H 2 (ra,p)2n 
by neutrons of the order of 90 Mev, has given quantitative support to 
the equivalence of (nri) and (pp) forces (C19). 

e. H 3 and He 3 in Nuclear Reactions. The differential cross sections 
at ~10 Mev for the reactions H 3 (d,n)Hc 4 and He 3 (d,p)IIe 4 show marked 
asymmetries which are very similar for both reactions. These observa- 
tions have been interpreted (A18) as also supporting the equality of (nri) 
and (pp). 

f. Excited Levels in Mirror Nuclei. In each of the mirror nuclei 
3 LiJ and 4 BeJ, the first excited level has / = i" and appears to form a 2 P 
spin doublet with the ground level I = |~. The excitation energies 
differ by only 10 per cent, most of which may be accounted for by magnetic 
interactions. This evidence (B130) and an analogous but more compli- 
cated situation in the several heavier mirror nuclei (A10, T19, E14) add 
further support to the hypothesis of charge symmetry, l (nri) = l (pp). 

7. Summary of Central Forces 

Thus far in this chapter, we have dealt analytically only with the 
ordinary (nonexchange), central (nontenjspr) forces. These forces 
Hetween individual nucleons can be represented by many "explicit poten- 
tial shapes (rectangular, Gaussian, exponential, Yukawa, . . .) or by 



346 The Atomic Nucleus [CH. 10 

their shape-independent approximation, which characterizes the effec- 
tive-range theory. 

a. Interactions at Energies Comparable with the Nucleon Binding 
Energy. Insofar as phenomena below the order of 10 Mev are con- 
cerned, these considerations now give quite satisfactory explanations for 
a wide variety of two-body phenomena, including: 

1. Small binding energy of the deuteron 

2. Large size of the deuteron 

3. Large cpithermal n-p cross section O-Q of free protons 

4. Small coherent scattering cross section of cold neutrons by para- 
hydrogen 

5. Coherent scattering of slow neutrons by bound protons in crystals 
0. Total reflection of neutrons from "neutron mirrors" 

7. Radiative capture of neutrons by protons 

8. Pliotomagnet.ic disintegration of the deuteron 

9. Photoelectric disintegration of the deuteron 

10. Singlet phase shift 5 in p-p scattering 

11. Instability of the di-proton and di-neutron 

12. Difference in binding energy of H 3 and He 8 

b. Charge Independence. All the dramatic surprises in this group 
of experiments have now been explained in terms of short-range triplet 
and singlet central forces between nucleons in S states. Within the 
accuracy of theory and experiment, the^ singlet forces between all pairs of 
nucleons are found to be equal, and we can write 

1 (PP) = l (nn) "charge symmetry" (7. la) 

and l (np) = l (pp) = 1 (nn) "charge independence" (7.1b) 

c. Spin Dependence. The total (np) force is clearly stronger in the 
triplet state than in the singlet state. This is a clear experimental result 
which is almost independent of any type of theory, because the ground 
level of the deuteron has angular momentum 1=1 and is a *5 level, 
whereas the 1 S level is unbound. The conclusion 

*(np) > i(np) (7.2) 

is supported by the theoretical interpretation of all n-p collision phe- 
nomena. 

d. Exchange Forces. The nuclear force must be some type of sat- 
urated force in order that heavy nuclei may have a binding energy pro- 
portional to the number of nucleons. This is most simply achieved by 
invoking exchange forces of the Majorana type, for which the nucleon- 
nucleon force is attractive in S states and all other states of even-i but 
changes sign and is repulsive for P states and all other states of odd-Z. 
To the extent that the low-energy interactions involve only S states, the 
introduction of exchange forces produces no change in the theoretical 
interpretation based on ordinary forces. 

From other evidence, the actual central force appears to be neither 
a pure ordinary force nor a pure exchange force but instead contains a 



7] Forces between Nucleons 347 

proportion of ordinary force plus enough exchange force to provide a 
saturation mechanism. The introduction of exchange forces does not 
produce any significant change in the bulk of the low-energy predictions, 
and it is in accord with the small P-state repulsion seen in the p-p scatter- 
ing data of Fig. 5.4. 

The physical picture of the exchange force lies in a meson field sur- 
rounding the nucleon and in the actual exchange of a meson between two 
nucleons which approach each other very closely. For visualizing the 
relationship between the range of the nuclear force and the mass of the 
meson, a simple consideration due to Wick (W45) is useful. Imagine 
that a proton spontaneously emits a IT meson of mass A/,, according to 
the reaction 

p n + Mr 

This would violate conservation of energy because an amount of energy 
Afi ~ M *c z has appeared spontaneously. Thus, classically, the reaction 
cannot take plane. But we can imagine the " virtual' J emission of the TT 
meson, followed by its recapture within a time AJ, according to 

p<=n + M r (7.3) 

Now if AJ is a sufficiently short time so that the uncertainty principle 

AE A* ~ h (7.4) 

is satisfied, we have no experimental way of contradicting Eq. (7.3), 
because any experiment performed in a time A/ will disturb the energy 
of the system by at least Al? ~ h/At. We then ask how far away from 
the proton the meson could travel in the time A/ and still get back without 
its individual journey being detectable. The distance b traveled in time 
AJ is b ~ v AJ, where v is the velocity of the meson. The uncertainty 
A in energy must be at least as large as the rest energy M T c 2 if the virtual 
emission is to take place without violation of conservation of energy. 
Then 

(7.5) 



AJ 6 " 
The maximum value of v is c; hence Eq. (7.5) becomes 

6 *jk-s()S* (7 ' 6) 

where (h/moc) = 2,426 X 10~ 13 cm is the Compton wavelength of the 
electron. Then, for a meson of mass M* = 273m , the travel distance 
of a meson in virtual emission is about 

b~\A X 10- 13 cm (7.7) 

which agrees more than qualitatively with the observed range of the 
nuclear forces. 

We may note in passing how this "meson field" is simply related to 



348 The Atomic Nucleus [GH. 10 

the more common electromagnetic fields of classical experience. Con- 
sider the Yukawa potential, Eq. (2.21), and identify the range b of the 
potential with the rationalized Compton wavelength of the meson, that 
is, b = h/M T c. If we reduce the mass of the field particle M T toward 
zero, then the e~ (r/b} term approaches unity, and the Yukawa potential 
becomes the conventional l/r potential of electrodynamics, in which the 
field quantum hv does have zero rest mass. The Yukawa potential is 
thus seen to be a more generalized potential in which the field is associ- 
ated with a particle whose rest mass is finite. 

Problems 

1. State very briefly the principal types of evidence which show that the (np) 
binding force in nuclei is (a) short-range, (b) nonelectric, (c) nongravitational, 
(d) nonmagnetic, and (e) spin-dependent. 

2. What experimental evidence shows that the (nn) and (pp) forces are sub- 
stantially equal? 

8. Effects of Tensor Forces 

Among the low-energy data there are two experimental results for 
which central forces, with or without exchange, are completely inade- 
quate. With central forces, in a spherically symmetric S state, the 
deuteron should have a magnetic moment p d = Mn + M P and an electric 
quadrupole moment Q = 0. The experimentally observed departures 
from these conditions are small, but they are also very certain (Chap. 4, 
Sec. 5). 

a. The Tensor Operator. In order to obtain a representation of these 
nonspherical effects, one adds to the central force a small amount of a 
noncentral force. Then the total potential has the form (R6, R29, B68) 

U(r) = tft(r) + Z7.(r)di d 2 + U*(r)S u (8.1) 

where l/i(r), Ui(r), and C7 8 (r) = ordinary functions of r [such as one of 

the potential wells of Eqs. (2.18) to 
(2.21)] 

di and d 2 = spin operators of the two nucleons 
di d 2 = 1 for a triplet state and 3 for a singlet 

state 

The noncentral character of the interaction is contained in the tensor 
operator 

[3(d 1 >r 12 )(d 2 -r 12 )] 



. 
12 



. ( . 

d s ) (8.2) 



which gives a dependence on the direction of the spin vectors di and d 2 
relative to the separation ri 2 between the two nucleons. 

Note that 5i 2 is a scalar. It can be shown that when 173(7-) is finite 
the total angular momentum / and the parity remain good quantum 
numbers but that the orbital angular momentum L is no longer a constant 



8] Forces between Nucleons 349 

of the motion. There also emerges from Eq. (8.2) the valuable general- 
ization 

Si2 = for all singlet states ( J S, 1 P, 1 D, . . .) (8.3) 

Hence all tensor effects are to be found in triplet states only. 

b. Magnetic and Electric Moments of the Deuteron. The effect of a 
finite tensor force on the low-energy data was first studied in detail by 
Rarita and Schwinger (R6), who used a rectangular potential well of 
radius 6 = 2.8 X 10~ 18 cm for both the central and tensor potentials. 
With this rectangular potential, the magnetic dipole moment and the 
electric quadrupole moment of the deuteron emerged correctly, without 
disturbing markedly the theoretical match with the other low-energy 
data, by assuming that the ground state of the deuteron is an admixture 
ofjibout 4 per cent 3 D state plus SCTper cent 3 S state. 

Otter potential shapes and ranges have been studied by many investi- 
gators. These are summarized by Feshbach and Schwinger (F48), who 
have made detailed calculations for the low-energy phenomena, using a 
Yukawa shape in both the central and tensor potentials. With these 
potentials, the magnetic and electric moments of the deuteron can be 
matched, within experimental and theoretical uncertainties, by an admix- 
ture of 4 1.6 per cent 3 D state in the ground state of the deuteron. 
The experimentally determined triplet scattering length 3 a and effective 
range 3 r of Eq. (3.47) are found to require that 

b t > b c (8.4) 

where bt and b c are the triplet ranges of the tensor and central Yukawa 
potentials, Eq. (2.21). 

c. Binding Energy of H 3 , He 3 , and He 4 . These same potentials can 
be applied to the three-body problem, H 3 and He 3 , and the four-body 
problem He 4 , by variational methods. It is well established generally 
that the observed binding energy of these nuclei is distinctly smaller than 
is predicted by any purely central interaction which will at the same time 
match the properties of the deuteron and the low-energy scattering data. 
Also, any admixture of noncentral force reduces the theoretical binding 
energy and brings it closer to the experimentally determined values. 

The Yukawa potential shapes studied by Feshbach and Schwinger 
(F48) have been applied to H 3 and He 8 by Pease and Feshbach (Pll) 
and to He 4 by Irving (16). It is found that the following potential is 
one which matches all the low-energy scattering data, as well as the 
properties of the deuteron, and the binding energy of H 3 , He 3 , and He 4 

-(r/6e) 

singlet: *U(r) = -U e ^ (8.5) 

(r/bc) 

,-(r/6) p-(r/bt) 

triplet: 3 E7(r) = - U. - U. S (8.6) 



where 

central: U c = 47 Mev b c = 1.18 X 1Q- 18 cm / 7 ^ 

tensor: U, = 24 Mev b t = 1.70 X 10~ 13 cm l ; 



350 The Atomic Nucleus [CH. 10 

Note that the central parts of the triplet Yukawa potential 8 J7(r) and the 
singlet Yukawa potential 1 l '(r) are equal. This means that the well- 
established spin dependence of nuclear forces 

\np) > l (np) 

can be ascribed entirely to the additional tensor force, which is effective only 
for the triplet state. 

Note that the depth of the tensor potential U t is about half that 
of the central potential U CJ but that the tensor range b t is greater than 
the central range b c . The small effects of the tensor force are to be associ- 
ated with its small percentage admixture, which for Yukawa wells is of 
the order of 3 per cent D state in H 2 , H 3 , He 3 , and He 4 , and the balance 
S state (F48, 16). 

d. Exchange Tensor Forces. Exchange interactions of the Majorana 
type leave the attractive forces unaltered in states of even-Z (S, D, . . .) 
and only change the sign of the force to a repulsion for states of odd-/ 
(P 9 Fj . . .). The low-energy data involve a P state only in the photo- 
electric disintegration of the deuteron (*S > 3 P), where the P-state inter- 
action was taken as negligible in Eq. (4.20), and in the small (< 1 per 
cent) anomaly in the p-p scattering, Fig. 5.4. Therefore the inclusion or 
omission of exchange forces in Eqs. (8.5) and (8.G) is a matter of relative 
indifference in the low-energy domain. For simplicity, exchange is ordi- 
narily omitted here, and no dire consequences result even though the 
potentials in Eqs. (8.5) and (8.6) are unsatu rated. 

9. High-energy n-p and p-p Scattering 

The (np) and (pp) two-body interactions for states of nonzero angular 
momentum I can be explored in scattering experiments at very high 
energies. For example, to achieve an angular momentum as large as 
2h in an n-p collision whose impact parameter is as small as 1 X 10~ 13 cm 
requires an incident kinetic energy of at least 320 Mev. Such energies 
are so much greater than those encountered in a heavy nucleus that there 
may be little direct connection between the high-energy scattering and 
conditions within an average nucleus. 

Both n-p and p-p scattering experiments have now been carried out 
at energies up to several hundred Mev. The results have been, in the 
words of Blatt and Weisskopf, " strange and unexpected," and were 
definitely not predicted from the low-energy data. We shall indicate 
here the general character of the experimental findings. The interpreta- 
tion of these high-energy scattering experiments is as yet by no means 
complete, but one or two important and clear facts about the nature of 
nuclear forces have already emerged. 

a. n-p Scattering above 10 Mev. Incident neutrons up to 20-Mev 
laboratory kinetic energy are scattered isotropically (in C coordinates) 
by protons (B8). At energies of 27 Mev, deviations from spherically 
symmetric scattering are clearly measurable (B126). At still higher 
energies the angular distribution of the scattered neutrons takes on a 



9] 



Forces between Nucleons 



351 



characteristic "valley" shape. This is the prominent experimental fea- 
ture of Fig. 9.1 which shows the differential cross section for neutrons of 
40-, 90-, and 260-Mev laboratory kinetic energy, as scattered by protons 
(HI, K15). The pronounced minimum in the vicinity of 90 is to be 
associated with the effects of exchange forces and not with scattering in 
higher-angular-momentum states. At 90 Mev, for example, some 90 per 
cent of the total cross section is still attributable to /S-state scattering 
(C21). 

Exchange Forces in High-energy n-p Scattering. The influence of 
exchange forces and of ordinary forces on high-energy scattering can be 




40 80 120 160 

Neutron scattering angle & (C coordinates) 

Fig. 9.1 Angular distribution of neutrons scattered by protons, for incident neutrons 
of 40-, 90-, and 260-Mev laboratory kinetic energy. [From Jastrow's (J8) compilation 
of the measurements by Hadley, Kelly, Leilh, Segrb, Wiegand, and York (HI, K15).] 

understood qualitatively in the following way. We have seen from the 
low-energy collision data that the depth of the n-p potential well U Q is 
of the order of, say, 20 to 50 Mev, depending on the shape of the well. 
This depth is a measure of the strength of the interaction between a 
neutron and proton. It therefore is a rough measure of the maximum 
energy transfer, in laboratory coordinates, between a colliding neutron 
and proton. At incident kinetic energies which are greatly in excess of 
l/o, only a limited fraction of the incident kinetic energy can possibly 
be transferred in the collision, if ordinary (nonexchange) forces govern the 
interaction. Then, by Eq. 28 of Appendix B, the center-of-mass angle 
through which the neutron can be deflected is limited to small values. 



352 



The Atomic Nucleus 



[CH. 10 



For ordinary forces there should be substantially no scattering of the 
neutron through large angles, if the interaction is weak compared with the 
mutual kinetic energy. Thus we are led to predict an angular distribu- 
tion like that shown in Fig. 9.2. 

For an exchange force the situation is entirely different. Now the 
interaction mechanism involves an exchange of identity between the 
incident neutron and the struck proton. With weak interaction the 
particles which travel predominantly along in their initial direction have 
exchanged their identity and are protons after the interaction. The 
particles which emerge from the collision at large angles, such as 90 to 
180, are struck particles, which exchanged their identity during the 
collision and emerged as neutrons. This situation is also illustrated in 
Fig. 9.2. 



I 



Pure ordinary force 



90* 



Pure exchange force 



180" 




\/ 

Exchange/ X^'dinary 

x ^ *^Jr 



90" 



180 



90 



180 



Fig. 9.2 Schematic behavior of the angular distribution of neutrons scattered by 
protons, when the interaction energy is weak compared with the incident kinetic 
energy. The observed angular distribution represents a mixture of ordinary and 
exchan ge forces . [ After Fermi (F41 ) . ] 

Berber Force. The observed angular distribution has neutron maxima 
both at (as for an ordinary force) and at 180 (as for an exchange force) 
and therefore represents a mixture of both types of force. Reasonable 
fits with the observed neutron angular distribution have been obtained 
by Christian and Hart (C21) and others by using a half-and-half mixture 
of ordinary and exchange force. This mixture is the so-called Berber 
force and is represented analytically by the operator 

4(1 + Pa) (9.1) 

where P M is the Majorana exchange operator whose value is +1 for 
Z = 0, 2, 4, ... (attractive force) and is 1 for I = 1, 3, 5, . . . 
(repulsive force). The Berber force is therefore zero for odd states 
(Pj F, . . .). Although this gives a reasonable match with the high- 
energy n-p scattering, the Berber force does not contain enough exchange 
force to give a repulsion in odd states, and hence it cannot account for the 
saturation which is definitely seen in the binding energy of heavy nuclei. 
In contrast with n-p scattering below ~10 Mev, the effective range 
for high-energy scattering is dependent on the assumed shape of the 
potential well. Analytically (p. 501 of R4) this corresponds to the addi- 
tion of a shape-dependent term Pk*r\ to the right-hand side of Eq. (3.20). 



9] Forces between Nucleons 353 

Christian and Hart have used several potential shapes, all with Majorana 
space exchange (Serber force) in the central-force part of the interaction. 
With pure central forces, the predicted distribution near 90 is too flat. 
The addition of some tensor force, either with or without exchange, makes 
the theoretical curve less flat near 90. Although this recipe matches 
the observed n-p scattering distribution, it fails completely when applied 
to high-energy p-p scattering. 

The fundamentally important result of the high-energy n-p scattering 
studies has been the most suggestive experimental demonstration of exchange 
forces in a two-body interaction. 

b. p-p Scattering above 10 Mev. In p-p scattering the identity of 
the two particles produces three effects which are not present in n-p 
scattering. 

1. For each proton scattered at any angle tf (C coordinates) the 
partner is scattered at TT #. Therefore the experimental results always 
display fore-and-aft symmetry about 90 and are ordinarily reported for 
only to 90 even when the measurements involve a wider angular 
range. 

2. Coherence between the scattered amplitudes 



for the two identical protons makes the singlet and triplet p-p cross 
sections inherently four times larger than the corresponding n-p cross 
sections. 

3. The Pauli principle excludes all odd-Z singlet states and all even-Z 
triplet states. Therefore the p-p scattering can take place only from 
1 S, 8 P, 1 D, *F, . . . states. Among these possibilities, the 3 F state is 
still ineffective at ~320 Mev, because for I = 3 the colliding protons are 
widely separated, compared with the effective range of the nuclear forces. 
Among the remaining 1 S, 3 P, 1 D interactions, only the 3 P can partake 
of a tensor interaction or can display a repulsion if Majorana space 
exchange is mixed in adequate amounts with a central force. 

The experimental results for p-p scattering up to 340 Mev differ 
sharply from what would be expected on a basis of the n-p results at 
comparable energies. 

1. The angular distribution is roughly isotropic between 20 and 90. 

2. The absolute magnitude of the observed differential cross section 
clearly exceeds the maximum possible S-state value d<r = X 2 dfl, where 
X = h/MV, which is predicted by Eq. (5.1) for angles & ~ 90 where 
coulomb scattering is unimportant. 

3. The expected variation of da with l/E p , where E p is the incident 
proton kinetic energy in laboratory coordinates, is seen in the domain 
19 < E p < 32 Mev (C44), but da is substantially independent of E p for 
E p > 100 Mev (C7, CIS, 09, M70, H21). These general characteristics 
appear definite. The pioneer work at Berkeley in the energy domains 

E p ~ 30 Mev (C44, P5) 

E p ~ 120, 164, 250, 345 Mev (C15) 



354 

Las been "confirmed" for 



E p ~ 75, 105 Mev 
E p c^ 147 Mev 
E p ~ 240 Mev 
E p ^ 435 Mev 



The Atomic Nucleus 



[CH. 10 



at Harvard (B57) 

at Harwell (C7) 

at Rochester (O9) 

at Carnegie Tech (M70, H21) 



Illustrative experimental results are shown in Fig. 9.3 (angular variation 
ot dr) and Fig. 9.4 (energy variation of da). 



ID 

14 


/*** - 

- 29 Mev $ 


o | 12 


* * 


i 2 


P-P 


^10 


~~ ~ 


ll $ 


' 75 Mev ----*-<s-<r--*~ *- " 

?AO Mow _ _ _ * M f 




"345 Mev - ^- i -T -"" " 


2 



i i i i i i i i i 



20 40 60 80 100 
Scattering angle & (C coordinates) 

Fig. 9.3 Angular variation of p-p scatter- 
ing at high energies. In general, the abso- 
lute values of the cross sections reported 
from Berkeley are ~20 per cent smaller 
than the initial results from some other 
laboratories. The important finding is 
that the trends of da with angle, and with 
energy (Fig. 9.4), are found to be the same 
from all laboratories. 







I ' I ' I 
p-p at tf =90 



20 



15 



10 



100 200 300 400 500 
Incident proton energy in Mev (lab coord) 

Fig. 9.4 Variation of the differential 
p-p scattering cross section at tf = 90 
(C coordinates) with incident proton 
kinetic energy E p (laboratory coordi- 
nates). 



The absence of the expected minimum in da (tf) for p-p scatter- 
ing at 90 cannot be explained (C22) by potentials of the types 
used by Christian and Hart (C21) for ra-p scattering unless the 
hypothesis of charge independence, 1 (np) l (pp), is given up. This 
breakdown of the standard forms of potential well, with recipes involving 
admixtures of central and tensor forces, with and without exchange, has 
stimulated a theoretical search for some new formulation which will pre- 
serve charge independence and at the same time will fit the p-p and n-p 
scattering data at all energies. 



59] 



Forces between Nucleons 



355 



c. Short-range Repulsion between Nucleons. One method of pre- 
serving charge independence is the introduction of a new form of poten- 
tial well. Jastrow (J8, J9) has made progress toward the discovery of a 
potential which will fit all experimental results by adding a very short-range 
repulsive core to the interior of the standard forms of potential well. Such 
a composite potential is shown schematically in Fig. 9.5, where it can be 
seen that a spin-independent core radius can result in an effective hard- 
core repulsion in singlet states and in its essential absence in the triplet 
states. 

In order to reduce the mathematical difficulties, Jastrow has tested a 
composite potential which represents an impenetrable spherical core of 





Constituents Singlet Triplet 

Fig. 9.5 The addition of a spin-independent short-range repulsion *U to the spin- 
dependent attractive potentials 1 U (singlet) or *U (triplet) can result in a single! 
potential C7 + 1 U with an effective hard-core repulsion, and a triplet potential 
U + *U with a negligible hard-core repulsion. [After Jastrow (J8).] 

radius r , surrounded by a standard exponential well with Serber exchange 
for the singlet interaction. Thus 



singlet: U(r) = + 



r < r 



0.6 X 10~ 13 cm 



r 



(9.2) 



For the triplet interaction, the hard core is assumed to be negligible. 
Good semiquantitative agreement with the high-energy n-p and p-p scat- 
tering is obtained using a triplet potential containing exchange in both a 
central and tensor part, and given by 



triplet: C7(r) = - 



where 



i(l + P H ) + (0.3 + 0.7P*)1.84S 12 ] (9.3) 



I7o 
r 



375-Mev singlet depth 
69-Mev triplet depth 
0.60 X 10~ 13 cm hard-core radius 
= 0.40 X 10~ 18 cm singlet exponential range 
S 6 = 0.75 X 10~ 13 cm triplet exponential range 

Qualitative Effects of the Repulsive Core on High-energy p-p Scattering. 
At sufficiently high energies, the singlet S-wave phase shift is negative 
because of the singlet repulsive core. States of higher angular momen- 
tum still experience an attractive force due to the outer part of the poten- 
tial well. In this energy region interference between the 1 S and 1 D 
scattered amplitudes tends to suppress the forward scattering and to 
increase the scattering at 90. In addition, the tensor scattering of the 



356 The Atomic Nucleus [CH. 10 

IP interaction builds up the intensity in the vicinity of 55. Thus, 
qualitatively, the angular distribution of p-p scattering takes on thfe flat 
character which is seen experimentally, and also the absolute cross sec- 
tions are of the order of 4 millibarns/steradian, as observed. Although 
the repulsive-core model best predicts the observed single-scattering cross 
sections, it does not appear possible at present to match the experiments 
on p-p double scattering with this same potential (O8). 

Qualitative Effects on the Repulsive Core on High-energy n-p Scattering. 
In contrast with p-p scattering, the n-p scattering involves both 3 S and 
*S states, and the triplet scattering enjoys three times the statistical 
weight, of singlet scattering. Thus the addition of the repulsive core to 
the singlet interaction produces little over-all effect, because the n-p 
scattering is dominated by the triplet interaction. 

Effects of the Repulsive Core on the Low-energy Data. Some of the 
effects of a hard-core singlet interaction on low-energy scattering have 
been calculated for rectangular and Yukawa potential wells (H73). In 
general, the results are found to depend on the shape of the assumed 
potential well. Introduction of the repulsive core tends to reduce the 
range and increase the depth of the attractive part of the potential well. 
There is as yet no indication that parameters for the repulsive-core model 
cannot eventually be found which will simultaneously satisfy the low- 
energy and high-energy data. This work remains for the future. 



CHAPTER 11 

Models of Nuclei 



The complex interrelationships between nucleons when they aggre- 
gate to form medium and heavy nuclei will continue to defy precise 
analysis for a long time to come. In the absence of an exact theory, a 
number of nuclear models have been developed. These utilize different 
sets of simplifying assumptions. Each model is capable of explaining 
only a portion of our experimental knowledge about nuclei. 

If it is assumed that in the ground level and in the lowest excited 
levels of a nucleus the nucleons have very little interaction, then the 
independent-particle models emerge. We shall discuss the shell model 
as an example from the broad group of independent-particle models. 

The extreme opposite view is that of very strong interaction between 
all the nucleons in a nucleus. As representatives of the strong-interaction 
models, some aspects of the liquid-drop model and of the statistical model 
will be examined. 

1. Summary of Experimental Evidence Which Should Be 
Bepresented by the Model 

The main experimental characteristics of nuclei, which we should 
like to see described by as few models as possible, may be summarized 
briefly, as a base line for the appraisal of presently available models. 

1. Nuclear angular momenta I of ground levels 

For even-Z even-N nuclides, 7 = 0. 

For odd-Z odd-N nuclides, 7 = 1, 2, 3, . . . . 

For odd- A nuclides, 7 = *, f , . . . . 

Mirror nuclei have equal 7. 

Extremes of triads have equal 7. 

2. Magnetic dipole moments p, and their approximate two-valued rela- 
tionship with 7, as summarized in Schmidt diagrams (Chap. 4, Figs. 4.1 
and 4.2) 

3. Electric quadrupole moments Q, and their systematic empirical vari- 
ation with Z or N (Chap. 4, Fig. 5.4) 

4. Existence of isomers, and their statistical concentration in . the 

357 



358 The Atomic Nucleus [CH. 11 

regions of N or Z = 40 to 50 and 70 to 80 (the so-called "islands of 
isomerism," Chap. 6, Fig. 6.1) 

5. Relative parity of nuclear levels, as seen in decay and 7 decay 
(Chap. 6, Table 3.2) 

6. Discontinuities of nuclear binding energy and of neutron or proton 
separation energy, as seen for particular values of N or Z, especially 
50, 82, and 126 (Chap. 9, Figs. 3.2 and 4.3) 

7. Frequency of stable isotones and isotopes, especially the statistical 
concentration for particular values of A r and Z (Chap. 8, Fig. 3.1) 

8. Pairing energy for identical nucleons, as seen in the occurrence of 
stable, nonadjacent, isobars (Chap. 8, Fig. 3.3) 

9. Substantially con start {density of nuclei, with radius R oc A * (Chap. 2) 

10. Systematic dependence of the neutron excess (N Z) on A 5 for 
stable nuclides (Chap. 8, Fig. 3.4) 

11. Approximate constancy of the binding energy per nucleon B/A, 
as well as its small but definite systematic trends with A (Chap. 9, 
Fig. 3.1) 

12. Mass differences in families of isobars and the energies of cascade 
|9 transitions (Chap. 8, Figs. 3.2 and 3.3) 

13. Systematic variation of a decay energies with N and Z (Chap. 2, 
Fig. 6.2) 

14. Fission by thermal neutrons of U 235 and other odd-JV nuclides 
(Chap. 11, Sec. 3) 

15. Finite upper bound on Z and N of heavy nuclides produced in 
reactions and the nonexistence in nature of nuclides heavier than U 238 

16. Wide spacing of low-lying excited levels in nuclei, in contrast with 
the close spacing of highly excited levels (Chap. 11, Fig. 4.1) 

17. Existence of resonance-rapture reactions, such as (n,,y). Con- 
stancy of the fast-neu tron-capture cross section for A > 100, except for 
its anomalously small value for the isotones in which N = 50, 82, or 126 
(Chap. 8, Fig. '2.2) 

Experimental items 1 through 8 are well represented by the inde- 
pendent-particle shell model. The liquid-drop model is built to account 
for items 9 through 15. Item 16 forms the main basis of the statistical 
model. Item 17 finds its best representation in the strong-interaction 
models (liquid-drop and statistical) but must draw on the independent- 
particle model also, in order to represent the effects of N = 50, 82, and 
126. 

2. The Nuclear Shell Model 

In 1932, Chadwick's discovery of the neutron opened the way for the 
development of models of nuclear structure. Drawing heavily upon 
analogies with the extranuclear electronic structure of atoms, Bartlett, 
Guggenheimer, Elsasser, and others developed early individual-particle 
models (F54) involving closed shells of 2(21 + 1) neutrons or protons, 
where I is the angular-momentum quantum number of the nucleons. 



2] Models of Nuclei 359 

The order of filling of shells having various I values was expected to differ 
from that found in the electronic structure, because of the different 
type of forces involved. With LS coupling, the theoretical order of levels 
for simple potential shapes was able to account for the then known 
nuclear discontinuities, or "magic numbers," only as far as O 18 , where 
the first p shell closes. Because of the small absolute values of the 
nucleon magnetic moments, jj coupling was not expected theoretically. 

From roughly 1936 until 1948 interest in nuclear models turned away 
from individual-particle models arid centered around the development of 
Bohr's idea of a liquid-drop nucleus and around Wigner's (W47) uniform- 
model, supermultiplet, and isobaric-spin concepts. 

Attention was drawn forcefully by Maria Mayer in 1948 to the 
accumulation of experimental evidence for closed shells in nuclei at the 
higher magic numbers, especially 50, 82, and 126 identical nucleons 
(M23). The liquid-drop model and the uniform model are inherently 
incapable of predicting such discontinuities. Attention swung sharply 
back to the individual-particle models. Important contributions were 
promptly made by Feenberg, Hammack, Nordheim (F20, F21, N22, 
F18), and many others. 

It remained for Mrs. Mayer (M24) herself, and independently for 
Haxel, Jensen, and Suess (H24), to take the important step of introducing 
jj coupling. By assuming strong spin-orbit forces for individual nucleons, 
a sequence of independent-particle states emerges which matches the 
experimentally known "magic numbers." The justification for intro- 
ducing a strong spin-orbit interaction and its .//-coupling scheme lies only 
in its success in matching experimental facts, which has been noteworthy. 
No adequate theoretical basis for jj coupling in nuclei has been found, 
although the introduction of tensor forces appears to hold promise (K9) 
for a possible future explanation. 

a. Assumptions in the Independent -particle Model. In contrast with 
the situation with atoms, the nucleus contains no massive central body 
which can act as a force center. This deficiency is circumvented by the 
bold assumption that each nucleon experiences' a central attractive force 
which can be ascribed to the average effect of all the other (A 1) 
nucleons in the nucleus. On this assumption, each nucleon behaves as 
though it were moving independently in a central field, which is describ- 
able as a short-range potential well. Secondly, this potential is assumed 
to be the same for all values of L 

The Weak-interaction Paradox for Low-lying States. In the assumed 
central potential, each nucleon is imagined to be capable of describing 
an orbit of well-defined energy and angular momentum, in a manner 
analogous to the behavior of atomic electrons. This condition implies 
that each nucleon can describe at least several revolutions without being 
disturbed or scattered in collisions with other nucleons. The assumed 
"mean free path" between collisions therefore has to be at least several 
times the nuclear radius. In such a model, the interaction between 
individual nucleons has to be weak. This assumption seems to be in 
clear conflict with the well-demonstrated strong interaction between 



360 



The Atomic Nucleus 



[CR.ll 



nucleons, as seen in scattering experiments, and in nuclear reactions gen- 
erally. To an incident nucleon, the struck nucleus is not "transparent," 
as would be implied by a mean free path exceeding the nuclear radius, 
but on the contrary is nearly opaque. All incident particles are scattered 
or captured. The mean free path for an incident nucleon is therefore 
short, compared with the nuclear radius. 

Weisskopf (W24) has drawn attention to the Pauli exclusion principle 
as a means of resolving the weak-interaction paradox. Amongst indi- 
vidual nucleons within a nucleus in its ground level, or a level having small 
excitation energy, the expected strong interaction may be present but 
unable to manifest itself, because all the quantum states into which the 
nucleon might be scattered are already occupied. Contrariwise, an 
incident nucleon can be scattered or captured into a previously unoccu- 
pied, and highly excited, quantum state. Thus it is possible to accept 
the model of weak interaction between the constituent nucleons within 
a nucleus at low excitation energies, without denying the inherently 
strong character of the interaction between free nucleons. 

b. The Sequence of Nucleon States for the Ground Levels of Succes- 
sive Isotopes and Isotones. The value of the independent-particle model 
lies mainly in its ability to give a nearly correct energy sequence for 
nucleon states having different values of I. It turns out that the order 
of the nucleon states is quite insensitive to the detailed shape of the 
assumed potential, so long as the potential decreases rapidly outside the 
nuclear radius. 

A simple rectangular well having a great depth f/ and a radius 
about equal to the nuclear radius R is a sufficiently good representation 
of such a short-range force. The wave functions for independent par- 
ticles within such a well obey the radial wave equation [Chap. 2, Eq. 
(5.75)] f or r < R and are zero at the well boundary r = R, as well as 
outside the well r > R. The allowed energy states then correspond 
to the sequence of solutions of the radial wave equation (Bessel functions) 
which have zero values at r = ft. 

Each state with orbital angular momentum I is degenerate (same 
energ} r ) with respect to m r , since m does not occur in the radial wave 
equation. Therefore, in each state of a given I, there can be (21 + 1) 
identical nucleons when spin is neglected or 2(21 + 1) identical nucleons 
if the energy is independent of spin orientation. The order of energy 
states for the deep rectangular well turns out to be 







'f 






*-j 


~r 




Occupation number 2(21 + 1) 


?, 


6 


10 


?! 


14 


6 


18 


















AeKreeate number of nucleons 22 (2J +1) 


2 


8 


18 


20 


34 


40 


58 



where the letter gives the I value and the integer prefix gives the radial 
quantum number, as denned in Chap. 4. This sequence fails to give any 
indication of a closed shell at 50 nucleons and fails even more clearly 
for still larger nucleon numbers. 



2] Models of Nuclei 361 

Sequence of States in the Spin-orbit-coupling Model. Additional 
assumptions are needed if the sequence of energy states is to match the 
empirically known " magic numbers" 50, 82, and 126. It was noted 
independently by Mayer and by Haxel, Jensen, and Suess that this match 
could be obtained by postulating strong spin-orbit coupling for nucleons. 
Then, for the same / value, the energy of the j 1 = I + \ state may be quite 
different from the energy of the j 1 = I ? state. As presently visualized, 
the "shell model," or " spiu-orbit-coupling model," or "^"-coupling 
model," involves the following assumptions (M25, M26), in addition to 
those which are inherent in every independent-particle model: 

1. For the same value of the orbital angular momentum , the j 
I + T state ("parallel" orbit and spin) is deeper lying, or more tightly 
bound, than the,/ = I ? state. 

3. The energy separation between j - I + ? and j = I -* increases 
with increasing values of Z, being then approximately proportional to 

' 



3. An even number of identical nucleons having the same I and j will 
always couple to give even parity, zero total angular momentum, and 
zero magnetic moment. 

v ^ An odd number of identical nucleons having the same I and j will 
always couple to give odd parity if I is odd and even parity if I is even, a 
total angular momentum j, and a magnetic moment equal to that of a 
single nucleon in the state j [Chap. 4, Eqs. (4.9) and (4.10)]. 

5. There is an additional binding energy, or pairing energy, d associ- 
ated with double occupancy of any state I, j, by two identical nucleons. 
In any nucleus, the pairing energy is greatest for states of largest j. 
This extra binding energy fi for an even nucleon compared with an odd 
nucleon is approximately proportional to (2j + I)/ A. 

The primary purpose of assumptions 1 and 2 is to match the higher 
magic numbers 50, 82, and 126. Assumptions 3, 4, and 5 are refine- 
ments which provide agreement with the experimentally known values of 
parity, nuclear angular momentum 7, and magnetic dipole moment /x, for 
the ground levels of nuclei as well as for many low-lying excited levels. 

Figure 2.1 illustrates the succession of states in a very deep rectangular 
potential well and the general nature of their splitting by the spin-orbit 
coupling, according to assumptions 1 and 2. The level order is indepen- 
dent of the depth and radius of the well, as long as it is deep (t/ ~ 30 
Mev) and narrow (b ~ R). The same level sequence is given by many 
other short-range potentials, such as a three-dimensional harmonic oscilla- 
tor potential U = UQ + ar z (B48, H24), or hybrids of oscillator and rec- 
tangular-well potentials (M6). 

In accord with the Pauli exclusion principle, each state is permitted 
to contain "a maximum of (2j + 1) identical particles, corresponding to 
the number of possible values of m,j (Chap. 4). The occupation numbers 
(2j + 1) are given in Fig. 2.1 for each state and shell. Note that for any 
I value the total occupation of the j = I + i and the j = I TF levels 
always has the value 2(22 + 1), which is independent of the assumed 
coupling scheme. 



362 



The Atomic Nucleus 



[CH. 11 



4s 

3d 

if / 


. + 


Occupation number of 
identical nucleons 

State 1 Shell | Total 


/ " T 


N 


X, 


4 
f 6 

8 

r 10 


44 
82 




**^_ . nn - 


j, 2f ^ 


*/ 


^ r 


Ih ^ 


2 /, 


^^ 


;' 3. " 


*s^ , _ 


12 

r ^ 

4 

6 
f 8 


^32 


1 IT 


'i 2d ^ 


U J ^ 


Edi 1 " 


" "^ 


tf i 



x" r 


S 


v r^ r* 


/v 

10 1 

r 2 U 

L_J_I____ 28 




^ p. 

E- - DJ- 


'^v 

r> u ^ 




f 


^x 

\- 2s 


^^ 


8 a 

on 


"z 


Et,- 1 - 


?_]_)_!.._ 


Id ^ 


*J 




^x. 


^^ d* 


J _ R 


z" 


-p^- 


r J ) 

222 


Iff 


-p- 

n + 


Rectangular 
potential 
well 


i 
Splitting by 

spin-orbit 
coupling 



VI 



IV 

Ilia 
III 

II a 
II 



Fig. 2.1 Order of energy states (denoted by their radial and orbital quantum num- 
bers) for identical nucleong in an independent-particle model using a deep rectangular 
potential well (left). Center, the empirical splitting of j I + -J- and j = I -$ 
states, which is attributed to spin-orbit coupling. The energies are not to scale. 
The exact order of the states is subject to variations, particularly with respect to 
the crossovers shown by bent arrows. For convenience, the parity of each state is 
indicated by the superscript, ( ) denoting odd for p, /, h states and (+) denoting 
even parity for s, d, g, i states. "~ - 

Major Closed Shells. Figure 2.1 shows that the higher magic numbers 
50, 82, and 126 can be obtained from the spin-orbit- or ^"-coupling hypoth- 
esis by asuming that the major shells close with a j = I + state and 
that the next shell begins with the corresponding j = I ? state. For 
example, the state of the last odd proton in the Z = 50 shell is shown by 



2] Models of Nuclei 363 

the nuclear moments of odd- A isotopes of indium, Z = 49. These prop- 
erties are 

49 InJ 1 4 3 :/ = | M = +5.49 

49 In!! 5 : 7=| ii = +5.50 

and in accord with the Schmidt limits [Chap. 4, Eq. (4.9)] the 49th proton 
is in a g$ state, which is j = I + with I - 4. After the Z = 50 shell 
is filled, the state of the first proton in the next shell is shown by the 
nuclear moments of odd- A isotopes of antimony, Z = 51. Here we find 

QV,123. T ? ,. I O CK 

BlOIDyo J "2" A* ~~ "T i m\.t*j 

and therefore this 51st proton is in a gi state, which is j = I -j with 
I = 4. 

Crossovers within Major Shells. Within each shell the exact order 
of the energy states is somewhat flexible. Some adjacent levels lie very 
close to one another in energy, and their actual order in any given nucleus 
may depend upon factors which are as yet unknown. The magnitude 
of the spin-orbit splitting may cause states which arise from adjacent I 
values in the same shell to cross over. Several apparently frequent 
instances of such crossovers, or inversions of the elementary order of 
energy levels, are shown in Fig. 2.1 by the bent arrows. As an example, 
consider again the 51st proton but in another odd- A isotope of antimony 

siSbft 1 : 7=| P = 3.36 

whose moments correspond to a rfj state, rather than to the g\ state 
shown by the 51st proton in 6 iSb 123 . In these two Sb isotopes, the shell 
just above 50 protons can therefore begin with either a g% or a d g proton, 
and the relative energy of these two states depends upon some factor 
other than the proton number Z, possibly the neutron number N. 

Minor Shells. Figure 2.1 shows also how minor discontinuities in 
nuclear properties, e.g., the possible cases at 14, 28, and 40 identical 
nucleons, can be accommodated as subshells in the jj-coupling model. 

Pairing Energy in the Shell Model. Assumption 5 states that the 
pairing energy 6 is finite and increases with j. This is physically the 
same pairing energy which we noted in our discussion of binding energies 
in Chap. 9. The shell model provides no information on the'absolute 
separation of nuclear energy levels nor on binding energies. We may 
wonder why the pairing energy is pertinent to the shell model. 

The pairing energy is invoked in the shell model in order to account 
for the observed nuclear moments. Although Fig. 2.1 shows that shell V 
(between 51 and 82 identical nucleons) has places for 12 particles with 
; = V, no nuclide is known which actually has a ground-level nuclear 
angular momentum 7 = Y-. The inference is that the nucleons do not 
always fill up the lowest states first in a shell. 

As an example of the action of pairing energy, consider the nuclide 

H Tc}i*: 7=-J M = -0.74 
in which the values of 7 and M show that the 71st neutron is in an ty state. 



364 The Atomic Nucleus [CH. 11 

In Fig. 2.1 we may count up the available states in shell V and note that 
the 71st particle must be in hy. if the states fill up in order, regardless of 
the presence or absence of a crossover between s and A v . The experi- 
mental values of / and n show then that for ground levels the Ay state 
always contains an even number of identical nucleons. In the case of 
Te 125 , this is accomplished by drawing one neutron out of the Sj state in 
order to complete the pairing in the Ay state. 

A very approximate calculation by Mayer (M26), assuming a short- 
range attractive potential between identical nucleons, gives an inter- 
action energy per pair of identical nucleons which is proportional to 
(2j + 1)/-A, and an interaction energy of zero for any odd nucleon. 
Then the pairing energy 5 is approximately proportional to (2j + I)/ A, 
and the pairing of identical nucleons is energetically favored for states of 
large j over states of smaller j. This is in accord with the observation 
that for ground levels the /i v state, if confronted with occupancy by an 

___^___^ odd nucleon, will always rob a state 

T < ay of smaller j, in order to make up an 

M4 \ 0.088 Mev even number of nucleons in Ay. 

| t Excited Levels in the Shell Model. 

"* S Some of the low-lying excited levels 

Ml <[ 0.159 Mev of odd- A nuclei correspond to occu- 

8 |W J stab|e pation of higher-energy states by the 

T 123 odd nucleon. Figure 2.2 shows the 

52 &71 known excited levels of Te 123 , whose 

Fig. 2.2 The excited levels of Te 128 observed angular momenta and pari- 
corrcspond to an ocoupancy by the odd ties are consistent with occupancy 
neutron of a d g and an fcy state, within by fhc 71st neutron o f the fli state 
the same shell V as the ^ ground state ( nd leyel) ^ rf ^^ (mciM 
The asterisk denotes an assignment of M } * d ^ 

and ? based on direct measurements of ,, _ , , k .7 JT ' / , 

7 and M (104-day isomenc level). All these 

states are available in shell V. 

Recall also the decay scheme of 49!^^ > 48CdJJ 1 , given in Fig. 8.5 
of Chap. 6. The g l state for 4 9ln nl shows that in this case (and in shell 
IV) the proton pairing energy does not overcome the lower states. In 
the ground level of the decay product 4aCdJJ 1 , the 63d neutron occupies 
the s$ state (due clearly to the effect of neutron pairing energy), and the 
excited states shown in Fig. 8.5 correspond to occupancy by the 63d 
neutron of other states (d 5 and } ) in the same shell. In addition, two 
other low-lying levels are known in Cd 111 , and the angular momentum 
and parity of these correspond to the rfj and h^ states which are also 
available in shell V. 

Excited levels may also correspond to occupancy by the odd nucleon 
of states belonging to the ground-level configurations of higher shells. 
For example, 4gIn llB has known excited levels (G25) for which the state 
of the 49th proton is p^, d 3 , dg, and t/j. The p\ state is available in shell 
IV, while the others are typical of ground-level configurations in shell V. 

Domain of Success of the Shell Model. The successes of the present 
shell model extend through the first eight items of experimental informa- 



3] Models of Nacki 365 

tion listed in Sec. 1 of this chapter. These are the phenomena associated 
with the magic numbers, or the "periodic system for nuclei," and with 
nuclear moments. The shell model, with strong spin-orbit coupling, 
gives the first satisfactory representation of the angular momentum, 
parity, and magnetic dipole moment of the ground levels and the low- 
lying excited levels of nuclei. This model also gives the best representa- 
tion found so far of the "dynamic" electric and magnetic moments which 
account for the 7-ray transition probabilities between different levels in 
nuclei (Chap. 6). Besides the original literature, a number of useful 
reviews (F18, F19, F54) may be consulted for additional details. 

Problems 

1. Predict the following characteristics of the ground levels of (a) izMg" and 
(fe) 29Cu 63 : (1) state of the odd nucleon, (2) total nuclear angular momentum, 
(3) nuclear magnetic dipole moment, (4) sign of the nuclear electric quadmpole 
moment, and (5) parity of the ground level. Explain the probable cause of any 
important discrepancies between your predictions and the following measured 
values of the moments 





7 


M 


Q 


12 Mg26 


5 


-0.96 


Not reported 


29 Cu 





+2.22 


-0 1 



2. The observed nuclear moments of 8 sBi 209 are: 7 = f , /* = +4.1, and 
Q = 0.4 X 10~ 24 cm 2 . What are the expected values on the independent- 
particle model? Comment on any significant discrepancies. Why would you 
expect Bi 209 to have an unusually low cross section (^0.003 barn) for the capture 
of 1-Mev neutrons, as compared with an ' 'average 7 ' heavy nuclide (^0,10 barn)? 

3. The Liquid-drop Model 

The liquid-drop model provides reasonable explanations for many 
nuclear phenomena which are inaccessible to the shell model. In the 
main, these phenomena are items 9 through 15 of the tabulation in 
Sec. 1, involving the masses and binding energy of nuclear ground levels; 
the energetics of ft decay, a decay, and nuclear reactions; the cross sec- 
tions for resonance reactions ; and the energetics of nuclear fission. 

The liquid-drop jnodel is the antithesis of the independent-particle 
models. JThe interactions between nuclepns are assumed tp^ be. strong 
instead of weak. Nuclear ieyels__are i j-epresratedjs_quantized states 
of the~M^earBytienras a whole and jiotjis states of a single particle in 
an average fielcT TKeTrqurd-dr^~^^lj)ri^ginated in Bohr's _conceptj)f 
the comp oundnSucTeus nTmicIear reactions. When an incident particle 
is captufed~by Vlafget nucleus, Jts "e'nerf^l^p^ 
by all the nucledhs. " TEe captured particle has a mean free path in 
nuclear -"matter which is much smaller than the nuclear radius. To 
account for such behavior, interactions between nucleons have to be 



366 The Atomic Nucleus [CH. 11 

gtrong, and the particles cannot behave _iiuJpeaderit^_ witJi negligible 
cross sections for collisions and interactions with their neighbors. 

Application of the strong-interaction, liquid-drop model to the cross 
sections for nuclear reactions will be considered in Chap. 14. We shall 
devote our attention in this section mainly to those restricted aspects 
of the liquid-drop model which are fruitful in quantitative discussions of 
nuclear masses, the nuclear energy surface, and the energetics of spon- 
taneous and induced nuclear reactions. Primarily, this involves the 
development and use of a modern yersioiLpf Weizsacker^s i_(W26) so-called 
semiempiricdl masjijj&m^a . 

a. Qualitative Basis of the Semiempirical Mass Formula. The mass 
M of a neutral atom whose nucleus contains Z protons and N neutrons is 

M = ZM W + NM n - B (3.1) 

where the binding energy B is made up of a number of terms, each of 
which represents some general characteristic of nucJei, as seen in the 
empirical data on the binding energy of stable nuclei. Thus 

B = Bo + B! + B 2 + - (3.2) 

A 

( Under a reasonable set of simplifying assumptions, we can develop a 
quantitative model which describes the binding energy B of the ground 
levels of all but the lightest nuclei, say, A > 30. The initial assumptions 
are: 

1. The nucleus is like a droplet of incompressible matter, and all 
nuclei have the same density. 

2. The distinction between the triplet (up) and singlet (up) force is 
ignored; forces between nucleons are considered to be spin-independent 
as well as charge-independent. If the coulomb force is turned off, 

(np) = (nn) = (pp) 

3. These nuclear forces have a short-range character and are effective 
only between nearest neighbors. Each nucleon interacts with all its 
nearest neighbors.*} 

4. Additional assumptions will be introduced later to provide refine- 
ments in the model. 

. Volume Energy. The firstjjppr^jna^oiij^^and the largest term 
in the binding energy, isTdentified as due to the saturated exchange force. 
We have seen (Chap. 9) that the average bii\ding_energy_per nuclemi is 
approximately constant (10 per cent) in all nuclei (.4 > 1C). Then 
we write 

Bo = a v A volume energy (3.3) 

where the arbitrary constant a v is to be evaluated empirically. The 
subscript v connotes "volume 11 energy. (With equal justification, what 
we here call volume energy is often called the " exchange energy.") 

Surface Energy. Those nucleons which are visualized as being at the 
nuclear surface have fewer near neighbors than nucleons which are deep 
within the nuclear volume. We can expect a deficit of binding energy 



3] Models of Nuclei 367 

for these surface nucleona. We^ interpret th^exchange energy term B 
as a "volume energy" representing the^ binding of nucleons which are 
totally within the nuclear volume. Then we_deduct_a.,ci?rrection term 
for the nucleons which^bnstifute_the nuclear surface. The radius of the 
nucleus is R = Rt>A* under the assumption of constant density. If the 
range of the nuclear forces is &, we can take the effective radius of a 
nucleon as about fe/2 if the nucleons are presumed to be essentially in 
contact with each other. Then the volume of a nucleus would be 

A (3.4) 

and the effective radius of one nucleon 6/2 is about equal to the nuclear 
unit radius R Q ^ 1.5 X 10-" cm. 
The surface area of the nucleus is 



4irB a [_= 4*RIA* (3.5) 

Then the number of nucleons on the surface would be approximately 

^ (3.6) 

TT/tg 

and the fraction of the nucleons which are in contact with the surface 
is of the order of 

2 - T> 

Thus for light nuclei nearly all the nucleons are at the surface, while for 
heavy nuclei about half the nucleons are at the surface and half are in the 
interior of the nucleus. We jntroducc & nfigativ^-Gorref4ion term BI 
representing the loss of binding energy by the nucleons at the surface 

B! = a,A l surface energy (3.8) 

where a. is an arbitrary constant to be evaluated from empirical data. 
The subscript s means "surface energy." Occasionally this surface 
energy is referred to as "surface tension/' by analogy with these two 
concepts in ordinary liquids. It should be remembered, however, that 
"surface energy" is generally a larger quantity than "surface tension" 
even though the two do have the same physical dimensions (M35). 

Coulomb Energy. The only known long-range force in nuclei is the 
coulomb repulsion between protons. We have seen in Chap. 2 that in 
evaluating the coulomb energy for nuclei we are justified in regarding 
the total nuclear charge Ze as spread approximately uniformly through- 
out the nuclear volume. Again assuming a constant-density nuclear 
radius, RvA*, the loss of binding energy due to the disruptive coulomb 
energy is 



B 2 = - ^ -j 5= -a e - coulomb energy (3.9) 

here a e is to be evaluated and the subscript c designates coulomb energy 



368 The Atomic Nucleus ICH. 11 

Asymmetry Energy. Another deficit in binding pnergy depends on 
"the neutron excess (N _Z) and is proportional to_C/V _Z) Z /A. This 
"asymmetry energy" Is aTpurely quantum-mecEamcaT effect, In contrast 
with the simple classical effects of surface energy and coulomb energy. 

Among the lightest elements, there is a clear tendency for the number 
of neutrons and protons to be equal, as in 6 C 12 , 7 N 14 , 8 16 . This is properly 
interpreted as showing that the (up) force can dominate (nri) and (pp) 
forces. But it is the triplet (np) force which is involved in the N = Z 
relationship for the lightest nuclides. In the liquid-drop model for 
heavier nuclei we are neglecting the difference between the (np) triplet 
and the (np) singlet. 

Heavy nuclei always contain appreciably more neutrons than protons. 
If we were to attempt to build a heavy nucleus out of equal numbers of 
protons and neutrons we should find it violently unstable, because the 
large disruptive coulomb energy could not be overcome by the available 
(np), (nri), and (pp) attractive forces. It is necessary to introduce a 
neutron excess (N Z) to provide enough total attractive force to 
dominate adequately the coulomb repulsion. At the same time one must 
not add too many neutrons or instability is again achieved. 

For a medium or heavyweight nucleus, of predetermined mass num- 
ber A, the approximate mass, based only on Eqs. (3.1) to (3.9), 

M = ZM* + NM n - B = ZM* + (A - Z)M n - B 

- Z(M n - M H ) - a,A + a>A* + a c + - (3.10) 



has only two terms which depend on Z. These terms are of opposite 
sign, so that by differentiation a value of Z can be found for which M 
is a minimum. If this equation really represented all the dominant 
effects, then we would have to expect ridiculously small values of Z. 
For example, for A = 125, Eq. (3.10) gives a minimum M, hence great- 
est stability for Z = 3. Clearly , some important term is still missing, 
and its sign must be such as to increase Z for a given A. 

The physical phenomena which have been neglected so far are the 
quantization of the energy states of the individual nucleons in the nucleus 
and the application of the Pauli exclusion principle. If we put Z protons 
into a nucleus, these will occupy the lowest Z energy states. If we add 
an equal number of neutrons N = Z, these neutrons will occupy the same 
group of lowest-energy states. If we now add one or more excess neu- 
trons, these (N Z) neutrons must go into previously unoccupied 
quantum states. In general, these will be states of larger kinetic energy 
(KE) and smaller potential energy (PE) than those already occupied. The 
binding energy of each nucleon is the difference (PE KE) between Us 
potential and kinetic energies. Hence these (N Z) excess neutrons 
will have less average binding energy than the first 2Z nucleons which 
occupy the deepest-lying energy levels. If there should happen to be 
more protons than neutrons, the (Z N) excess protons would have 



3] Models of Nuclei 369 

to go into higher-energy states, in a completely similar way. The reason- 
ing is then independent of the sign of (N Z). 

The form (N Z) 2 /A of the asymmetry-energy term can be "de- 
rived" in a variety of ways, depending on what assumptions one is 
willing to make (W26, F41, B48, W48). In every case the asymmetry 
term expresses the physical fact that, in a quantized system of neutrons 
and protons, any "excess" nucleons are pushed up to levels which they 
occupy alone. They are thus deprived of the fullness of binding which 
was implied in Eq. (3.3). 

'*The simplest approach to the form of the asymmetry energy is 
probably the following: If the \N Z\ excess nucleons are regarded as 
producing a deficit of binding energy because they are "out of reach 1J 
of the other nucleons (quantically), the fractipjLjrfJJxejuclear volume 
so affected is \N Z\/A, and the total deficit is proportional to the 
product of these, or 

(N - Z)' (A - 2Z) . 

# 8 = CL O -. = a a -. V*-J1) 

A . -. 4. ., 

where the asymmetry coefficient a a is to be evaluated empirically. 

Wigner's "uniform model" of nuclei (W47, W48), from which emerges 
a reasonably satisfactory account of the mass difference of isobars in the 
domain 16 < A < 60 (14, F71, W17), goes over smoothly for larger A 
(W48) into the semiempirical mass formula of the liquid-drop model. 
In Wigner's theory, the isobaric-spin quantum number T, when pro- 
jected onto the f axis in a hypothetical isobaric-spin space, has the value 
T{ = v(N ~~ Z), or one-half the neutron excess. For large A, the differ- 
ence between the total potential energy and kinetic energy contains the 
term T*/A, which again gives an asymmetry energy proportional to 
(N - ZY/A. 

In heavy nuclei, the asymmetry energy will be found later (Table 3.2) 
to be of the order of one-quarter as large as the coulomb energy. The 
presence of this "unbinding," or disruptive, energy term, containing 
(N Z) 2 , greatly favors proton numbers Z which are comparable with N. 

From a purely empirical point of view, it can be shown readily that 
the systematic dependence of (N Z) on A, as shown in Fig. 3.4 of 
Chap. 8, can only be had if in Eq. (3.11) the coefficient a a is positive 
and if the exponent in the numerator is exactly 2. Also empirically, the 
parabolic relationship between Z and the masses M z of a family of iso- 
bars, as shown in Figs. 3.2 and 3.3 of Chap. 8, only emerges if the expo- 
nent of (N - Z) in Eq. (3.11) is exactly 2. 

Pairing En&rgy. AUMpur enerjfj^tOT^^ involved a 

smooth vanation of the ^tqtal Hndmg gjgrgy every time Z or N changes. 
This is cuiilrai'srtjOwo^sets of empirical facts: first f the finite pairing 
energy 6 between odd-A and even- A nucleLand^secpjpd, the anomalously 
large bhidfflgjBnergy _rfjoudeL_which contain a "magic number" of 
neutrons oTprotons. These facts fail to appear in the!Tc[iird=drop-inodel 



370 The Atomic Nucleus [CH. 11 



because we have omjf.t^jj^_n^ nf the nucleons and have based 

the development, up to now, on spin-independent forces. To correct 
for this omission, we add the pairing_eneryJ .as .another correction term 
B 4 for the totallunfeuQii^rgy., Conventionally, the pairing-energy cor- 
rection is usually taken as zero for odd- 4 nuclides (F38, B96, K35). 
Then for even-^4 nuclides 

for even-Z even-Af /* 10^ 

aforodd-Zodcl-AT V ' ' 

When d is regarded as a correction to the mass rather than to the bind- 
ing energy, the sign of d is given by 

") 
for odd-.4 ( 

for even-Z even-.V \ (3.13) 

for odd-7 odd-TV ) 

From the shell model, the pairing energy 5 appears to be roughly pro- 
portional to (2j + \)/A. As no information on angular momenta j is 
implied in the liquid-drop model, the average pairing energy should be 
expressed as some smooth function of A, Fermi's (F41) empirical value 

5-a^ (3.14) 

where a p is an approximately constant empirical coefficient, has been 
used widely and is in consonance with the general trend of increasing j 
with increasing A in the approximate expression (2j + !)/-!. Equation 
(3.14) is only a rough representation of 5, as will be seen later in Fig. 3.4. 

In a refined treatment of the pairing energy, Kohman (K35) has 
given quantitative recognition to slight differences between those odd-<A 
nuclides which have odd-Z and those which have odd-JV. In a given 
heavy odd- A nucleus the (N Z) excess causes an unpaired neutron to 
lie in a higher state than does an unpaired proton. This leads to a slight 
difference e between the pairing energy for neutrons and for protons, 
which Kohman evaluates empirically by replacing 5 = for odd- A 
nuclides by 5 = +e/2 for odd-Z even-AT nuclides, and by 5 = c/2 for 
even-Z odd-TV nuclides. In the present discussion we shall omit this 
interesting (C46) refinement. 

Closed Shells. The Ijauid-drpp _model takes no cognizance of shell 
structure. Therefore the extra binding energy (~1 to 2 Mev) of nuclei 
which contain fully closed shells of neutrons or protons is not represented 
in the semiempirical mass equation. For those who desire it, a "coeffi- 
cient of magicity" could be added as a final correction term #&, to be 
used only^wheii N or Zj= 20, 28, 40~ 50, 82, "or 12(> (see Table 3.4). 
More commonly) one simply notes tfiafr thfcse Tracftdes^Tir have an 
abnormally small mass wten compared with their isobars. 

b. Empirical Evaluation of the Coulomb and Asymmetry Coefficients. 
Assembling the results of the previous paragraphs, we have for the semi- 
empirical mass formula of the liquid-drop model 



3] Models of Nuclei 371 

M (Z,A) = ZM* + (A - Z)M n - B 

(A - Z)M n - a v A + a.A* + a c ^ 

1 

-L. * "lim 

5 (3.15) 



where 5 is defined by Eq. (3.13) or (3.32). 

The five empirical coefficients are to be evaluated by comparison of 
Eq. (3.15) with data on the masses of stable nuclides and the energetics 
of nuclear reactions. In principle, the five constants, determined from 
five masses or reactions, will then serve to predict hundreds of other 
masses and reactions. In practice, Eq. (3.15) gives an extremely good 
average representation of nuclear energetics over a wide range of A. 

The Coulomb Energy of Nuclei. The nuclear unit radius 72 , as 
derived from those types of experimental evidence which involve coulomb 
effects, has been discussed in detail in Chap. 2. The unit radius which 
is usually chosen as appropriate to the semiempirical mass formula is 
the coulomb-energy unit radius, Eq. (2.14) of Chap. 2, which is 

fto = (1-45 0.05) X 10- 13 cm 10 < A < 240 (3.1t>) 
Then the coulomb coefficient a c in Eq. (3.9) becomes 

3 (c 2 /W 2 ) 2 3 2.82 X 10- cm ft _ 1 .. 

a < = 5- -ST" moc = s ISB^TF^ ' 51 Mev 

= (0.595 0.02) Mev = (0.64 0.02) X H)-* amu (3.17) 

The Neutron Excess in Stable Odd-A Nuclides. The asymmetry- 
energy term is evaluated by adjusting its coefficient a a so that Eq. 
(3.15) will predict stability against ft decay for the naturally occurring 
stable nuclides. 

For any odd-,4, the correction for pairing energy 8 is taken as zero. 
Then Eq. (3.15) becomes an analytical relationship between the masses M 
and the nuclear charge Z of any group of odd- A isobars. Equation 
(3.15) is quadratic in Z] hence for each fixed value of A there is some 
particular value of Z for which M is minimum. The Z value which 
corresponds to minimum mass M is called the nuclear charge of the moat 
stable isobar, denoted Z . By setting 



'dM\ 

,dZ /A 



= (3.18) 

Eq. (3.15) gives the following relationship, for odd- A, 

2a c ^ - 4o (A ~ 2Zt) = M n - M* (3.19) 

J\. \. 

On rearrangement, this becomes 

a a 1 / iA \ / A \ \JVLn MB) tn o^^ 

a c ~ 2 (A^WJ - (A~^ ] - (3>20) 



372 



The Atomic Nucleus 



[CH. 11 



In actual nuclei Z is an integer, but the charge Z of the "most stable 
isobar" was obtained by minimizing M; hence Zo is generally some hypo- 
thetical noninteger charge. The atomic number Z of the actual stable 
nuclide of odd mass number A is the integer which is nearest to Z . 
Therefore Z must lie within the bandwidth Z 0.5. In Eq. (3.20) a 
mean value of a a can be obtained by averaging the (Z Q ,A) functions 
over a number of stable nuclides. 

In evaluating Eq. (3.20) we find that the first term, Z A } /2(A - 2Z ), 
is predominant. The second term acts as a correction of about 5 per 
cent for heavy nuclides to 10 per cent for light nuclides. Therefore the 
value of the ratio a a /a e is substantially independent of the value of a r 
chosen for the correction term. Using a c = 0.595 Mev from Eq. (3.17), 
the dimensionlesa numerical coefficient of A/(A 2Z ) is 



(M n - Jf H ) _ 0.78 Mev 



4a c 



4 X 0.595 Mev 



0.33 



Table 3.1 shows a representative calculation of an average value of 
a a /a e . Note how the individual values vary more or less randomly from 
the average value. This is due in part to variations in Z Zo and in 

TABLE 3.1. EVALUATION OF THE ASYMMETRY-ENERGY COEFFICIENT a 0| FROM 
Z AND A OF STABLE ODD-A NUCLIDES, THROUGH EQ. (3.20) 



Z A 


A - 2Z 


ZA\ 
A -2Z 


A 
A -2Z 


0. 

a e 


33 As 75 


9 


65.2 


8.3 


29.9 


35 Br 79 


9 


71.6 


8.8 


32.9 


35 Br 81 


11 


59.5 


7.4 


27.4 


41 Nb 93 


11 


76.5 


8.5 


35.4 


45 Rh 103 


13 


76.3 


7.9 


35.5 


53 I 127 


21 


63.8 


6.1 


29.9 


55 Cs 133 


23 


62.3 


5.8 


29.3 


65 Tb 159 


29 


65.9 


5.5 


31.1 


67 Ho 165 


31 


65.1 


5.3 


30.8 


69 Tm 169 


31 


68.1 


5.5 


32.3 


73 Ta 181 


35 


66.7 


5.2 


31.7 


77 Ir 191 


37 


69.0 


5.2 


32.8 


77 Ir 193 


39 


65.9 


4.9 


31.3 


79 Au 197 


39 


68.7 


5.1 


32.7 


83 Bi 209 


43 


68.0 


4.9 


32.4 







Avg. -67.5 




Avg. - 31.7 



part to true nuclear effects. The value a a /a c ~ 32 can be used with 
Eq. (3.20) for the determination of Z for any A. Then calculations 
of the type shown in Table 3.1 can be repeated, using Z instead of Z. 
When this is done (F17), the average value of a a /a c remains essentially 
unchanged, and the fluctuations in a a /a c are reduced but not eliminated. 
The important physical fact is that the "local values 9 ' of a a /a c for various 
values of A do possess true variations of the order of 2 to 5 per cent from 
the average value. 



3] Models of Nuclei 373 

We conclude that the ratio of the asymmetry-energy coefficient a a to 
the coulomb-energy coefficient a, is given on the average by 



5= - 32 1 (3.21) 

a c 

Then if a c = (0.595 0.02) Mev, we have 

a a = (19.0 0.9) Mev 

= (20.4 0.9) X 10-' amu (3.22) 

Two relationships which have general utility may be obtained by 
rearrangement of Eq. (3.19) and substitution of the empirical values of 
a a and a c . The first is a general expression for Z , the nuclear charge 
of the most stable isobar having odd mass number A, which reads 



A Fl + 
2 L 1 



(M n - M 



(a c /4a fl )A' ' 

Z = ___ - - ; (3.24) 

1.98 + 0.0155A* 

This equation is an analytical refinement of the rough rule A ~ 2Z. 
The second generalization is an analytical expression for the neutron 
excess N Z = A 2Z , which is 



_ 2 .. - (M - M )M 

[ 1 + (a e /4a a )A< \ 

/0.0078A' - 0.0103\ 
~ A \ 1+0.0078,1' ) 

- 0.0078^ 132A (3-26) 



The terms containing A 1 are both small, and their variations with A 
tend to cancel, so that over the mass range 60 < A < 210, a good 
approximation (4 per cent) is 

A - 2Z c- 0.0060A* (3.27) 

Equation (3.27) agrees in form and magnitude with the empirical varia- 
tion of the neutron excess N Z with A*, as was seen in Fig. 3.4 of 
Chap. 8. 

c. Equation of the Mass Parabolas for Constant A. The parabolic 
relationship between isobaric mass M(Z,A) and atomic number Z is 
contained in the semiempirical mass formula of Eq. (3.15). In order to 
simplify the nomenclature, we rewrite Eq. (3.15) as 



M (Z,A) = Z(M* - M n ) + A(M n - <O + a.A* + a c j + a a A 

-a fl 4Z + a a 
a aA + 0Z + yZ* S (3.28) 



374 The Atomic Nucleus [cfe. 11 

where a = M n - I a, - a fl - -^J , (3.29) 

ft = -4a a - (M n - MH) s (3.30) 

* 



(+6 for odd-# odd-N:even-A 
for odd-Z even-AT : odd- A , 
Oforeven-2odd-JV:odd-A t 3 ' 3 ^ 
6 for even-Z even-JV : even- A 

For constant A, Eq. (3.28) is the equation of a parabola. The coefficients 
a, ft, 7 have dimensions of energy (or of mass). We note that ft is inde- 
pendent of A, a is nearly independent of A, and 7 varies approximately 
inversely with A. The coefficients for surface energy a and volume 
energy a v are contained only in a. These are accessible to empirical 
evaluation, then, only when A varies, as in a decay systematics, nuclear 
reactions such as (a,p) or (y,ri) or fission, or a sequence of exact atomic- 
mass values. The coefficients a c and a a /a c have been evaluated, on the 
average, so we can at once write the average values of ft and 7, which are 



(M n - Jlf u )J 
= -[4(19.0 0.9) +0.78] 
= -(77 4) Mev .4 > 60 (3.33) 

We can expect local variations of at least 5 per cent about this mean value. 
Note that 99 per cent of ft comes from the asymmetry term 4a a . Like a a , 
ft tends to be larger than average for small values of A, say, A < 60. 
The average value of 7 is 



A ft f 

7U>a I .. 

"V 1 



A* \ 
4a a /aJ 



with about 5 per rent fluctuations expected in local values. 

The coefficients a and 5 could be evaluated here if accurate mass data 
were available for a number of middleweight and heavy elements. In 
the absence of such data, we can turn our attention to differential forms 
of Eq. (3.28). Then local values of ft, 7, and 5 can be obtained by com- 
parison with the energetics of nuclear reactions in which A does not 
change, e.g., in (p,n), (n,p), (d,2n) reactions, and in ft decay. 

Local Values of the Energy Coefficients. In the simplified notation of 
Eq. (3.28) the charge Z of the most stable isobar is 

= (3.35) 

or Z = ^ (3.36) 

27 



3] Models of Nuclei 375 

Equation (3.36) is the algebraic equivalent of Eq. (3.23). In the average 

2 7mv Z = -jfl.v = 77 Mev = constant for all A (3.37) 

For odd- A (hence 5 = 0), the mass M(Z ,A) of the hypothetical "most 
stable isobar" is given by Eq. (3.28), with ft = 2-yZo, and is 



M(Z*,A) = aA - 2-yZoZo + yZl 
= aA - yZ\ odd- A 



(3.38) 



On the same basis, the mass M(Z,A) of a real nuclide, with an integer 
value of Z, is given by Eq. (3.28). Then a A can be eliminated between 
Eqs. (3.28) and (3.38), giving 



M(Z,A) - 



= PZ + yZ z + yZl 

= -2yZZ + yZ z + yZ\ 

= y(Z - Z ) 2 odd-A 



(3.39) 



Equation (3.30) is the parabolic mass relationship for odd-A isobars, 
with vertex at Z , M(Z ,A), as shown in Fig. 3.1. 




M(Z ,A)- 



Fig. 3.1 The parabolic relationship between the masses M(Z,A) of odd- A isobars, 
Eqs. (3.39) and (3.42). The two possible values of the ft disintegration energy Qp are 
shown in boxes, where Z is the atomic number of the initial nuclide. 

Transitions between Odd- A Isobars. Reactions in which Z > (Z + 1), 
at constant A, such as (p,n) 7 (d,2n), and ft- decay, will involve an energy 
release which for ft- decay only [see Eq. (3.54a) for Q (pl n)] is given by 

Q,- - M(Z,A) - M(Z + 1, A) = y[(Z - Z ) 2 - (Z + 1 - Z ) 2 ] 

- 27(Z - Z - i) for odd-A (3.40) 

In a similar way, the energy release for all Z > (Z 1) reactions, at 



376 The Atomic Nucleus [CH. 11 

constant A, such as (n,p), 0+ decay, and electron-capture transitions, is 
related to that for 0+ decay. 

Q0+ = M(Z,A) - M (Z - 1, A) - 7 [(Z - Z ) 2 - (Z - 1 - Z ) 2 ] 

= 2 T (Z - Z - i) for odd-A (3.41) 

Both types can be summarized as 

Q? = M(Z,A) - M (Z 1, A) 

- 27[ (Z - Z) - i] for mid- A (3.42) 

where the + sign in + (Z Z) is to be used with the + sign in Z 
(Z 1). The graphical implication of Eq. (3.42) is indicated in Fig. 3.1. 
For any odd- A, two measured Q values suffice to determine the local 
values of both unknowns Z and 7. For example, the ft decay schemes of 
Te 131 and I 131 involve complex ft spectra and 7 radiation, for which the 
total energy release, or Q values, are 

6 2 Te 181 -> ft- + B3 I m + (2.16 0.1) Mev (3.43) 

sal 111 -> ft- + 64Xe 181 + (0.97 0.01 ) Mev (3.44) 

Expressing the energetics of these reactions in the form of Eq. (3.42), 
we have 

for 52TC 1 ": (2.16 0.1) = 2y(Z Q - 52 - i) (3.45) 

for 53 I m : (0.97 0.01) = 2 7 (Z - 53 - i) (3.40) 

The local solution for 7 is given at once by the difference between these 
two equations, 

7 = i[(2.16 0.1) - (0.97 0.01)] = 0.60 0.05 Mev (3.47) 

while the local solution for Z follows from the quotient of the two equa- 
tions and is 



z - 52 - 5 + - 52 - 5 + (1 - 8 ai) 

= 54.3 0.1 (3.48) 

The stable isobar of A = 131 is actually xenon, Z = 54. 

Prediction of Reaction Energetics among Odd- A Isobars. These empir- 
ical local values of 7 and Z , for A = 131, should be compared with pre- 
dictions based on 7. v and 0. v for which Eqs. (3.33), (3.34), and (3.36) give 

7.v(A = 131) = 0.70 0.04 Mev 
Z (A = 131) = 55 3 

The rather wide uncertainties implied here are reflections of the con- 
servative view taken in Eq. (3.21) regarding (a a /a c \ v . For the predic- 
tion of local values of Z , the use of y nv and Q mv is seen to be of little value. 
However, if the energetics is known for any one reaction at constant 
mass number A, then usefully accurate estimates of the energetics for all 
isobars at mass number A can be made, using 7.*. For example, if we 



3] 



Models of Nuclei 



377 



assume that at A 131 only Eq. (3.46) is known, then the energetics 
of the Te 1 ' 1 ft decay can be predicted, using only ?, v . By generalizing 
Eqs. (3.45) and (3.46) we can write 

Q,-(Z lt A) = Q ft -(Z 2 ,A) + 2 7 . V (Z 2 - Z x ) (3.50) 

Then <MTe') = Q,-(F 81 ) + 2(0 JO 0.04) (53 - 52) 

' = 0.97 + (1.40 0.08) 

= 2.37 0.08 Mev (3.51) 

We see that such predictions have an inherent uncertainty of only about 
0.1 Mev. 

Transitions between Even-A Isobars. Isobaric masses for even-^4 
nuclides follow the same pattern as for odd- A, except for the introduction 




z 



Fig. 8.2 Energetics of the even- 4 mass parabolas, Eqs. (3.52) and (3.54). The four 
forms of the ft disintegration energy Qp are shown in boxes, where Z is the atomic 
number of the initial isobar. 

of the pairing energy 5. Then in Eqs. (3.28) to (3.38), 0. v , T.V, and Z 
remain unchanged. The mass of the hypothetical "most stable iso- 
bar/ 1 Eq. (3.38), becomes 

M(Z ,A) = <*A - yZl - 6 even-A (3.52) 

the negative pairing energy 5 being chosen so that M (Z 0| A) will have 
the smallest possible value. Then the parameters M (Z , -4) and Z locate 
the vertex of the lower, or even-Z even-AT, mass parabola, as shown in 
Fig. 3.2. 

In place of Eq, (3.39) we obtain, from Eqs. (3.28), (3.52), and 



378 The Atomic Nucleus [CH. 11 

the even-A mass parabolas 

M (Z,A) - M (Z ,A) = 7(Z - Z ) 2 

, (25 for odd-Z \ A /0 eox 

+ I } even-A (3.53) 

1 for even-Z) ' 

The reaction energy Q, for Z (Z 1) at constant even-A, becomes 

= M(Z 9 A) - M(Z 1, A) 

' I O* *. ^JJ *7 1 

even-A (3.54) 



The + sign in (Z Z) is to be used with the + sign in Z > (Z 1), 
that is, for 0~ decay, (p,ri) and (d,2/i) reactions, etc., while the sign 
corresponds to Z > (Z 1) transitions such as /8+ decay, (w,p) reactions, 
etc. When heavy particles are involved, their mass differences must, of 
course, be added into Eq. (3.54). For example, 

Q(p,n) = Qe- - (M n - M H ) (3.54a) 

d. Determination of Local Values of 6, Z , and 7. Historically, the 
original evaluations of the most stable charge Z , and the pairing energy 
6, were based on the catalogue of known stable nuclides (F38, B96, F17, 
K33). 

Limits of (3 Stability. The main features of the variation of Z with 
A can be determined from the systematics of stable nuclides. In a 
Z vs. A diagram, the path of Z is determined within rather narrow limits 
by the atomic numbers Z of the known stable nuclides, Fig. 3.3. For all 
odd- A nuclides Zo is confined to the narrow region 

(Z - i) < Z < (Z + i) odd-A (3.55) 

For even- A, the odd-Z odd-JV nuclides are unstable, so that we need 
study only the even-Z even-JV species. Among these, each value of A 
may correspond to one, two, or three stable isobars. Figure 3.3 shows 
the Z and A values for all stable nuclides, in a form devised by Kohman 
(K33) to emphasize the limits of ft stability. 

The breadth of the variations in Z which can correspond to stable 
even-Z even-AT nuclides is contained implicitly in Eq. (3.54). For all 
0-stable nuclides, Q & < 0. Then from Eq. (3.54), 

Z* - Z\ (3.56) 



a 



<T/^ 2 
for stable even-Z even-2V nuclides, while 

1 . , 



Z " ~ Z| (3>57) 

for unstable even-Z even-JV nuclides. Thus 6/7 is bracketed for all A. 

Inspection of Fig. 3.3 shows that for constant A the approximate half 
width of the region of stability is 5/7 ~ 1.5 for heavy and middleweight 
elements and is smaller for the lightest elements. 



3] 



Models of Nuclei 



379 



Energetics of ft Decay. For even- .4 isobars, Eq. (3.54) shows that 
the energetics of any reaction Z > Z 1 is determined by Z and three 
parameters: (1) the pairing energy 5, (2) the charge of the "most stable 
isobar" Z , and (3) the energy coefficient 7 which is defined by Eq. 
(3.31). All three parameters 5, Z , and 7 are functions of A. Each 
varies smoothly over a broad domain of A and also exhibits local varia- 
tions. For any particular A , the energetics of at least three independent 




50 



250 



100 150 

Mass number A 

Fig. 3.3 ^-stability diagram of the naturally occurring nurlides. The vortical scale 
is (Z 0.4A) instead of Z, in order to compress the diagram into a rectangular shape 
and to enhance the local variations. The line of "greatest stability" Z passes 
always within Z -J of each stable odd- A nuclide and otherwise is adjusted to pass 
about midway between the outer limits of ft stability which are set by the even- -4. 
(even-Z even-JV) nuclides. In Fig. 3.1 of Chap. 8 the line of stability Z is the cen- 
tral line Z of this diagram. [Adapted from Kohman (K33).] 

reactions am needed for the numerical determination of the three local 
parameters, 5, Z , and 7. 

As an illustration, the decay schemes for three radioactive isobars at 
A = 106 have been studied carefully, with the results (H61) 

44 Ru 106 -> |8- + 4 B Rh 106 + 0.0392 Mev 

4 5 Rh 106 - 0- + 4 6 Pd 106 + 3.53 Mev (3.58) 

47 Ag I06 -> p+ + 4 6 Pd 106 + (1.95 + 1.02) Mev 

These provide the data for three simultaneous equations in 5, Z , and 7, 
based on Eq. (3.54) 

44 Ru 106 -> p-i 0.0392 = 2y(Z Q - 44 - 0.5) - 25 
4 5Rh 106 - 0-: 3.53 = 27(^0 - 45 - 0.5) + 25 (3.59) 

4 7 Ag 106 -> 0+: 2.97 = 27(-Z + 47 - 0.5) + 26 



380 



The Atomic Nucleus 



[CH. 11 



The simultaneous solution gives, for the local values at A = 106, 

5 = 1.25 Mev Z = 46.19 7 = 0.752 Mev (3.60) 

As experimental data on decay and nuclear reactions accumulate, 
these methods have been applied systematically by various workers. 
Figure 3.4 shows the numerical data on the pairing energy 5, plotted as 
5/7 for 44 < A < 242, as computed by Coryell (C46) from 0-decay data. 
From the same survey, the local values of Z are found to follow the 
general trends seen in Fig. 3.3 and to be influenced clearly by shell 
structure. 




40 



60 



80 100 



120 140 160 180 
Mass number A 



200 220 240 



Fig. 3.4 The pairing energy a, expressed as S/y, for A > 40. The circles are indi- 
vidual local values computed from 0-decay energetics by Coryell (C46). The solid 
line represents the trend of these individual values. The dotted lines provide com- 
parisons with Feenberg's (F17) evaluation of 6/7 from the limits of ft stability, and 
with Fermi's (F38, F41) analytical approximation fi - 33.5/4* Mev. [Adapted from 
Coryell (C46).] 

e. Total Binding Energy for Stable Nuclides. We return to the full 
semiempirical mass formula, Eq. (3.15). The coefficients a, (of "volume 
energy") and a. (of "surface energy") are still to be evaluated. When 
this has been done, Eq. (3.15) will give predicted values for 

1. The atomic mass M(Z,A) and total nuclear binding energy B of 
any nuclide having A > 40 

2. The energy release, or Q value, for nuclear reactions in which A 
changes, for example, (a,d), . . . 

3. The energetics of a decay 

4. The energetics of nuclear fission 

In order to evaluate the two remaining coefficients a v and a., we require 
a minimum of two independent experimental data concerning any 
phenomena in which A is not a constant. We elect, arbitrarily, to use 
mass values. 

Masses of Stable Nuclides. It is convenient to express the observed 
neutral atomic mass M in terms of the average binding energy per 



3] Models of Nuclei 381 

nucleon B/A. Then, from Eq. (3.2) of Chap. 9, we have 

= (Mn - 1) - (Mn - M H ) -? - M ~ A (3.61) 



The corresponding theoretical value, from Eq. (3.15), is, for odd- A, 

(|) _ ._..* (,_) (3.62) 

VA/.1 A* A a \ A/ 

with a e = (0.595 0.02) Mev and a a = (32 l)a e = (19.0 0.9) Mev, 
as previously evaluated. 

Table 3.2 shows the masses of a few odd- A nuclides, selected to avoid 
proximity to magic numbers of Z or N. Any two masses suffice for a 

TABLE 3.2. EXPERIMENTAL VALUES OP MASS AND BINDING ENERGY, 

EQ. (3.61) 

Compared with the values calculated from the semiempirical mass formula, Eq. 
(3.62), with energy coefficients as shown above each energy column. 





Volume 


Surface 


Coulomb 


Asymmetry 












BoM 


Bi/A 


Bt/A 


Bi/A 








Refe. 


Z A 


a. 


a. Ml 


a*fA\ 


- - T)' 


(BM) .i 


(BMW 


MP 


for 




a. - 


a. 


a c 


da 








Afexp 




14.1 Mev 


13 Mev 


0.595 Mev 


19 Mev 










8O 17 


14.1 


5.05 


0.87 


0.07 


8.11 


7.75 


17 004 533 


(L27) 
















7 




168 33 


14.1 


4.06 


1.44 


0.02 


8.58 


8 50 


32 981 88 


(C35) 
















4 




25 Mn 55 


14.1 


3.42 


1.78 


0.16 


8.74 


8 75 


54 955 8 


(C36) 
















1 




29 Gu 65 


14.1 


3.22 


1.92 


0.22 


8.74 


8 75 


64.948 35 


(C36) 
















6 




53 I 127 


14.1 


2 59 


2.62 


0.52 


8.37 


8.43 


126 945 3 


(Hll) 
















1 




78 Pt 195 


14.1 


2.24 


3.20 


0.76 


7.90 


7 92 


195 026 4 


(B4) 
















8 




97 Bk 245 


14.1 


2.08 


3 66 


0.82 


7.54 


7.. r 2 


J4. r > 142 


(B4) 
















1 





determination of a, and a,. The masses of Cu 66 and I 1 " 7 were deter- 
mined in the same laboratory, using the same standard*-. Simultane- 
ous solution of Eq. (3.62) for Cu 65 and I 127 , whose mussi-s correspond to 
(B/A) np = 8.75 Mev and 8.43 Mev, gives a, = 13.0 M^v und r = 14.28 
Mev. If, as a check, we compare I 127 with a ho:'vi*'v mi.'Jide Pt 195 
(whose mass has been determined in another luliornt >*; ' ;>[. (3.62) 
leads to the values a s - 11.7 Mev and a v = 13.90 Mev. ':"!-. difference 
appears to be well outside the reported uncertainties in :u- aiasses of 
Cu 86 , I 127 , and Pt 196 and may be taken as a rough index of th" Tue varia- 
tions of a v and a . We adopt as representative mean values l^r A > 40 



a. = (13 1) Mev 
a v = (14.1 0.2) Mev 



(3^64) 



382 



The Atomic Nucleus 



[CH. 11 



Table 3.2 also gives the observed and theoretical B/A for a few 
lighter and heavier nuclides. Note that over the entire range of A 
from S 33 to Bk 24B , the semiempirical mass formula, with the coefficients 
given in Table 3.2, predicts average binding energies B/A which are 
everywhere within 1 per cent of the observed values. This is a remark- 
able achievement for so simple a theory. Equation (3.15) can therefore 
serve as a smoothed base line against which local variations in M and 
B/A can be compared. In this way, discontinuities due to the shell 
structure of nuclei become prominently displayed, as we shall see shortly. 

Evaluation of Components of the Total Binding Energy. Table 3.2 
also lists the separate contributions of the four energy terms, volume, 
coulomb, surface, and asymmetry, for odd- A nuclides. The pairing 




30 60 



270 



90 120 150 ISO 

Mass number A 

Fig. 3.6 Summary of the scmiempirical liquid-drop-model treatment of the average 
binding-energy curve from Fig. 3.1 of Chap. 9. Note how the decrease in surface 
energy and the increase in coulomb energy conspire to produce the maximum observed 
in B/A at A ~ 00. For these curves, the constants used in the semiempirical mass 
formula are given in the last line of Table 3.3. 

energy, for even- A nuclides, is best determined from Fig. 3.4 and added 
in as an empirical local value. 

Figure 3.5 shows the separate contributions of each of the four energy 
terms to the average binding energy per nucleon B/A, for all A. The 
initial rise of B/A with A, which we first noted empirically in Fig. 3.1 
of Chap. 9, is seen to be attributable mainly to the decreasing importance 
of surface energy as A increases. At still larger A, the importance of the 
disruptive coulomb energy becomes dominant, causing a maximum in 
B/A at A ~ 60 and a subsequent decline in B/A at larger A. Through- 
out the entire range of A above A ~ 40 the semiempirical mass formula 
matches the observed binding energies within about 1 per cent. 

Summary of Evaluations of Energy Coefficients. Table 3.3 collects the 
evaluations of the energy coefficients of Eq. (3.15). For comparison and 



3] 



Models of Nuclei 



383 



reference purposes, the values used by the principal contributors to this 
field are also given. Among these, Fowler and Green avoided the con- 
ventional practice of determining o c from the coulomb-energy difference 
of light mirror nuclei, Eq. (3.16), and determined all four energy coef- 
ficients from a least-squares adjustment to the mass data for 0-stable 

TABLE 3.3. COMPARATIVE EVALUATIONS (IN MEV) OF THE ENERGY 
COEFFICIENTS OF THE SEMIEMPIRICAL MASS FORMULA 





Volume 
a v 


Surface 
a. 


Coulomb 
a e 


Asymmetry 
a a 


Pairing 
d 


1936 Bethe and Bach erf 
(B48) 


13 8G 


13 2 


0.58 


19.5 




1939 Feenberg (F16). . 
1939 Bohr and Wheeler 
(B96) 
1942 Fiuegge (M22) . . . 


14 66 


13 3 

14 
15 4 


62 

59 
602 


(Table of -y) 
20.5 


(Table) 


1945 Fermi (F38, F41) .. 
1950 Metropolis and Reit- 
wiesnerj (M43) . . . 
1947 Feenberg (F17) 


14 

14.0 
14.1 


13 

13 
13.1 


583 

0.583 
585 


19 3 

19.3 
18 1 


33.5/A* 

33.5/4* 
(Graph) 


1947 Fowler || 


15 3 


16 7 


69 


22 6 




1949 Friedlander and Ken- 
nedy (F69) 


14 1 


13 1 


585 


18 1 


132/A 


1949 L. Rosenfeld (R36) . 
J945 Canadian National 
Research Council 
(P36) 


14 66 
14 05 


15.4 
14 


602 
61 


20 54 
19.6 




1953 Coryell (C46) 
1954 Green (G46) 
1955 Eqs. (3.17), (3.22), 
(3.63), (3.64) ... 


15.75 
14.1 
2 


17 8 
13 

1 


71 
0.595 
0.02 


(Table of T ) 
23.7 
19.0 
0.9 


(Graph) 
(Fig. 3.4) 



t Constants were fitted for even-N even-Z. Pairing energy recognized but not 
evaluated. 

t These voluminous ENIAC computations of M(Z,A) for every conceivable 
value of Z and A use Fermi's 1945 constants, including M n 1 .008 93 amu and 
M n - M H - 0.000 81 amu. 

The particular values given here are for the incompressible fluid model, as used 
by all others in this tabulation. Feenberg (F17) also studied extensively the effects 
of finite nuclear compressibility. 

|| All four coefficients determined from a least-squares fit with packing fractions 
by Mattauch and Fiuegge (M22) for Ne 20 , S", Fe", Kr", Xe, Gd', Hg", U" 
(1947, unpublished). 

nuclides. This procedure leads to a larger value for a c and corresponds 
to a smaller nuclear unit radius in the neighborhood of fl ^ 1.2 X 10~ 1B 
cm for the heavy nuclides. All but these two determinations fall clearly 
within the uncertainties assigned in Eqs. (3.17), (3.22), (3.63), and (3.64), 
in which our general objective has been to set up an average base line with 
which local variations and shell discontinuities can be compared. The 



384 



The Atomic Nucleus 



[CH. 11 



remarkable fact is that so simple a theory, with only four adjustable 
constants for odd-^4. and one additional parameter for even-^A , can match 
the broad general behavior of the mass and binding energy of nearly a 
thousand stable and radioactive nudides. The important point then 
becomes the extent and cause of local variations from this smooth base 
line. 

f. Effects of Closed Shells. The semiempirical mass formula pro- 
vides a base line from which shell effects can be quantified. 

Energetics of Decay arid a Decay. In any group of isobars, the 
parabolic variation of mass M(Z) with nuclear charge Z (Figs. 3.1 and 
3.2) is an accurate representation if no isobar contains a magic number of 
neutrons or protons. However, the mass of any isobar which has Z or 
N = 20, 28, 50, 82, or 126 will lie about 1 to 2 Mev below the mass pre- 
dicted by the smooth parabolic relationships of Eqs. (3.39) and (3.53). 

TABLE 3.4. DECREASE AM IN NEUTRAL ATOMIC MASS FOR AN ISOBAR 

CONTAINING A CLOSED SHELL OF NEUTRONS OR PROTONS 

[As compared with the expected mass M(Z,A) of Eqs. (3.39) and (3.53) with local 
values of y and Z Q . Corycll (C46).] 



Z 


AM = AQ0, 
Mev 


N 


A-W = AQ0, 
Mev 


20 


1.1 


20 


1.1 


28 


1.1 


28 


1.0 


50 


1.5 


50 


1.9 


82 


(1.5) 


82 


1.8 







126 


0-8) 



This state of affairs is illustrated graphically for the case of A = 135 by 
Fig. 3.2 of Chap. 8. There the atomic mass of b3 I\l* lies clearly below 
the parabola which correlates the masses and ft decay energies of its 
isobars. 

From a systematic survey of ft decay energetics Coryell (C46) has 
obtained estimates of the magnitude of the shell discontinuities, as 
shown in Table 3.4. There are analogous discontinuities in the energetics 
of a decay, as has been shown clearly by Perlman, Ghiorso, and Seaborg 
(PI 5) and by Pryce (P36). The heaviest stable element bismuth 
undoubtedly owes its one stable isotope saBi^S to the closed shell of 126 
neutrons. All other bismuth isotopes are unstable. 

Neutron Separation Energy. As in Chap. 9, we define the neutron 
separation energy S n as the work required to remove the last neutron 
from a nucleus Avhich contains AT neutrons and Z protons. Then 

S n (Z,N) = M(Z, N - 1) +M n - M(Z,N) 
= B(Z,N) - B(Z, N - 1) 

SS) (3 - 65) 

The predicted value (S n )i is obtained from Eq. (3.65) by using B from 



3] Models of Nuclei 385 

Eq. (3.15). This is to be compared with experimental values (S n ) MP 
computed from the Q values of (y,n), (d,H 3 ), (d,p), an ^ (n,y) reactions 
as well as interlocking ft decay and a decay energetics. The differences 

AS n = OS n ) p - OSn)o* (3.66) 

are plotted in Fig. 3,6 as a function of N. Sharp discontinuities are 
evident. The 127th neutron is loosely bound and has a separation 
energy which is about 2.2 Mev less than that of the 126th neutron. A 
discontinuity of about the same size is seen between the 83d and 82d 
neutron, and between the 51st and 50th neutron. From the present 
data, no equally abrupt change in S n is seen at N = 28. 



+3 



3-1 



*:.i. x r i nr ! 



ti? 



50 82 126 

III U I I 1 LJ-l 



40 60 80 100 120 140 

Neutron number JV 

Fig. 8.6 Observed neutron separation energies (S n ) P compared with (5) c .i pre- 
dicted by the smooth variation of the semiempirical mass formula, using Fermi 1945 
coefficients (Table 3.3) in Eq. (3.65). Discontinuities of the order of 2 Mev are evi- 
dent for N - 50, 82, and 126. Evidence for a shell closure at N = 28 is inconclusive. 
[ Adapted from Harvey (H22).] 

g. Stability Limits against Spontaneous Fission. When the nucleus 
is visualized as a droplet of incompressible liquid, the main features of the 
ground-level energetics are quite well represented by Eq. (3.15). How- 
ever, the liquid-drop model fails to give an acceptable representation of 
the excited levels of nuclei. The excitation energy has to be visualized 
as due to surface vibrations, which correspond to periodic deformations 
of the droplet. The energy of these oscillations is proportional to the 
surface tension, hence to the surface-energy coefficient a., It can be 
shown (B68) that the lowest permissible mode of surface vibration cor- 
responds to an excitation energy which is many times greater than the 
observed excitation energy of low-lying excited levels in nuclei. The 
liquid-drop model cannot match the observed close spacing of nuclear 
excited levels, even if an additional parameter, corresponding to a finite 
compressibility of nuclear matter, is introduced in the model. 

However, the possibly superficial resemblance between the nucleue 
and an oscillating drop of incompressible liquid does lead quite directly 
to a plausible model which describes the stability limits of very heavy 
nuclei and the energetics of the nuclear-fission process. 



386 The Atomic Nucleus [CH. 11 

Energy Available for Nuclear Fission. The maximum binding energy 
per nucleon occurs in nuclei which have A ~ 60 (Fig. 3.5). In heavier 
nuclei, say, A > 100, the total binding energy of the A nucleons can be 
increased by dividing the original nucleus into two smaller nuclei. Thus, 
if U 238 is divided into two nuclei, having mass numbers A = ^f 1 , the 
binding energy per nucleon will increase from B/A ~ 7.6 to B/A ~ 8.5 
Mev/nucleon. This is an increase of M).9 Mev/nucleon, or some 210 
Mev for division of the single U 238 nucleus. 

In general, the division of any nucleus (Z,A) into two lighter nuclei 
is energetically advantageous if A > 85. These spontaneous fission reac- 
tions do not take place in the common elements because they are opposed 
by a potential barrier, which we shall discuss presently. 

The division of a nucleus (Z,A) into halves (Z/2, A/2) is called 
symmetric fission. From the semiempirical mass formula Eq. (3.15) the 
energy Q released in symmetric fission is 

(3.67) 



= -0.260a.A* + 0.370a c ~ (3.68) 

A 1 

Q = -3AA* + 0.22 -? Mev (3.69) 

A 1 

In Eq. (3.68) we have not specified the odd or even character of Z and A 
and have therefore omitted a possible small contribution from the pairing 
energy B. Applying Eq. (3.69), we have for the energy release on sym- 
metric fission of 9 2 U 288 

Q= -3.4(238)' + 0.22 

= -130 + 300 = 170 Mev (3.70) 

This is less than the 210 Mev estimated from the change in B/A, because 
the B/A values corresponded to stable nuclides. The fission frag- 
ments (Z/2, A/2) will have too large a neutron excess for stability. 
They will release additional energy in several forms, including an aver- 
age of about 2.5 "prompt" neutrons per fission, and a cascade of two or 
three ft disintegrations in each fission fragment. Symmetric fission does 
take place, but asymmetric fission is more probable. The energy released 
is only slightly different. 

In Eq. (3.70) we note that the positive energy release Q is the result 
of a large diminution in coulomb disruptive energy (300 Mev), which 
overrides a smaller change in the opposite direction ( 130 Mev) due to 
the increased ratio of surface to volume. The energetics of nuclear 
fission is seen to depend mainly on the interplay between coulomb energy 
and surface energy. 



3] 



Models of Nuclei 



387 



Potential Barrier Opposing Spontaneous Fission. The origin and 
behavior of the potential barrier can be visualized more clearly by revers- 
ing the fission process. We shall consider the mutual potential energy 
between two fission fragments which approach each other from a large 
distance and finally coalesce to form a 9 2U 288 nucleus. 

In Fig. 3.7, assume for simplicity that each fission fragment has the 
mass number A/2, nuclear charge Z/2, and radius R = Ro(A/2)*. 
When the separation r between the centers of two particles is large com- 
pared with their radii R, their mutual potential is simply the coulomb 
energy E c = e z (Z/2) 2 /r. When r decreases until the two particles are 
nearly touching, r > 2R, nuclear attractive forces begin to act. Then 
the mutual potential energy is less than the coulomb value, as indicated 
between positions b and c in Fig. 3.7. 



200 - 




O CD 00 G> G 



(d) 



(c) 



(6) 



(a) 



Fig. 3.7 Representation of the potential barrier opposing the spontaneous fission of 
U Z3H . Pictorially, the conditions at a, b, c, and d are illustrated on a reduced scale 
below. 

If the particles remained spherical, and no attractive forces entered, 
then the coulomb energy when the two spheres just touched, that is, 
r = 2R, would be 



Remembering that a e = 3e z /5R<> ^ 0.595 Mev, Eq. (3.71) becomes 

E c = (- a c J j = 0.262a c -^ ~ 210 Mev (3.72) 

for spheres (Z/2, A/2) in contact. This is shown as the extrapolated 
Bo curve at r = 2R in Fig. 3.7. Actually, the coulomb energy for 



388 The Atomic Nucleus [CH. 11 

undeformed spheres is just equal to Q ^ 170 Mev at position 6 in Fig. 3.7, 
where r ^ 2.5R for the case of U 288 . 

As the two particles come closer together, r < 2fl, the nuclear 
attractive forces become stronger and the two halves coalesce into the 
(Z,A) nucleus, whose energy of symmetric fission, d in Fig. 3.7, is below 
the barrier height. 

The nucleus (Z,A) will generally be essentially stable against spon- 
taneous fission if its dissociation energy Q is a few Mev below the barrier 
height. Experimentally, fission can be induced in U 288 by adding an 
excitation energy of only a few Mev. The threshold for the (T,/), or 
"photofiflflion" reaction, in which the required excitation energy is 
acquired by the capture of a photon, is only 5.1 Mev for U 28B . There- 
fore the barrier is only about 5.1 Mev above Q. U 288 does show a half- 
period of about 10 18 yr for spontaneous fission, or about 25 fissions per 
hour in 1 g of U 218 . The probability of a decay is about 10 7 times as 
great. 

Many experimental and theoretical aspects of spontaneous fission 
have been summarized by Segr& (S25). 

Stability Limits for Heavy Nuclei. A rough estimate of the mass and 
charge of a nucleus which is unstable against spontaneous fission can be 
had by finding (Z,A) such that Q for symmetric fission is as large as the 
coulomb encfrgy E c for undeformed spheres (Z/2, A/2) in contact. A 
nucleus will be clearly unstable if 

Q > E c (3.73) 

Upon substituting Q and E c from Eqs. (3.68) and (3.72), the condition 
for absolute instability becomes 

-0.2600.^* + 0.370a c ^ > 0.262a e ^ (3.74) 

which reduces to the inequality 

> 2.4^ = 53 (3.75) 

A a e 

This is an upper limiting value, because it ignores the possibility of finite 
penetration of the barrier. The important point here is the character 
and dimension of the critical parameter Z 2 /A and its dependence solely 
on the relative effective strengths of the forces associated with the coulomb 
energy ( Z 2 /A*) and with the surface energy ( A 1 ). 

A much better estimate of the critical value of Z*/A is based on the 
modes of oscillation of a drop of incompressible fluid, under the joint 
influence of short-range forces, as represented by surface tension, and the 
long-range coulomb forces [Bohr and Wheeler (B96) and others]. Oscil- 
lations of the type illustrated in Fig. 3.8 will be unstable and will result in 
division of the drop if small displacements from sphericity increase the 
total binding energy of the constituents of the drop. 

The volume of the sphere jrA 1 is the same as that of the ellipsoid 



3] Models of Nuclei 389 

?Trab z , if the drop is incompressible. The maj or and minor semiaxes of the 
ellipsoid can be represented by a = R(\ + e) and 6 = 7Z/(l + e)*, where 
measures the eccentricity and e = for the sphere. Then the surface 
energy can be shown to be 

J5. = 47rfl 2 (l + Ik 2 + - ) X surface tension 

= a.A*(l + |e 2 + - - ) (3.76) 

while the coulomb energy of the ellipsoid is 



Thus the ellipsoidal form has less disruptive coulomb energy, because 
the charges are farther apart on the average. Contrariwise, the ellip- 
soidal surface area is greater than that of the sphere; hence the deforma- 
tion involves some loss of binding energy, because the surface deficit is 
increased. 





Fig. 3.8 Oscillation of an incompressible liquid drop, for considerations of stability 
against spontaneous fission. 

Taking only the terms in e 2 , we see that the sphere (Z,A) ceases to 
be the stable configuration and undergoes fission when nudged, if 



<3 - re> 

which reduces to the inequality 

^ > 2^ = 44 (3.79) 

A a c 

Nuclei for which Z*/A < 44 will be stable against small deformations, 
but a larger deformation will give the long-range coulomb force a greater 
advantage over the short-range forces which are represented by surface 
tension. It may be expected that nuclei which are essentially stable 
against spontaneous fission will have values of Z 2 /A which are clearly 
less than the limiting value. 

In Fig. 3.9 the fissionability parameter Z*/A is plotted against Z for 
representative elements. The heavy nuclide 98 Cf 246 decays primarily by 
a-ray emission, with a half-period of 1.5 days. As would be expected 
from its Z*/A value of 39.0, it also undergoes spontaneous fission, against 
which its partial half-period is only about 2,000 yr. 

Fig. 3.10 shows the empirical correlation between the fissionability 
parameter Z*/A and the measured values of the partial half-period for 



390 



The Atomic Nucleus 



[CH. 11 



50 



40 



30 



20 



10 




20 



40 60 80 

Atomic number Z 



100 



120 



Fig. 3.9 The fissionability parameter Z*/A for some representative nuclides. The 
limiting value, Z 2 /A 2a s /a c 44, from the incompressible liquid-drop model is 
shown dotted. All the nurlidcs shown above Z 90 exhibit spontaneous fission but 
not as their major mode of decay. 



Partial half- per tod for spontaneous fission (years) 
5 . S S 5. S. o 

i (DM e 


Th 23 


'\ 

V*238 

Uranium 


^,234 








- 


- 


\] 


)Pu 239 






- 


- 


2' 
PI 


242 2JS, 
4^X^ ~ 

utonium 


^ 


oCf : 


49 


- 


oOdd-A 
Even-A 


Y^ 

250/ 


^242 

s^ 

25t y^ 


Fm 


- 


- 




'I 

Curiu 


m 

25 

1 


/^ 

J 


\ 


- 


- 1 month 
-Iday 






Califor 


lium 
Fermium 


Z N v 


\ - 



Z 2 /A 

Fig. 3.10 Syatematics of spontaneous fission in heavy even-Z nuclides. The partial 
half-periods which have been reported thus far extend from >10 20 yr for Th" 2 to 
3 hr for Fm 266 . Empirically the logarithm of the envelope of the partial half-periods 
for the even-4 isotopes of even-Z elements is a linear function of the fissionability 
parameter Z*/A. The partial half-periods arc several orders of magnitude greater 
for the odd- A isotopes than for the even-yl isotopes. [Adapted from Seaborg and 
MlLahnratorn (S24. 020^1 and K K fTude TTnPT..on^ft_P^r MQA9M 



3] Models of Nuclei 391 

spontaneous fission, in heavy even-Z nuclides from Z = 90 to 98. This 
systematic relationship was first pointed out by Seaborg (S24, G20) and 
is useful in predicting the spontaneous-fission rates for undiscovered 
nuclides. In general, the odd- A nuclides have longer partial half-periods 
than the even-Z even-# nuclides, for the same value of Z 2 /A. It 
appears from Figs. 3.9 and 3.10 that nuclides having Z = 100 may be 
reached, or exceeded somewhat, before spontaneous fission becomes the 
major mode of decay. 

Excitation Energy for Induced Fission. Returning to Fig. 3.7, we 
note that a relatively small excitation energy (^5 Mev in the heaviest 
nuclides) (K28) is sufficient to induce fission. This excitation energy 
can be supplied in many ways. For example, immediately after the 
capture of a thermal neutron the compound nucleus has an excitation 
energy equal to S n , the neutron separation energy from the ground 
level of the compound nucleus. If the probability of fission from this 
excited level competes favorably with deexcitation by 7-ray emission, 
then fission may be a principal consequence of the neutron capture. 

We have seen that S n is appreciably greater for even-JV' than for 
odd-AT nuclides. This is a consequence of the pairing energy 8. The 
excitation energy in the compound nucleus (Z y A) immediately following 
the capture of a thermal neutron is 

S n = M(Z, N - 1) + M n - M(Z,N) (3.80) 

where the neutron number of the target nucleus is N I. Note that 
the neutron separation energy S n from the ground level of the compound 
nucleus (Z,JV) is the same as the Q value for the (n,y) reaction on the 
target nucleus (Z, N I). 

Consider the excitation energy produced when a thermal neutron is 
captured by an odd-N target (say ozU 236 ) compared with an even-AT 
target (say 92U 238 ) of approximately the same mass number. It is easy 
to show, by inserting Eq. (3.15) into Eq. (3.80), that the excitation 
energy in 

92 U 23B + n -> U 236 

is substantially 26 greater than the excitation energy in the compound 
nucleus formed by 



This relationship is quite a general one. If the mass numbers of two 
nuclides are nearly alike, the excitation energy is 26 greater in a com- 
pound nucleus which has even-N than in one having odd-N. In the 
domain of A ^ 236, the pairing energy d is of the order of 0. 5 Mev, so 
that the difference in excitation energy 25 is about 1 Mev. The fact 
that U 28B undergoes fission with thermal neutrons but U 23S requires 
bombardment by fast neutrons of ~1 Mev in order to undergo fission is 
attributed to this expected difference of 25 in the excitation energies. 

Asymmetric Mass Yield in Low-energy Fission. Symmetric fission, 
as contemplated in Eqs. (3.67) et seq., is actually an uncommon mode of 



392 



The Atomic Nucleus 



[CR.ll 



cleavage when nuclear fission takes place from a level which has low-to- 
moderate excitation energy (C47, W42, M57). 

The fission of U" 6 by thermal neutrons (~Tir ev) has been most 
exhaustively studied, because of its present practical importance in 
nuclear reactors. When U 2 " captures a thermal neutron, the resulting 
excited nucleus U" 6 transforms to its ground level by 7-ray emission in 
only ~15 per cent of the cases. Predominantly, the excited U 2 ' 6 nucleus 
undergoes fission. The immediate products of this fission process gener- 
ally are two middleweight nuclei (the so-called " fission fragments") and, 
on the average, 2.5 0.1 prompt neutrons which are emitted instantly 
by the fission fragments. 

The prompt neutrons have a continuous energy spectrum, with a 
maximum intensity near 1 Mev, and an approximately exponential 
decrease in intensity at higher energies, such that the relative intensity at 
14 Mev is only about 10~ 4 of the intensity at 1 Mev (B97, H52, Wll). 
Each of the fission fragments is "neutron-rich," because N/Z is appreciably 
greater for /3-stable heavy nuclei than for 0-stable middleweight nuclei. 

TABLE 3.5. ASYMMETRY CHARACTERISTICS OF THE MASS-YIELD CURVES FOR 

LOW-ENERGY FISSION OF SEVERAL TARGET NUCLIDES 

[From Turkevich and Niday (T31)] 





Most 


Most 


Ratio of 


Mass 


Ratio of 


Target 
nuclide 


probable 
light 
mass 


probable 
heavy 
mass 


most 
probable 
masses 


width 
at half 
height 


peak-to- 
trough 
yields 


TV" 


91 


140 


1.54 


14 


110 


U 1 " 


93 


137 


1.48 


14 


400 


U" 8 


97 


138 


1.42 


15 


600 


U" 8 


98 


140 


1.43 


17 


100 


Pu 


99 


138 


1.39 


16 


140 



On the average, a fission product undergoes about three successive ft 
transformations before becoming a stable middleweight nuclide (W12). 

The mass yield in nuclear fission ia the sum of all the independent 
fission yields of nuclides having the same mass number. Figure 3.11 
shows this percentage yield of isobars, between mass number A ~ 70 and 
A ~ 160, which results from the fission of U 28B by thermal neutrons 
(P24, Rll). The outstanding characteristic of Fig. 3.11 is its double- 
peak mass distribution. This "asymmetric fission" is characteristic of 
all cases of low-energy fission. It is also observed in the mass-yield 
curves for the spontaneous fission of U 288 and Th 282 , which have been 
evaluated by mass-spectroscopic measurements on the Xe and Kr iso- 
topes which have accumulated in ancient Th and U minerals (F53, 
W28). 

Table 3.5 shows the magnitude of the asymmetry characteristics for 
the low-energy fission of U 28 ', U m , and Pu*" by thermal neutrons, and 
for the low-energy fission of Th" 2 and U 2 " by "pile neutrons," i.e., by 



53] 



Models of Nuclei 



393 



slightly moderated prompt neutrons from the fission of U 215 by thermal 
neutrons. 

The dramatic asymmetry of the mass-yield curves for low-energy 
fission gradually disappears as the excitation energy of the fissioning 
nucleus is increased. This trend is illustrated by the dotted curve in 
Fig. 3.11, which shows the mass-yield distribution for the fission of 
excited TJ 218 compound nuclei which are produced by bombardment of 
Th 282 by 37-Mev a particles (N9). The threshold for this reaction, Th 282 




10 60 80 100 120 140 160 

Mass number, A, of fission product 

Fig. 3.11 Mass yield in the low-energy fission of U" 5 by thermal neutrons (solid 
curve). The asymmetry is reduced when fission takes place from more highly excited 
levels of the compound nucleus. The dotted curve shows the mass yield for Th 1 " + 
37-Mev a particles, which also involves the compound nucleus U" fl . [from the 
Plutonium Project Reports (P24) and Newton (N9).J 

(a,/), is about 23 Mev; hence the compound U 2 " nucleus is of the order 
of 14 Mev more highly excited than the U" 8 formed from U 2 ' 6 plus 
thermal neutrons. This is sufficient to reduce the peak-to-trough ratio 
from 600 to about 2. For very-high-energy fission, e.g., that of Bi 209 
bombarded by 400-Mev a particles, symmetric fission predominates 
(862). 

A wide variety of tentative theories have been advanced in efforts to 
match the asymmetric mass yields which are seen in low-energy fission. 
These have been summarized and their inadequacies discussed by Hill 
and Wheeler (H53), in connection with their discussion of the so-called 



394 



The Atomic Nucleus 



[CH. 11 



collective model of nuclei, which blends somewhat the individual-particle 
and the liquid-drop models of nuclei. At present, a simple and wholly 
acceptable theory of asymmetric fission remains a task for the future. 



Problems 

1. Consider the semiempirical mass formula of Eq. (3.15), but with the asym- 
metry term written as a a (A 2Z) n /A, with the arbitrary exponent n (instead 
of 2). 

(a) For any given odd value of A, derive an expression for the neutron excess 
(A 2Zo) for minimum neutral atomic mass. 

(6) Show that n must equal 2 for heavy nuclei, by assuming Z substantially 
proportional to A, and recalling that (A 2Z ) is empirically proportional to A*. 

2. (a) Show that the experimentally observed parabolic relationship 



(M z - 



= y(Z 



of Eq. (3.39), between mass Mz and atomic number Z, for odd- A isobars, will 
emerge from the semiempirical mass formula Eq. (3.15) only if the exponent 
n, in the asymmetry term (aJA)(A 2Z) n , has the value n = 2 exactly. 

(b) Would a parabolic relationship emerge if there were no asymmetry term, 
i.e., if a a = 0? 

3. A portion of the very complicated decay scheme of s-Ge 77 is shown below 
[from A. B. Smith, Phys. Rev., 86: 98 (1952)], together with the simple decay 
scheme of 3 iAs 77 . 



12 hr 




38 hr 




A 



34 Se 77 Stable 



(a) From this evidence alone, evaluate the local values at mass number 
= 77 for the semiempirical mass parameters Z and 7. 

(b) Plot a reasonably accurate graph of the neutral atomic mass differences 
(M z - Mz ) in Mev, against Z, for A = 77 and Z = 32 to 36. 

(c) Predict the decay energy of aeKr 77 and the maximum energy of the positron 
spectrum which it can emit in those transitions which go directly to the ground 
level of asBr 77 . Compare with recent measurements of the Kr 77 decay spectrum. 

(d) Predict the Q value for the nuclear reaction Se 77 (d,2n)Br 77 . 

4. The principal modes of decay of As 74 are as shown. Evaluate the pairing 
energy d for the last neutron and proton in Ge 74 and Se 74 . 



2m c : 



(49*) 
0.69 Mev 



0.596 Mev;, 




0.635 Mev 



3] Models of Nuclei 

6. (a) Look up the disintegration scheme for the dual ft decay of Cu 64 . 



395 



From 

the ft + and &~ disintegration energies, determine the pairing energy 5 at A = 64. 
Locate the resulting value of d/7v on Fig. 3.4. 

(6) Do likewise for the dual P decay of 4 r,Rh 102 . 

6. Examine the chain of fission products having mass number 140: Xe 140 * 
Cs > Ba - > La > Ce 140 . From the properties of Ba 140 , La 140 , and Pr 140 , as given 
below (H61), evaluate the parameters Z 0} 7, and 6 of the nuclear energy surface 
at A = 140. Predict the decay energy of Xe 140 and Cs 140 . Comment on any 
"magic-number effects." 



=12.8 d 



140 



59 



Pr 140 : r=3.4m 




2m c 2 



,Ce 140 (stable) 



58 

7. The reaction V(p,n)Cr fil has a Q value of -1.534 + 0.003 Mev, as deter- 
mined from the threshold energy. The 5.8-min 0- emitter Ti 51 is reported to 
Rive E m ** = 2.14 Mev m &~ transitions to an excited level at 0.32 Mev in V B1 . 
From these data alone: 

(a) Determine the local value of 7, and compare with 7 BV at A = 51 . 
(6) Determine the most stable nuclear charge Zo at A = 51 and compare 
with the line of /9 stability in Fig. 3.1 of Chap. 8. 

(c) Determine the decay energy for electron-capture transitions of Cr B1 and 
whether positron decay is permitted. 

(d) Predict the positron decay energy Qp+ of Mn &l and compare with reported 
values. 

8. 7-ray photons of sufficiently large quantum energy can be reliably detected 
without interference from low-energy radiations by the use of so-called "threshold 
detectors/ 7 For example, copper becomes radioactive when exposed to 7 rays 
of sufficiently high quantum energy, which produce 2oCu 64 by the photonuclear 
reaction 



(a) Make appropriate use of the concepts of separation energy and of the 
quantitative formulation of the liquid-drop model to predict (say, 0.5 Mev) 
the minimum photon energy which can produce the 12-hr Cu 64 activity in metallic 
copper. 

(6) The measured value of the threshold photon energy for this reaction is 
10.2 + 0.2 Mev [McElhinney et al., Phys. Rev., 75: 542 (1949)]. Point out 
clearly what components of the nuclear binding energies cause this to exceed 
the average binding energy per nucleon, which is about 8.5 Mev in Cu 6B . 



396 JTAe Atomic TVucfeiw [CH. 11 

(c) Qualitatively, would you expect the separation energy of a neutron from 
the stable nuclide aoZn 87 -to be appreciably greater or less than that of 2 gCu 65 ? 
State why clearly but briefly. 

9. (a) Show that the Q value of a (d,a) reaction on a target nucleus M (Z t A) is 



Q(d,a) = B a B d - 2a v + a.[A* (A 2)'] 

~Z 2 (Z - 1) 2 1 \2(A - 2Z) 2 1 f 25: -* even ' Z odd 

\ ~~ TA oU ~ fla A o^~ ~^ I ~26: A even, Z even 

(A - 2) J L ^(^ - 2) J [ 0: A oddj z ally tL;-nr 

where B a and 5 d are the binding energies of the a particle and the deuteron. 

(6) Evaluate the local values of a v , a,, and 8 from the measured Q values for 
the following reactions, by assuming that a c and a a are known. 

Mg 2B (d,a)Na 23 : Q = 7.019 0.013 Mev 
Al 27 (d,a)Mg: 6.694 0.010 

Si 2B (d,a)Al 27 : 5.994 0.011 

Si 80 (d,a)Al 28 : 3.120 0.010 

P sl (d,a)Si": 8.158 0.011 

10. (a) Derive from the semiempirical mass formula a general expression for 
the Q value of a (d,p) reaction on a target nucleus M(Z,A). Evaluate the coeffi- 
cients a v and a. by utilizing tabulated data on the Q value of (d,p) reactions 
(e.g., references B4, H22, and V3a). 

(6) Will the Q values of (d,p) reactions give independent inforrriktion on the 
pairing energy 5? 

(c) Will the Q values of (d,n), (y,n), and (n,y) reactions give information on 5? 

11- Show that the average binding energy per nucleon can be represented by 

B a. Z 2 / 2Z\ 2 _ 



Assuming that only a c and a a are known, derive for odd- A nuclides an expression 
for a. as a function only of a c , a a , and A (the mass number for which B/A has 
its empirical maximum), that is, 

a. = a v (a ei a ai AiO 

and evaluate a. in Mev. 

12. (a) Reevaluate the coefficients a c , a a , a., and a, under the assumption 
that the nuclear unit radius is R = 1.1 X 10~ 18 cm. Note from Eq. (3.21) 
that a a /a e will remain unchanged. Evaluate a a and a,, using the masses of 
Cu", I 127 , and Pt 19B as given in Table 3.2. 

(6) Estimate the masses of 8 2Pb 208 and giBi 101 , using the coefficients derived 
above, and making allowances for the closed shells involved in both nuclides. 
Compare with the values reported by Richards, Hays, and Goudsmit (R17), 
from time-of-flight measurements, based on C 12 = 12.003 895: 



208.0416 0.0015 amu 
Bi 209 = 209.0466 0.0015 amu 

IS. From the considerations involved in Eq. (3.67) et seq., evaluate the mini- 
mum mass number A for which symmetric fission would result in a net release 
of energy. Ans.: A ~ 85. 

14. Show that the neutron separation energy S n is substantially 26 greater 



4] Models of Nucki 397 

in an even-Z even-JV nuclide, such as U 236 , than it is in a nearby even-Z odd-JV 
nuclide, such as U 289 . What consequences does this generalization have in the 
utilization of normal uranium as a nuclear fuel? 

15. Estimate the partial half-period (!Tj) flB81011 for spontaneous fission of 9 4Pu 240 . 
What fraction of the Pu 240 nuclei will be expected to decay by spontaneous fission 
if Pu 240 is ^-stable and has a half-period for a decay of (T^) a = 6,580 yr? Ana.: 
~5 X 10" yr; ~1 X lO" 8 . 

16. If you were setting out to produce a /J-stable isotope of the element Z = 102, 
approximately what mass number A would you probably have to produce? 
For this nuclide, what would be the approximate partial half -period against 
spontaneous fission? Ana.: A ~ 266; T\ ~ 10 s yr. 

4. Statistical Model of Excited Levels 

The lowest excited level in most nuclei usually lies well above the 
ground level, a separation of the order of several hundred kilovolts 
being common. We have seen, for example, in the case of Cd 111 (Fig. 8.5 
of Chap. 6), that the angular momentum and parity of the lowest-lying 
excited levels in many nuclei are described correctly by the shell model, 
with jj coupling. In odd- A nuclides, the shell model represents these 
excited levels as due to the independent motion of the single odd-nucleon, 
which then occupies some higher quantum state. In its present stage of 
development, the shell model does not make general predictions about 
the magnitude of the excitation energy for these discrete low-lying levels. 

At moderate excitation energies, in the domain, say, of 2 to 6 Mev, 
the excited levels are found to be more closely spaced (A10). In this 
region, a generally acceptable theory is not presently known. At still 
higher excitation energies, say, above about 6 Mev, the energy difference 
between successive levels is often as small as 10 to 100 ev. This is the 
energy domain of many nuclear reactions, and especially of the radiative 
rapture of slow neutrons. In these (n t y) reactions the compound nucleus 
is formed with an excitation energy equal to the neutron separation energy 
S nj say, 6 to 8 Mev in all but the lightest nuclides. 

At these higher excitation energies the independent-particle model 
becomes entirely inadequate. The very formation of a compound 
nucleus, whose subsequent behavior is found to be independent of its 
mode of formation, shows that the incident particle interacts strongly 
with all the nucleons in the target nucleus. These more highly excited 
levels are best viewed as quantized levels of the compound nucleus as a 
whole. The individual nucleons in the excited compound nucleus have 
strong interactions. They share the total excitation energy in some way 
which is too complicated for detailed analysis. One turns then to the 
metho^ of statistical mechanics and thermodynamics for aid in develop- 
ing a qualitative model of the energetics of highly excited nuclear levels. 

a. Bound and Virtual Levels. Bound levels are those for which the 
excitation energy E is insufficient to permit dissociation of the excited 
nucleus by particle emission (n, p, a, . . .). Deexcitation of a bound 
level can occur only by y-ray emission or by its competing internal 
conversion. 



398 



The Alomic Nucleus 



[CH. 11 



Figure 4.1 shows the presently known excited levels in N 14 . Ener- 
getically, any level above 7.542 Mev (which equals the mass difference 
C 13 + H 1 N 14 ) is capable of dissociating by the emission of a proton, 
even though selection rules may prevent it for certain levels. Any level 
whose excitation energy exceeds the lowest dissociation energy, in this 



-12.3 



10.545 Mev 




il'L23 1126 11.38 
-11.04- 
10.43 



7.542 Mev 



13 7 Mev 



13.23 n 15 



9.49 



9.18 



8.70 



862 
8.06 




-55- 
5.0 



4.80 
39 



2 31 Mev 



OMev 



11.613 Mev 
B 10 +He 4 



10.264 Mev 
C 12 +H 2 



N 14 

Fig. 4.1 The known excited levels in N 14 and tho mass difference (in Mev) between 
various possible dissociation products arid the ground level of N 14 . The bound levels 
are those below 7.542 Mev. Deexcitation ol these can take place only by 7-ray 
emission or its ever-present competitor, internal conversion. Above 7.542 Mev all 
levels are called virtual because dissociation by particle emission is energetically pos- 
sible. The gaps near 10 Mev represent a region which is not yet explored experi- 
mentally. [Adapted from Ajzenberg and Lauritsen (A10).] 

case, 7.542 Mev, is called a virtual level. The virtual levels have many 
competing modes of deexcitation. For example, a virtual level at 
E ~ 15 Mev in N 14 could transform by 7-ray emission or internal con- 
version to any lower-lying level, or by p, n, d, or a emission. In the 
cases of particle emission, the transition could be to the ground level or 
to an excited level in the product nuclei C 18 . N", C 12 , and B 10 . Clearly, 



1] Models of Nuclei 399 

the higher the excitation energy, the larger is the number of possible 
competing modes of decay, or "exit channels/ 7 

b. Level Width. The width of a nuclear level is a completely 
quantum-mechanical concept. When the probability of deexcitation is 
included, a particular level can be represented by a- wave function of the 
form 

\ = -(n iT/2)r/ 



where the energy (IT iY/2) is complex. The probability ty^* of 
finding the level intact after a time t will be proportional to r~ rr/ *. Thus 
r/h is the probability of decay per unit time, or the reciprocal of the 
mean life T of the level. The energy T is called the width of the level 
and is related to the mean life r by 

Tr = ft (4.2) 

This means that, because of its finite lifetime, the level cannot be said 
to have a perfectly sharply defined energy, and the uncertainties in 
energy and time are related as in the Heisenberg uncertainty principle, 
AA7 AZ > h. Inserting the numerical value of /, En- (4.2) becomes 

r _ OOO.X 10 - ey 

r(sec) 

Thus a bound level whose mean life against 7 decay is 10~ 12 sec will have a 
width of only ~10~ 3 ev. Even ground levels have a nonvanishing widlh 
because of the finite, though very small, probability of their transforma- 
tion by fission, decay, or other nuclear reaction. 

It has been possible to measure both r and r for the same nuclear level, 
in spite of the smallness of h. The width T has been measured for 
a number of excited levels. In N 14 , for example, the levels whose excita- 
tion energy is in the domain of 11 to 13 Mev have measured widths of 
~200 to 20 kev (Chap. 13, Table 4.1). The mean life of such levels is 
of the order of 10~ 20 sec. 

Partial Width. Any level which has a number of possible competing 
modes of decay will have a corresponding number of "partial widths" 
T,, each corresponding to the probability of decay by a particular mode. 
Then the total width of the level, which corresponds to the total probability 
of decay, is the sum of all the partial widths, or 

r = r, + r a + r a + - - - (4.4) 

c. Relationship between Average Level Spacing and Level Width. 

As th^ excitation energy increases, it is found experimentally that the 
level width T increases and that the average spacing between levels 
decreases^ The levels are riot uniformly spaced in energy, but for pic- 
torial purposes it is convenient to think of an average energy spacing, or 
average "level distance," D in any particular region of excitation energy. 
Recurrence Time for an Excited Configuration. Weisskopf (W22) has 
suggested a, very simple semiclassical model, from which one may visual- 
ize the main features of a relationship between the average spacing D 



400 The Atomic Nucleus [CH. 11 

of levels in a highly excited nucleus, the level width F, and the trans- 
parency T z of the nuclear barrier at this excitation energy. Suppose 
that, within a very large group of equally spaced levels, all having the same 
value of total angular momentum and parity, the total energy of the 
nth level is W 9 = W + nD. Then a wave packet which represents a 
relatively well-defined grouping and motion of the nucleons can be 
constructed from a linear combination of the wave functions of a number 
of adjacent stationary stales whose spatial dependence is <p n . The total 
wave function will be of the form 




(4.5) 

where e~ lWnt/h is the tin* .pendent part of the wave lu^ction for each 
of the neighboring states <p n . The time dependence of the total wave 
function is seen from Eq. (-4.5) to be such that |v| 2 has the same value 
at time t and at a later time t + 2-n-h/D. Thus the wave packet does 
riot represent a stationary configuration but rather one which repeats 
itself, with a recurrence time A given by 



(4.6) 

According to Bohr's correspondence principle, the motion for highly 
excited states, with very large quantum numbers, approaches a corre- 
sponding classical motion, which we can describe in this case as oscillation 
between the walls of a potential discontinuity at the edge of the nucleus. 
If we consider, for example, a spacing of D = 10 cv, then the recurrence 
time from Eq. (4.G) is ~10~ 16 sec. In such a time interval, a nucleon 
traveling at the order of one- tenth the velocity of light (a few Mev of 
kinetic energy) would travel some 10~ 6 cm, or about a million nuclear 
diameters. This serves to emphasize, in a semiclassical way, the enor- 
mous complexity of the nuclear motion in a highly excited level. By way 
of contrast, the lowest-lying levels, with spacings of a few hundred kilo- 
volts, would have much shorter recurrence times and therefore shorter 
recurrence path lengths, corresponding roughly to only some tens of 
nuclear diameters. This would be compatible with the orbital motion 
of independent particles, as visualized in the shell model for low-lying 
nuclear levels. 

Transparency of the Nuclear Surface and Coulomb Barrier. Now we 
must recognize that these configurations are not exactly stationary 
states but that they have a finite probability of decay, which is T/h per 
unit time. For definiteness, we may think of a level at some 8-Mev 
excitation, which has just been produced by the capture of an incident 
nucleon. After a recurrence time A = 2irh/D this initial configuration 
will be repeated, and the nucleon would be back at the nuclear surface, 



4] Models of Nuclei 401 

with its original velocity reestablished, and ready to leave the nucleus 
if it is capable of penetrating the nuclear surface. However, there is at 
the nuclear surface a rapid change of potential, corresponding to the 
absence outside the nucleus of the strong nuclear forces. At this poten- 
tial step the de Broglie wave, which represents the nucleon, suffers 
reflection, and the nucleon returns to traverse once again its long path 
through the nucleus. 

Each time the nucleon reappears at the edge of the nucleus, it has a 
finite probability of traversing the potential step and escaping from the 
excited nucleus. This event would correspond to the deexcitation of the 
virtual level by dissociation. The concept of a well-defined quasi-station- 
ary virtual level is only possible if the reflection coefficient at the nuclear 
surface is large. Then the complicated motions visualized between 
recurrence times will be repeated many times, and a well-defined level 
could be found experimentally. The transmission coefficient is given 
by the familiar formula for a step barrier (Appendix C, Sec. 1, Prob. 6), 
which reduces to simply 4k/K when the wave number k outside the 
nucleus is small compared with the wave number K inside the nucleus. 
If the emerging particle is charged, there will be a coulomb barrier to be 
penetrated also. Symbolically, we can represent thd over-all trans- 
mission coefficient J a i for a particle a with emergent angular momentum I 
as 

T ol ~^/> al (4.7) 

A 

where P a j is the appropriate Gamow-type penetration factor for the com- 
bined coulomb and centrifugal barrier. Then the excited level will have 
a probability of decay per unit time T/h which is given semiclassically 
by the transmission T flZ times the number of recurrences per unit time, 
\l At = D/2irh, from which the partial width T a i is 

* '- 

-'-2 

Reduced Width. The factors which characterize the interior of the 
nucleus may be conveniently grouped as a so-called reduced width 7^ by 
rewriting Eq. (4.8) as 



where the nuclear radius R is introduced so that the reduced width has 
the dimensions of energy and the first factor is dimensionless.t Experi- 

f The particular definition of reduced width given here is due to Blatt and Weiss- 
kopf (p. 390 of B68). It differs only by the factor i/R from the reduced width 
defined earlier by Wigner, which has dimensions of Mev-centimeters and is usually 
denoted in the periodical literature by the symbol 7*, where 7* r |/2&Po* s #7oi- 



402 



The Atomic Nucleus 



[CH. 11 



mental data compiled by Wigner (W50) show that the ratio y/D is 
roughly constant while 7 and D individually vary over a range of '10 6 . 

For the important special case of the emission or capture of s-wave 
neutrons (I = 0, P a i = 1), the particle width T a i is seen to vary with A; 
and hence linearly with the velocity of the emitted or incident neutron. 
A well-defined resonance level for neutrons can be expected only when 
the width T a i is small compared with the level separation D t . We see 
from Eq. (4.8) that this implies k <3C A', that i.s, an external neutron 
momentum p = hk which is small compared with the momentum of a 
mu'leon inside the nucleus. Because the total width of the level is the 
sum of all the partial widths, the total width T can exceed the average 
separation Di even when k < K. 

d. Relationship between Excitation Energy and Average Level Spac- 
ing. As the excitation energy E increases, the total width I 1 of the 



ry 

4V-> 



t 



1 



Fig. 4.2 Energetics of neutron dissociation of a virtual level. The kinetic 1 energy K H 
nnd the wave number k of the emitted neutron arc given by Eq. (4.10), 

excited level increases. Two distinct factors contribute to this increas- 
ing breadth. First, as E increases, more competing modes of deexoitation 
are available to the level. Second, as E increases, the barrier trans- 
parency Tai increases for each mode involving particle emission. 

It is well demonstrated experimentally that the average spacing Di 
of the levels decreases as we go up in excitation energy. This would 
mean, in Eq. (4.8), that the partial level width T aJ would become narrower 
with increasing excitation energy, were it not for the fact that the increase 
in transparency !? overrides the decrease in level spacing DI as the 
excitation energy E increases. Physically, this can occur because a small 
change in E, and hence in i)/, can correspond to a very large change in k. 

The physical situation is shown in Fig. 4.2. For defmiteness, con- 
sider the nucleus Z A in a virtual level whose excitation energ}' is E. 
Suppose that a neutron is emitted. Then the maximum kinetic energy 
of the emitted neutron will be E S nj where S n is the neutron separation 



4] Models of Nuclei 403 

energy from Z A. But if the neutron emission leaves the residual nucleus 
Z (A 1) in an excited level E T , then the kinetic energy E n = p z /2M n of 
the emitted neutron will be reduced to 



As long as E n < E, the reduced width y a i and the level spacing DI will 
be only slightly influenced by small changes in E. On the other hand, 
E n and k vary rapidly with E, especially just above the dissociation 
energy, where E = S n . 

Evaporation Model and Nuclear Temperature. Where a large number 
of closely spaced levels are involved, the methods of statistical mechanics 
may be applied as a guide to the possible behavior of highly excited 
nuclei. The emission of a neutron from an excited nucleus then becomes 
analogous to the evaporation of a molecule from a heated liquid droplet. 
The usual thermodynamic concepts of temperature, entropy, and heat 
capacity are applied to the nucleus. A clear summary of several forms 
of the theory has been given by Morrison (M57). 

A general formulation of the statistical model has been developed by 
Weisskopf (W20, W25, W21, B68). For highly excited nuclear levels, 
the energy distribution of emitted neutrons is expected to be of the form 

n d(E n ) ~ const X E n c- E * /T d(E n ) (4.1 1) 

where n d(E n ) neutrons should be emitted within the energy range E n to 
E n + dE n . Here the quantity T is the nuclear temperature of the residual 
nucleus, evaluated at its maximum residual excitation energy 

(AV) m = E - S n 

(see Fig. 4.2). From Eq. (4.11) it is seen that T has the dimensions of 
an energy, as does its counterpart kT in the classical Maxwellian distribu- 
tion of the energies of molecules evaporated from a surface at temperature 
T. 

Figure 4.3 shows the general character of the expected energy distri- 
bution of neutrons "evaporated" from a highly excited nucleus. The 
statistical theory appears to be a good general guide for, say, A > 50. 
The smoothed, or continuous, spectrum represented by Eq. (4.11) should 
apply when the levels in the residual nucleus are so closely spaced that 
individual peaks in the neutron spectrum are not resolved. The prepon- 
derance of low-energy neutrons (small E n in Fig. 4.2) is to be attributed 
to the joint influence of the larger number of excited levels available in 
the residual nucleus when E n is small and E r is large, and to the relatively 
much smaller dependence of the transparency T n i on E n . The maximum 
of the energy distribution is therefore shifted toward values of E n which 
are small compared with its maximum possible value CEn)m = E S n - 

If the evaporated nucleons are protons instead of neutrons, then the 
coulomb barrier suppresses the low-energy protons, and the maximum 
of the expected distribution shifts to higher energies, as shown sche- 
matically in Fig. 4.3. 



404 



The Atomic Nucleus 



[CK. 11 




15Mev 
E n or E p Kinetic energy of emitted particle 

Fig. 4.3 Schematic energy spectrum for nucleons evaporated from a highly excited 
nucleus, as predicted by the statistical model. 

The application of these considerations to real nuclei can be illus- 
trated by Fig. 4.4, which shows the observed energy spectrum of neutrons 
emitted in the reaction Au 197 (p,n)Hg 197 when a thin foil of gold is trav- 

10 3 




12 Mev 



Fig. 4.4 Energy distribution of neutrons emitted in the reaction Au 197 (p J n)Hg 197 with 
16-Mev protons. E n is the kinetic energy of the emitted neutrons; E r = 14.6 E n 
Mev is the excitation energy of the residual Hg iu7 nucleus. The slope of the experi- 
mental points corresponds to a nuclear "temperature" of T = 0.8 0.1 Mev in the 
residual Hg 197 nucleus. [Adapted from Gugetot (G47).] 

crsed by 16-Mev protons (G47). The Q value of this reaction is 1.4 
Mev. Therefore the maximum neutron energy [compare Fig. 4.2 and 
Eq. (4.10)] is (tfO = E - S n = 16 - 1.4 Mev = 14.6 Mev. It will 
be seen from Fig. 4.4 that the smoothed distribution of unresolved 



4] Models of Nuclei 405 

neutron peaks is fairly well represented by an exponential function like 
that of Eq. (4.11). From the average slope of Fig. 4.4, the "nuclear 
temperature" of the residual Hg 197 is T = 0.8 0.1 Mev. 

Calculated Density of Levels in the Statistical Model. It also follows 
from Weisskopf's formulation of the statistical theory that, at large 
excitation energy , the density of excited levels 1/D, where D is the 
average distance between levels, should be roughly of the form 

A = Ce 2( **>* ^ Ce** lT (4-12) 

where C and a are parameters which are functions of the mass number A 
and which are to be adjusted empirically. Generally C is evaluated 
from the observed level density at low excitation (E ~ 1 Mev), and a is 
adjusted to represent the spacing of levels found from the resonance 
capture of slow neutrons (E ~ 6 to 8 Mev). 

Observed Density of (n,y) Resonance Levels. Several experimental 
methods are accessible for determining the average spacing of highly 
excited levels in real nuclei (D35). One method depends upon measuring 
the average (n,y) capture cross section for neutrons having a broad 
energy distribution centering about ~1 Mev. For each discrete excited 
level in the compound nucleus the capture cross section is described by 
the one-level resonance formula of Breit and Wigner (B118, F45, B141), 
which may be written [Chap. 14, Eqs. (1.10) and (1.13)] 

__*J 2/ c + l 1 TJ\ , J10X 



L(2/n + 1)(27, + 1)J (E n - So) 2 + (r/2) 2 

where X = rationalized de Broglie wavelength (X/2?r) of incident neu- 
trons 
/ = i = intrinsic angular momentum (spin) of bombarding 

neutron 

It = intrinsic angular momentum (spin) of target nucleus 
I c = angular momentum of compound nucleus 
E n kinetic energy of incident neutron 
EQ = kinetic energy of incident neutron which just forms excited 

level E of compound nucleus (E Q = E 8 n ) 
I\ = width of excited level for 7-ray emission 
F n = width of excited level for neutron emission 
F = total width of excited level 

When Eq. (4.13) is summed over a number of resonance levels, whose 
average spacing is D , the result for the case of s-wave neutrons (I = 0) 
is simply (F45, H69) 

Hughes, Garth, and Levin (H68, HG9) have measured the capture 
cross section a(n t y) of a large number of nuclides for the prompt neu- 
trons emitted during the fission of U 235 . These neutrons have a continu- 



406 



The Atomic Nucleus 



[CH. 11 



ous distribution of energy, with an effective value of ~1 Mev; hence they 
cover a number of adjacent resonances. The main effects are due to the 
capture of I = or s-wave neutrons. The observed (71,7) cross sections 
have been presented in Fig. 2.2 of Chap. 8. When these are inserted in 
Eq. (4.14), Do can be estimated, provided that the 7-radiation width 
T 7 is known. The experimental values of I\ are about 0.1 ev for A ~ 100 
at an excitation energy of E ~ 6 to 8 Mev, and T T decreases slowly as A 



10 5 



10 



.E 10 3 



1 



10 



10 








1 



1 



1 



1 



1 



20 40 60 80 



100 120 140 
Mass number A 



160 180 200 220 



Fig. 4.6 Average spacing I) Q of excited levels in "noirniaRic" nucJides, as determined 
from experimental values of (r(n,y) combined with Eq. (4.14). The experimental 
errors in <r(n,y) are of the order of 50 per cent, and so no significance is attached to 
deviations from the general trend. The actual excitation energy for each compound 
nucleus is shown numerically beside the individual points. The solid curve repre- 
sents Eq. (-1.12) with E = 8.0 Mev and Weisskopf s empirical evaluations of C and a 
from independent experimental data. [From HugheK, Garth, arid Levin (H69).] 

increases. Inserting these values, tho average level springs 7J shown 
in Fig. 4.5 are found for "normal" nucli