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BOOK
THE ATOMIC NUCLEUS
Robley D. Evans, Ph.D.
PROFESSOR OF PHYSICS
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
TATA McGRAWHILL PUBLISHINC COMPANY LTD.
Bombay New D*lhi
THE ATOMIC NUCLEUS
1955 by McGrawHill, 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 McGrawHill, Inc.
New York.
This edition can be exported from India only by the Publishers,
Tata McGrawHill Publishing Company Ltd.
Published by Tata McGrawHill Publishing Company Limited and
Printed by Mohan Makhijani at Rekha Printers Pvt. Ltd., New Delhi15.
Preface
This book represents the present content of a twosemester 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 oneyear 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 oneyear 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 centerof
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 firstyear 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 groundlevel 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,
aray spectra, ft decay, and radioactiveseries 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 ingratemeter fluHualiaiiH, and
the statistics of rapidly decaying sources.
Keferencr tables of many of the reasonably wellestablished 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
liieu 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 wellknown "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. Xray Scattering 7
3. Charge on the Atomic Nucleus. aRay Scattering . 11
4. Frequency of K and Lseries 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. Coulombenergy 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 aRay 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 Cornhit 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. Massspectrograph 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. SzilardChalmera 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. NeutronProton Scattering at to 10 Mev . 317
4. Electromagnetic Transitions in the np System . ... 330
5. The ProtonProton 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 Highenergy np and pp 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 Liquiddrop 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/uiitrAr'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
RADIOACTIVESERIES DECAY
Introduction. .... . ......... 470
1. Decay of a. Single Radioactive Nuclide . .... . 470
2. RadioactiveHenes 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 Radioactiveseries 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 aRay 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
0RAY SPECTRA
Introduction. . . ... ....... 536
1. Experimental Characteristics of the /3Ray 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. Quantummechanical Theories of Inelastic Collisions with Atomic Electrons 574
3. Comparison of Classical and Quantummechanical 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. Thicktarget Bromsstrahlung . 614
3. RangeEnergy 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. KleinNishina 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 yRay 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 Chisquare Test ............... 775
3. An Extension of the Chisqunre 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. Countingrate Meters . ... . ..... 803
4. lonizatinn Chambers . .......... . 810
5. Rapid Decay of a Single Radionurlide ........... 812
6. Radioactiveseries 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
CENTEROFMASS COORDINATES. AND THE NONRELATIVISTIC
ELASTIC COLLISION IN CLASSICAL MECHANICS
Introduction . ........ ...... 828
1. Relations between L (Laboratory! and C (Centerofmass) 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 Onedimensional 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 sixbook 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 atomicweight 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 Xray 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
0.0005mm 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 Xray 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 aray 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 Xray scattering, and with Moseley's (M60)
brilliant pioneer work on Xray 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 RutherfordBohr atom model and the existence of the small,
massive, positively charged atomic nucleus.
Soddy, Fajans, and others established the socalled 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 0ray 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 aray and two 0ray
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 positiveion 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 RutherfordBohr atom model.
The wavemechanical 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 urnilium.
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
Xray scattering
Isotopes, isobars identified
Discovery of stable isotopes of Ne 20  22 . . .
Atomic nucleus discovered by interpretation
of aray scattering results
Nuclear atom model "completed" by expla
nation of origin of spectra. Quantization
of atomic states .
Assignment of atomic numbers, from Xray
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
aray 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 protonproton 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 molecularbeam magneticresonance
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 wellknown 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 rareearth 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 rareearth
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. Xray 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 Xray 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
Xray transmission ///u, assuming only the validity of Thomson's theory
of Xrajr scattering.
Actually, at least two other phenomena contribute significantly to
The atom of oxygen the attenuation of lowenergy 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 RutherfordBohr 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 wideangle geome
fication. The RutherfordBohr 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
Xray 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 Xray 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 RutherfordBohr
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* Xray 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 higherenergy 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 17.5kev 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 highenergy 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 lowenergy 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 Xray beam strikes the slab at 30 with the normal?
3. Charge on the Atomic Nucleus. aRay 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 aroy 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 aray 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 aray
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
RutherfordBohr 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 ofmass 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 "zeromomentum coordinates," the
" ecu terof 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 twobody collision prob
lem to a onebody 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 onefifth
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/oVr, 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 centerofmass 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 inversesquare 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 laboratorycoordinate 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 centerofmass
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 centerofmass 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 headon 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 inversesquare
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 longrange forces,
such as the inversesquare 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 centerofmass 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 headon 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 onehalf 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
Xray 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 Xray 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 /vshell 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 aray
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 rays and the gold
nuclei is the inversesquare 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
tionscreen detector. Precision arayscattering 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 wavemechanical 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 wavemechanical
results [Chap. 10, Eq. (5.1); Chap. 19, Eq. (2.4)] are markedly different
from those of the classical mechanics. The wavemechanical 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 arayscattering 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.02Mev 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 1.02Mev 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 1mmsquare 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 aray 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 Lseries X Hays
a. Bohr Theory. Following the proof of the existence of atomic
nuclei hy the arayscattering 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 Xray 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 finestructure 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 Xray 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 Xray lines and thereby introduced a new era of Xray 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 Xray 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 overall 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 overall 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 rareearth elements all exhibit
similar chemical properties and stand in the same column of a Mendeleev
periodic, table. The total number of rareearth 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 rareearth 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 Xray 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 Lseries 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 Xray 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
jSray 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 electroncapture 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
16day isotope of vanadium, 2 .iV 48 . More rigorous experimental proof
was subsequently obtained from absorption curves (A23) and from Abel
son's bentcrystal spectrometer studies (Al) of the X rays of zinc (Z = 30)
which are emitted in the pure electroncapture 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 Xrayemission spectra is the internalcon
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 7ray emission and
which often predominates over 7ray emission if the nuclear excita
24
The Atomic Nucleus
[CH. 1
tion energy is small and the angularmomentum 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.
Internalconversion transitions are therefore accompanied by Xray
emission spectra. Neither internal conversion nor 7ray emission
involves any change in the nuclear charge, so that the Xray spectra are
characteristic of the element in which the actual nuclear transition took
place. For example, the 0ray 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 0.41Mev 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 Xrayemission
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 Lseries X rays of Th, 9 iPa, 92!!, 9 aNp, 9 4Pu, g B Am, and
96 Cm (B18).
Nuclear Isomers. Nuclear isomers are longlived excited levels of
nuclei, in which the decay by internal conversion and 7ray emission to
the ground level is measurably delayed (Chap. 6, Sec. 6). Many nuclear
isomers are sufficiently longlived 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 /3ray emitter (for example,
siSb 122 ). This Xrayemission property is useful in identifying nuclear
isomers, especially in those cases in which isomeric transitions are in
competition with 0ray 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 de, 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 0ray 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 Xray 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 socalled displacement law in 1914. In its original form the dis
placement law simply stated that any element which is the product of an
aray disintegration is found in the Mendeleev periodic table two columns
to the left of the parent radioactive element, while the product of a
/3ray 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 aray decay has chemical properties which are
indistinguishable from those of radium, in group II. This product,
mesothorium1, happens to be a 0ray emitter, and so is its daughter
product, mesothorium2. The product of these two successive trans
formations is radiothorium, which has chemical properties which put it
again in group IV. In seriesdecay 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 selfevident 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
bestknown 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, atomicnumber 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 Lseries Xray spectra as excited
by internal conversion or by electroncapture transitions.
Problems
1. A uranium target is bombarded with highenergy 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 Xray data, determine the atomic number Z of the atoms which
emit the 13.79kev 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 arayscatteriiig 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 aray 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 finestructure splitting of Xray 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) "coreand
neutralsatcllite " 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 shortrange 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 shortrange 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 constantdensity model, the nuclear radius R is given by
R = fl A* (11)
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 nuclearforce 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 jSray decay
(coulombenergy 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 /umesonic atoms
5. Finestructure splitting of ordinary
electronic Xray 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. Coulombenergy 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(Zl) (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 0ray 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 shortrange
nuclear forces between the nucleons is substantially the same in both
parent and product. These nuclei constitute a series of socalled 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 0decaying 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 shortrange 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 coulombenergy 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 tfray rnrrpy vs. the twothirds 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 coulombenergy radii which correspond to a constantdensity
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 discreteproton 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 smearedproton model of Eqs. (2.2) and (2.13) and the discrete
proton model of Eqs. (2.3) and (2.16) about equally well.
The coulombenergy 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 coulombenergy radii of nuclei having A < 41 follow the
constantdensity 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 coulombenergy 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 wavemechanical interpretation, lead to "electromagnetic radii' 7 which
are about 20 per cent smaller than these classical "coulombenergy
radii," for the same nuclei. These differences can be reconciled, at least
qualitatively, when wavemechanical 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 coulombenergy radii.
A second wavemechanical 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 overall 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 coulombenergy
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 wavemechanical "electromagnetic unit
radius" and the classical "coulombenergy 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 wavemechanical 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 constantdensity 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 discreteproton
model, and compare the correlation between ^ max and A for the discreteproton
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 "protonoccupied volume"
within the atomic nucleus. In the present section we shall examine only
those aspects of the isotopeshift 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 surfacecharged nucleus, or U v (r) for
the volumecharged nucleus. The decrease AFF in electron binding
energy can then be represented, for the volumecharged nuclear model, as
= e
t 2 (r)(V v  U)4*r* dr
(35)
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 volumecharged nucleus, are
compared graphically in Fig. 3.1, from which a qualitative idea of the
Distance from center of nucleus r >
rNuclear 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
isotopeshifted 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 hyperfincstructure components, because the nuclear moments are
zerovalued. [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 protonoccupied
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 onehalf 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 longrange coulomb force. Bombardment of nuclei by highenergy
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 10Mev 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
isotopeshift 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 wavemechanical 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, highenergy 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 highenergy 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 chargeexchange 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 neutrinoanti
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
Xray 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 finestructure splitting of the ^mesonic "Xray 11 spectra appears
to have been resolved in the experiments by Fitch and Rainwater (F52),
Tn electronic Xray spectra, the electron spin of T gives rise to the two
finestructure 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 /zmesonic levels
in Pb, a finestructure 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 jimesonic transition 2pj > Is, which corresponds to the
K a \ line in electronic Xray 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 finestructure 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 irmesonic /Cshell
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. Finestructure Splitting of Electronic Xray Levels in Heavy
Atoms. Each ordinary electronic Xray 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 /imesonic 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 finestructure 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 iK a2 Xray
finestructure 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 Xray energy measurements and in the theory of Xray
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 shortrange
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 socalled squarewell 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 squarewell 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 centertocentcr 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 socalled 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 Shortrange Forces. Clearly the discon
tinuities in this squarewell 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 roundedoff
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 shortrange 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 shortrange 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 longrange and shortrange 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 aRay 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 arayscattering 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 8.57Mev a rays which are
emitted by a source of ThC' are scattered classically by uranium nuclei.
In central collisions, these 8.57Mev 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
^B28 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 arayscatter
ing experiments. If the 4.2Mev 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 4.2Mev
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 wavemechanical 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 4.2Mev
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 halfperiod of U 28B . We
5] Radius of Nuclei 49
shall review the wavemochanical 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 5.3Mev 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 headon
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 8.57Mev 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 blackbody 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. 248259 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 wavemechanical 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 (521)
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 complexplane 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 = cfc*' (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 timedependent 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 (532)
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 conscrvationofencrgy
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 singlevalued, 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 Mevrm (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 ~ Mevsoc (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 overall 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 onedimensional 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 onetenth the velocity of light (about the velocity
of a 4Mev 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
" (555)
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^fIL^) (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 socalled penetration factor
P, which is merely the ratio of the probability densities on the two sides
of the barrier, i.e.,
P = {!! (565)
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 wavemechanical 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 wellknown 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 wavemechanical 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 socalled WentzelKramersBrillouin (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 aray
radioactive decay are based on the same general theory concerning the
transmission of nuclear potential barriers.
Wave Equation in Spherical Polar Coordinates. For the threedimen
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 threedimensional 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 wavemechanical 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 quantummechanical
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 angularmomentum 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)
jo
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 Zwave 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 swave 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 onedimensional 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 onedimensional 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 10A 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 = finestructure 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 coulombbarrier 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 aRay Emitters
We return now to reconsider the impasse which classical nuclear
theory faced in 1928. It was shown experimentally that 8Mev a rays
are elastically scattered by a pure coulomb field surrounding the uranium
nucleus but that the same nucleus emits only 4Mev 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 rtray 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 "evenZ 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 evenZ
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 halfperiods. 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 halfperiod for radioactive
decay T is given by
where the socalled 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,,rT (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 inld 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 evenZ 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 40fold change in the decay constant and halfperiod.
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
evenZ 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 halfperiod 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 constantdensity 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
Ifperiod 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 halfperiod
for radium varies with one's choice of
the nuclear unit radius R in the con
stantdensity model R = R A*. Note
that the halfperiod 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 aray 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 evenZ
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 iuy of the radii of evenZ even ,4 nuclei with the constantdensitx
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 aray 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 340Mev 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 halfperiod. 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
shortrange 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
adecay 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 /imesonic 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 socalled 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 eveiiZ and cveiiA. 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 16fold
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 /imesonic "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
Halfperiod
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
halfperiod" of about 1,740 yr, if the tot&l halfperiod is 1,620 yr.
3. Radium has been .found to emit a small percentage of 4.593Mev 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 8Mev a rays by
heavy nuclei follows the Rutherford scattering law (Chap. 1, Fig. 3.3).
This scattering is therefore due to a simple inversesquare 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 aray energy which would produce
anomalous scattering. Classically, anomalous scattering should set in
at an aray energy which is just large enough to bring the incident a ray
to the edge of the scattering nucleus. In a strictly headon collision,
this would occur when the initial kinetic energy ?MV* in the centerof
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 centerofmass 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 orray 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 aray 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 lowZ 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 phaseshift 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 quasistationary 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 longrange 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 shortrange forces, its principal contribution at low bombarding
energies is to swavc 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 evenZ 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 overall 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 aray
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 centerofmass coordinates. Deviations
from classical scattering are clearly evident at T a < 5 Mev, which is
only about twothirds 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 aray 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 swavc pene
tration of the coulomb barrier is sufficient to produce a detectable ampli
tude of nuclear potential scattering.
aRay 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 pwave 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 twothirds 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 discre 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 swave or dwave
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 overall 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 orrur 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 RAIIUS 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 constantdensity 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 8.6Mev a rays
(laboratory coordinates).
(a) Show that the numerical value of the parameter 2Zz/ 137/3 is about 0.6
for the 4.4Mev 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 aray 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 centerofmass coordinates is only onefifth 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 4.4Mev 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 4.4Mev 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 centerofmass coordinates is kept constant.
3. The elastic scattering observed when helium is bombarded by 1Mev to
4Mev 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 (pa) 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 centerofmass
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 highvoltage 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
barriertransmission concepts as applied to a decay stimulated Cockcroft
and Walton to believe that these lowenergy 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 Compoundnucleus 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 socalled 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 nuclearparticle 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 quasistationary 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 onetenth the esti
mated coulombbarrier 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 700kev 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 massenergy, 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 centerofmass coordinates
This plane wave is the sum of partial waves, corresponding to particles
whose angularmomentum 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 overall harriertransmission coefficient. For purposes
of visualization, we may write
T z = P,(D&(D (8.6a)
in which Pi(T) is a Gamowtype coulombpenetration 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 overMil transmission probabiluy 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 Gamowtype 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 wavemechanical 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
centerofmass 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 higherenergy 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 chargedparticle 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 constantdensity 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 wavemechanical 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 waveoptical
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 constantdensity model R = /?/!*. Again, as in the case of
chargedparticle interactions, this is a "nuclearforce 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, chemicalatomicweight
determinations had disclosed several elements which definitely do not
have integral wholenumber atomic weights on the oxygenequals16
(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 atomicweight 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 arayscattering 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
longlived excited nuclear level of UZ. Numerous other isomeric pairs
have been discovered among the artificially radioactive substances.
c. Discovery of Stable Isotopes by Positiveray Analysis. While
using a new parabola positiveray 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 mass20 neon parabola, and while it seemed that the
mass20 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 /3ray emitter,
iiNa" is the stable sodium isotope, and n Na 24 is a negatron /3ray 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 muchoverworked
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 liquiddrop 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 ThomasFermi 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 selfconsistent
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.
Positiveion 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 highvoltage discharge tube containing the appropri
ate gas at a low pressure. A small amount of neon in the tube will
stabilize the highvoltage 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
highvelocity 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 lowpressure ion source in which the
ions are produced by bombardment with lowvoltage 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 cylindricalcondenser filters (H45, W8) and of sector
magnets has been reviewed and extended by Bainb ridge (B4), by John
son and Nier (JIG), and others. Cylindricalcondenser sectors possess
optimum directionfocusing 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 20,000volt, low
pressure dischargetube ion source.
An alternative, but less exact,
energy filter is obtained by accel
erating low velocity ions obtained
from a lowvoltage 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) Cylindricalcondenser 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 lowvoltage 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 directionfocusing properties of a uniform magnetic field are of
special importance in mass spectroscopy [as they are also in 0ray spectros
copy (PI 8)] because they permit greatly increased beam intensities
without serious loss of resolution. The wellknown 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 wedgeshaped 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 socalled 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 sectortype 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 highermomentum 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) (36)
\ 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 wellknown 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 timeofflight mass spectrometer has been developed which
utilizes Eq. (3.9). Lowvelocity 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 directionfocusing elements (Fig.
3.1a and 3.1c), are known as doublefocusing instruments. Those
which have directionfocusing action in only one element (usually
because the electrostatic filter is of the accelerator type, Fig. 3.16) are
called singlefocusing.
Doublefocusing 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 directionfocusing 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 analjzer
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.
Doublefocusing 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
cylindricalcondenser 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
singlefocusing 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 singlecollec
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
vibratingreed electrometer, a vacuum
tube electrometer, or an electron multi
plier tube.
Figure 3.4 shows schematically a dou
blefocusing 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
cylindricalcondenser 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 ionsource 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.
Singlefocusing Mass Spectrometer. Many varieties of singlefocusing
mass spectrometers have been developed. Usually, these are similar to
3]
Mass of Nuclei and of Neutral Atoms
109
the doublefocusing 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 doublefocusing
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
socalled "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 massspectroscopic literature has been summarized from time to
time, with a view to deducing by leastsquares 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 pre1950 "host
values' 1 of doublet separation and derived masses, with probable errors, from least
squares analysis of available massspectroscopic results. The List line gives the
results for the same quantities as obtained purely from nuclearreaction 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, methaneoxygen doublet.
The direct massspectroscopic 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 assspec troscopic results are given at the top. Below these are shown two
evaluations calculated from differences between doublets containing sulfur toinpounds
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 propanecarbon 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 pre1050 massspectroscopic "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 selfconsistent mass tables,
from which accurate calculations of nuclear binding energy can be made.
Representative pre1950 and post1950 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 massspectroscopic 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 )
iC
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 0H
14.716
isl
Bainbridge (1948)
IOH
14.719
1
Roberts (1951)
H
1
1
4.765
Ewald(1951)
IOH
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 )
iCH
"
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 nuclearreaction mass scale to the heavy nuclides.
Until this is accomplished, it is necessary to have massspec, troscopically
determined secondary standards among the heavier nuclides. Then
nuclearreaction 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 nudearreaction 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
25cm 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 25cm 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 20,000ev 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 highenergy
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 massspectroscopic 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 nitrogencarbon 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 highspeed 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 wellknown 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 socalled 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 electronvolts, 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. 3739 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 massspcctroscopic doublet data
(J19).
Q Values of Reactions Compared with Massspcctroscopic Doublets.
When nitrogen is bombarded by highspeed 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 massenergy 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 massspec
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 massspectroscopic 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 lefthand 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 centerofmass 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 centerofmass 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 atomicmass differences can be obtained from measurements of
the energy of a, ft, and y rays, provided that the radioactivedecay 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'. Massenergetically, 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 Tray 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 /9ray 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 decayproduct atoms.
Energylevel 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 7ray energies plus the
rest mass m Q of the emitted ft ray.
The righthand side of Fig. 4.2 is the Al 28 decay scheme on the nuclear
massenergy 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 = 051 Mev of the ft ray and also the increase of one unit in
nuclear charge when the 0ray negatron is emitted. The vertical line
represents the kinetic energy 2.865 Mev of the electronneutrino pair, as
4]
Mass of Nuclei and of Neulral Atoms
123
measured by the maximum energy of the /3ray spectrum. The wavj r
line shows the final 7ray transition in Si' Jg .
The lefthand side of Fig. 4.2 shows the same transitions but drawn
on the more common atomic massenergy 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 overall mass balance, we can imagine that the
emitted /5ray 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 energylevel and mass diagram. An absolute mass scale for the
nuclear mas^encrgy diagram would equal the atomicimisH 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 atomicmass 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' = ziM' + 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 = ziM + 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  ziM = 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.
Energylevel 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 massenergy 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 atomicmass 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
positronneutrino pair, as evaluated by the maximum energy of the
positron 0ray 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
Electroncapture 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 massenergy
diagram
Nuclear massenergy
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 atomicmass diagram that among the prod
ucts of the decay process the total mass 2mo of the positron and one atomic electron
ran 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 overall 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 7ray 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 electroncapture (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  zi
+ To
T = T, +
(4.27)
Thus, to a good approximation, the electroncapture decay energy TQ
is equal to the difference in atomic mass Q E c between the isobaric parent
and product.
Energylevel 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 massenergy
diagram
r
0.478
Nuclear massenergy
diagram
"i
Fig. 4.4 Electroncapture 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 lowenergy neutrino group followed by a 0.478Mev 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 massenergy diagram, on the righthand 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 lowenergy neutrino is emitted, and the Li 7 nucleus
is left in an excited level at 0.478 Mev. A 7ray transition to the ground
level of Li 7 follows promptly.
Note that in an electroncapture 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 ftRay Emission.
Equation (4.27) shows that the electroncapture 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 massenergy
diagram
Nuclear massenergy
diagram
0.5 .E
Fig. 4.5 These schematic diagrams emphasize the difference between the energetics
of electron capture and positron ft decay. Electroncapture 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 electroncapture 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 energylevel 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 massenergy
diagram
Nuclear massenergy
diagram
Fig, 4.6 Principal dual 0decay and electroncapture transitions of Cu 84 . The
approximate relative abundance of the competing transitions is shown on the nuclear
massenergy 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
Massenergetically, 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 massenergy 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 massenergy 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
zzM 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 ccnterofmass coordinates. The lowerenergy a transition and
its 0.187Mev 7 rny occur in only about 6 per cent of the transitions of Ra Mi . The
atomicmass 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 clearreaction
data is the neutronhydrogen 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 centerofmass
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 ceriterofmass 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
(IO. =  ; 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 /8ray 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 neutronhydrogen 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 energylevel 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 magneticlens
spectrometers arranged end to end. The /3ray 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 0ray 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
twobody 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. Massspcctroscopic 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 nuclearreaction 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
nuclearreaction 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 massspectroscopic
doublets, as in Eq. (3.12). However, the accuracy of the Qvalue
determinations is now sufficiently high so that good atomicmass values
can be obtained for the light nuclides. In general, the massspectro
scopic masses are consistently larger than the nuclearreaction 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
centerofmass coordinates.
3. From the following nuclearreaction data (L26), calculate the atomicmass
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 0ray 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
0ray and yray 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 atomicmass 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
0ray transitions leads to the 12day isomer of Xe 131 . The percentages shown for
each y ray include the competing internalconversion 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 atomicmass difference
between K 40 and Ca 40 , in amu?
(6) What would be the maximum possible 7ray energy following an electron
capture transition in K 40 ?
(c) What is the actual kinetic energy of the neutrinos if the observed 7ray
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 0ray 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 massenergy 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 dd 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, massenergy diagram and show graphically the values of
Qi, Qf, T m ^ and (n  H').
12. Determine the atomicmass 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 massenergy 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
troncapture 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 pre1950 tables and are therefore subject to
some revisions because of changes in the mass sub standards.
Data on mass doublet separations, nuclearreaction 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), nuclearreaction 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 selfconsistent set of
atomicmass values derived entirely from nuclearreaction 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 post1951 massspectroscopic 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 nuclearreaction 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 NUCLEARREACTION 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 MAsssrECTBOSCopic 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 quantummechanical property. No satisfactory theoretical basis
was forthcoming until Dirac showed that the existence of electron spin
is a necessary consequence of his relativistic wavemechanical 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 quantummechanical 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 socalled vector model, which has long been used in optical
spectroscopy to translate the quantummechanical 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 quantummechanical
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 firstorder 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 orbitalangularmomentum 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 FermiDirac 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 integerspaced values m k = i, J.
Total Angularmomentum 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, integerspaced
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 halfinteger values,/ = 2, J, I
Magnetic Total Angularmomentum 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 socalled 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, ....
Isobaricspin 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 "isotopicspiri" and more recently the "isobaricspin" quantum
number (W47, 14). A common but arbitrary assignment for the total
isobaricspin 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 " isobaricspin 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 firstorder 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
Z711
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 mnemonically 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 socalled magic numbers are currently
considered as representing "closedshell" 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 rmclearangularmomentum 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 RussellSaundcrs, or
LS, coupling and the spinorbit, 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).
RusscUtfaundrrs Coupling (Lti). In this coupling scheme it is
assumed that there is a negligibly weak coupling between the orbital (I)
and the spin (s) angularmomentum 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
angularmomentum 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 gocalled spin midtiphts. Finally, the resultant L and S
couple together to form the total angularmomentum 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 integerspaced 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 lowlying excited level. The two levels taken together are a "spin
doublet" in KussellSaunders 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 angularmomentum quantum
number j for the individual nucleon, where j = I s. In turn, the total
nuclearangularmomentum quantum number 7 is a vector sum of the
individual j values. Hence jj coupling is also called strong spinorbit
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 nuclearangular
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 RussellSaundors 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 finestructure 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 energylevel 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 spinorbit interactions in nuclei has been sought but as yet with
out compelling success.
d. Total Nuclear Angular Momentum. The total nuclearangular
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 angularmomentum
vector, whose magnitude we de
note (W39, F41 ) for typographical
convenience as 7*, whore
(17)
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 /' ergsec 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* ergsec (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 pre1950 literature, the
nuclear gyromaynctic ratio, g) is equivalent to inserting a dimensionless cor
rection factor g in the righthand 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 = ^ = (215)
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 angularmomentum quantum
number 7.
The nuclear magnetic moment is taken as positive, if its direction
with respect to the angularmomentum 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
quantummechanical theory of the electron, the value g s = 2 also emerged
from this theory in a natural way.
By atomicbeam methods it is possible to obtain very precise measure
ments of the separation of optical finestructure 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 atomicbeam
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 finestructure
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 hyperfinestructure 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 electronspin gyromagnetic ratio are impor
tant for two reasons. First, many nuclear gyromagnetic ratios are
measured in terms of highprecision 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 onehalf 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 molecularbeam 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 liquidhydrogen 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 referencestandard 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 evenZ evenAT 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 evenZ evenJV 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 oddZ evenAT nucleus could be visualized as one odd proton
outside a closed "core" of evenZ even AT. On this singleparticle 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 oddZ evenN nuclei. Analogously, in evenZ oddAT
156
The Atomic Nucleus
[CH. 4
nuclei, the entire nuclear / and M are attributable to the one odd neutron,
in this singleparticle 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 evenA 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 .
OddZ oddTV 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 singleparticle model, these
oddZ oddTV nuclei consist of an evenZ evenTV core, with one proton
and one neutron outside. The angular momenta so far measured for
TABLE 4.1. SUMMARY OF THE FOUR "ODDEVEN" CLASSES OF NUCLIDES AND
THE GROSS RESULTS OF MEASUREMENTS OF THEIR GROUNDLEVEL
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 (oddZ
evenTV, Fig. 4.1) and when the odd nucleon is a neutron (evenZ oddTV,
Fig. 4.2).
b. Single particle Model, for Odd^4 Nuclei. If, with Schmidt, we
assume a singleparticle 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
OddZ, evenJV 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 oddZ evenA T nuclides. The solidline
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 
 
EvenZ, oddJV nuclides
^ 2
*J"
^wiH"
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 evenZ oddN 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 timeaverage values of the components normal to j* are zero. The
Angular
momentum
Magnetic dipole moments
Oddneutron /* s *=l 91 V3
Fig. 4.3 Vector diagrams (for I = 3) illustrating the composition of angularmom 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 socalled "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 vectormodel
diagram in Fig. 1.1 for / and /*. In this singleparticle 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* COH (l*j*) + M , cos (s*j*) (4.3)
The g factor for the single particle is, by Eq. (2.15),
5Jff ()
where /* is the observable component of the net magnetic dipole moment
and the nuclearangularmomentum 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~JJ\
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 socalled 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 SINGLEPARTICLE 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 100plus 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 oddZ 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 oddproton nuclides and zero for oddneutron 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 socalled
"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 symmetricalcore model to account for the observed electric
quadrupole moments of nuclei (Sec. 5). At least for the oddproton
nuclei, the singleorbit 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 singleparticle model for odd A nuclides, modified
both by quenching of freenucleon moments and by asymmetry of the
eveneven 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 socalled MargenauWigner (MW) limits. For
oddproton nuclei, the MargenauWigner limits do not follow at all the
empirical slope of about unity which is displayed in the Schmidt dia
grams. For oddneutron nuclei the MargenauWigner 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 singleparticle 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.
Molecularbeam 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 hyperfmestructure 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 singleparticle model than with the
uniform model.
c. OddZ OddN 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 oddZ oddN 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 oddZ oddAT class are radioactive, and they
decay by 0ray transitions to evenZ evenAT 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 oddZ oddN nuclide (Chap. 6).
Nordheim's Rule. It is often possible to infer the state of the odd
proton by comparison with a known oddZ 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>DZ ODDJNT 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 oddneutron (gi
nuclei.
0; >x fl = 1.91) and oddproton (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 socalled 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
(51)
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,
710
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 angularmomen
tum axis (footballshaped distribution). Negative moments correspond
to a flattened, or oblate, distribution (discusshaped 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 quantummechariically . 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 quantummechanical 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 quantummechanically 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 quantummechanical 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 constantdensity 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 welldefined radius. Farreaching 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 Pstate
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 socalled 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
OddZ
OddN
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
OddZ
OddN
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 coreplussingleparticle 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 evenZ ; 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 evenZ 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
socalled closed shells, corresponding in the shell model to the observed
"magic numbers," 2, 8, 20, 28, 40, 50, 82.
For oddZ QveuN 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 oddZ even AT nuclides,
Z corresponds to one proton less than a closed shell, then, by the socalled
"partial configuration method" (F18), the "hole" in the proton shell
behaves like a negatively charged particle. Thus for a sequence of oddZ
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 oddZ evenAT
nuclides and oddZ oddJV nuclides are plotted as circles against Z. Moments of
evenZ oddN nuclides are plotted as triangles. Arrows indicate the closing of
major nucleon shells. The solid curve represents regions where quadrupolemoment
behavior seems established (T2G, P27). The dashed curve represents more doubtful
regions. [Adapted from Townes, Foley, and Low (T26).]
evenJV 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 = +116 X 10~ 24 cm 2 ) and siSb 121
(Q = 0.3 X 10~ 24 cm 2 ) as Z passes through the wellestablished closed
shell of 50 protons.
For evenZ oddJV 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 oddjV 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 evenZ 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 evenZ
proton configuration is the closed shell Z = 8. Its core is the doubly
closedshell 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 oddZ evenJV 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 evenZ oddJV 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 oddZ 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 oddZ evenN 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 evenZ evenJV 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 quantummechanical
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 angularmomentum 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 fastneutron 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 fastneutron
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 quantummechanical 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 tothe "conservation of parity"
in the same sense and with the same rigor as the conservation of charge
and of angular momentum. Like the angularmomentum 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
socalled parity of a system of elementary particles, such as a nucleus,
atom, or molecule, is a fundamental property of the motion according
to the wavemechanical 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 righthanded or lefthanded,
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 motion of the
particle is said to have odd parity. Thus
iK x, y, z,s) = f(x,y,z,8) represents even parity .
iKz, 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 angularmomentum 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 oddparity particles, and any number of
evenparity particles, will have even parity. A system with an odd
number of oddparity particles, and any number of evenparity 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 fourcomponent 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 quantummechanical 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 coreandsingleparticle 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 singleparticle model, with jj coupling between
nucleons. This evidence will be examined in Chap. 11. Here we may
note that the singleparticle model forms a reasonable basis for the deter
mination of I for an oddnucleon 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 angularmomentum
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 wavemechanical 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, FermiDirac (antisymmetric) or EinsteinBose
(symmetric), and these two have the following principal characteristics.
a. FermiDirac Statistics. The wave function of a system obeying
FermiDirac 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
FermiDirac 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 FermiDirac
statistics. Generalizing these observations with the aid of Sec. c below,
we can expect that all nuclei of odd mass number have FermiDirac statistics.
b. EinsteinBose Statistics. A system whose wave function is sym
metric is said to follow EinsteinBose statistics. Interchange of two
identical EinsteinBose 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)
EinsteinBose 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 EinsteinBose statistics. From diatomic band spectra, H 2 , He 4 ,
C 12 , N 14 , O lfl , S 32 are known to obey EinsteinBose statistics. Generaliz
ing, we can expect that photons and all nuclei of even mass number have
EinsteinBose statistics. The IT meson, which is associated with the
binding forces between nucleons, is an EinsteinBose 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 FermiDirac 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 stepbystep 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 FermiDirac
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 stepbystep exchange having an even number
of stages. Because interchange of two such nuclei leaves the wave func
tion unaltered, these nuclei must have EinsteinBose statistics.
That the neutron is an elementary particle obeying FermiDirac
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 FermiDirac statistics,
the neutron must also.
Nucleons obey FermiDirac 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 FermiDirac statistics, and further that ele
mentary particles with any arbitrary integral spin (s = 0, 1, 2, . . .)
must obey the EinsteinBose statistics. This is a farreaching, funda
mental generalization.
Due to the conservation (vectorially) of angular momentum, all nuclei
having an odd number of nucleons also have an odd halfinteger total
angular momentum (7, 7, ?, . . .). Similarly, all nuclei having an even
number of nucleons have even halfinteger total angular momentum
(0, 1, 2, 3, . . .). In tabular form we have, then,
Mass number
Angular
momentum
Statistics
Odd . ...
/  i, i I. 
FermiDirac
Even
/  0, 1, 2, ...
EinsteinBose
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. FermiDirac statistics is now
frequently called simply Fermi statistics, and EinsteinBose statistics is
shortened to Bose statistics. Time has also brought spontaneous trans
formations in the corresponding adjectives, as denned by the identities
(FermiDirac particle) * (Fermi particle) fermion
(EinsteinBose 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 FermiDirac 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 GamowTeller 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 groundlevel moments.
1. Extranuclear Effects of Nuclear Angular Momentum and
Statistics
a. Number of Hyperfinestructure Components. The main features
of atomic spectra, including the socalled 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 molecularbeam magneticresonance
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
FabryP6rot interferometer.
The magnitude of the hyperfinestructure 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 angularmomentum 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 angularmomentum quantum number F. Thus F can
tuke on the series of integerspaced 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 jiiiiUijikl 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 rlKinm. Although nuclei have very small magnetic dipole
is. thr nia^noMc interaction energy between them and such huge
IM'P iNou^h to be measurable easily and constitutes the hyper
Lftiup 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 socalled 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*
(14)
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 totalangularmomeritum
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 + 71), (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 + 72)
(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 hypertinestructure 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
PaschenBack 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.
PaschenBack 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 spacequantized inde
pendently in the direction 7/o, with independent magnetic quantum
Hyperfine Zeeman PashenBack
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 PaschenBack effect of hyperfine structure.
Each level of a given mj (which corresponds to the ordinary Zeeman
effect of finestructure 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 hyperfinestructure region,
186 The Atomic Nucleus [CH. 5
and Am/ = 0, 1, or Araj = 0, 1, in the PaschenBack hyperfine
structure region.
BreitRdbi Formula. The region of intermediate fields (order of 1 to
1,000 gauss) has become of particular importance in recent years because
the atomicbeam magneticresonance 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, PaschenBack, 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 = hyperfinestrutrture 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 squareroot term in Eq.
(1.8) will yield a term linear in x, which defines the Zeeman splitting and
is the dominant fielddependent 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 BreitRabi 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
atomicbeam magneticresonance method permits Eq. (1.8) to be used
to determine the hyperfinestructure 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 zerofield hyperfinestructure splitting. Thus
x <3C 1 defines magnetic splittings which are much less than the zerofield
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 zerofield hyperfine structure (separation of levels of
different F) that the onset of the PaschenBack 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 zerofield, veryweakfield (Zeeman hyperfine structure), and
weakfield (PaschenBack 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 hyperfinestructure 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 JT O 10 )
tj 71 CLjf
where ^ 7l (0) = wave function at. zero radius for x electron with principal
quantum number n
a H = AV^Vwo = 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 finestructure 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 Tiestructure interval factor
a mid of the hyperfi liestructure separation Av are in excellent agreement
with the very accurate experimental values of AJ>, and independently of
/i//, which are obtainable by atomicbeam magneticresonance methods
(T8).
Diarnagnctic Correction. Whenever AV and ,u// are measured by a
magneticresonance 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 FermiThomas
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 HartreeFock 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). Hypcrfinestructure 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 relativeintensity relationships are analogous
to those of finestructure 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
angularmomentum quantum number of each nucleus, then the total
nuclearangularmomentum 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 EinsteinBose statistics (for which / isO, 1,2,
. . .). The ratio of the relative intensity of the symmetric lines to the
antisymmetric lines for nuclei obeying the FermiDirac 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
013)
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 rotationvibrational 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 BreitRabi 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 hyperfinestructure 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 BreitRabi 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 Hyperfinestructure 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 hyperfinestructure 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 hyperfinestructure separation factor a for
the normal slate of the neutral atom, hence by the nuclear magnetic
moment.
By the atomicbeam deflection method, Kellogg, Rabi, and Zacharias
(K14) measured in 1936 the hyperfinestructure 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 hyperfinestructure separations are too small to be measured by
optical methods. The atomicbeam 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 magneticresonance 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 angularmomentum 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, ergsec
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~ \ Tif
(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 integerspaced 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 radiofrequency domain. Studies
of nuclear, atomic, and molecular properties by Larmor resonance meth
ods have therefore come to be characterized as radiofrequency spectroscopy.
Three principal experimental arrangements are in current use: (1)
molecularbeam magneticresonance method (Rabi et al., 1938), (2) the
nuclear paramagneticresonance absorption method (Purcell et al., 1945),
and (3) the nuclear resonance induction method (Bloch et al., 1945).
194
The Atomic Nucleus
[CH. 5
Molecularbeam Magneticresonance Method. Magneticresonance
methods and radiofrequency 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 molecularbeam
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
berylliumcopper
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 molecularbeam 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 radiofrequency field, directed normal to H, which produces the
soughtf 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
singlefocusing mass spectrometer. Finally, the accelerated ions impinge
on the first plate of a BeCu 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 3yr 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 shorterlived radioactive
species.
With genetically similar molecularbeam 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 Paramagneticresonance 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 radiofrequency 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 lowerenergy 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 radiofrequency 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 longlived 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 radiofrequency 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 radiofrequency 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 socalled spinlattice 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 liquidair 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 magneticresonance methods
= 2.793/0.857 = 3.26.
Problems
1. Derive the Larmor precession frequency and kinetic energy from classical
electrodynamics.
2. Consider the hyperfinestructure 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 Hyperfinestructure Separa
tions. The relative separations of hyperfinestructure 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 hyperfinestructure levels when the energy difference between
the parent term and an adjacent electronic state is comparable with the
hyperfinestructure 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).]
hyperfinestructure spacings. Other states should be influenced approxi
mately in proportion to their finestructure 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 = hyperfinestructure 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 FabryPe'rot
clalon, in the X4 7 260 line of singly ionized osmium. The lines due to the evenZ
evenJV 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 hyperfinestruc
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 socalled ''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 molecularrotation 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
1cm wavelength are passed through a waveguide 16 ft long which con
tains IC1 vapor and acts as a 16ft 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 overall
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 microwaveabsorption 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
magneticresonance methods of radiofrequency 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 microwaveabsorption 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).
Molecularbeam Electricresonance Method. The electric moments of
molecules can be measured also by a molecularbeam resonance method,
in which all the fields are electric instead of magnetic. These electrical
resonance methods of molecularbeam 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 angularmomentum
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 groundlevel 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 7ray 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 72" 1
19J8 /
17.63
2.94
\
17.242
Li 7 +p
3.0 5
2.22 I
1.882J
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 centerof
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 halfperiod 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 quantummechanical
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
steinBose 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 17.6Mev 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 swave inter
actions (I = 0).
For aray 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 angularmomentum change of
I = A/ = 5
produces only about a 20fold reduction in the ar&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 aray 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 aray
spectra in terms of changes in nuclear angular momentum was of great
historical value. It led him later to initiate analogous considerations
regarding 0ray 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 0ray 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 0ray 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
Halfperiod
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 shorterlived group are "permitted," or "allowed" transitions, with a
selection rule (A/ = 0, no), while the longerlived group are "notper
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 0ray 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
firstforbidden, secondforbidden, thirdforbidden, 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
S6
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
0ray emitters. The maximum energy of the 0ray spectrum E mt * is plotted against
the partial decay constant X for the principal branch of the decay. The halfperiod
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 Halfperiods 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 halfperiod, and the other on the
shape of the /3ray spectrum. Both classifications are in acceptable
agreement with present theories of the lifetime and shape of forbidden
0ray spectra (Chap. 17).
The classification by "comparative halfperiods," or "ft value," was
introduced by Konopinski (K37) in connection with his theory of for
bidden 0ray 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 = halfperiod 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 socalled "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 0ray 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 socalled 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 firstforbidden 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 electronneutrino 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 nucleonclectron 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 (GT) selection rules."
The GamowTeller rules (G8) emerge when the intrinsic spin of the
transforming nucleori is introduced into the Hamiltonian which describes
the transformation. This couples the electronneutrino 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
GamowTeller rules they are emitted with parallel spins (triplet state)
in the nonrelativistic limit (p. 679 of B68). At present, CJamowTeller
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 GamowTeller 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
GamowTeller
(tensor and axial
vector)
Fermi
(scalar and
polar vector)
log ft
values
(M27)
Allowed
No
0, 1
36
(but not > 0)
Firstforbidden
Yes
0, 1, 2
0, 1
610
(not > 0, ^ > ?,
(not  0)
0<> 1)
Secondforbidden
No
2, 3; 0>0
1, 2,
>10
(not <> 2)
(not <> 1)
nthforbidden
n, (n + 1)
(n  1), n
n odd
Yes
n even
No
The GamowTeller rules are characteristic of a tensor or an axialvector
interaction, while the Fermi rules represent a scalar or an ordinary vector
interaction between the transforming nucleon and the electronneutrino
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 electronneutrino pair with successively larger values of
orbital angular momentum.
d. Shell Structure. From measurements of the /3ray energy and
halfperiod, log ft values are now available for over 140 odd4 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 angularmomentum 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 /3decay log ft values with assignments of oddnucleon states on the
sin glop article shell model with jj coupling, and with GamowTeller 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
Oddnucleon 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
singleparticle 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 singleparticle 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 wellverified 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 GamowTeller 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 GamowTeller 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 0ray
electronneutrino pair with respect to the emitting nucleus. Analogously,
the firstforbidden transitions all have A/ = 1, secondforbidden, AZ = 2,
etc.
The socalled lforbidden transitions (N23, M27, K39J constitute at
present a, small subgroup, closely related to the allowed unfavored transi
tions, and obeying the GamowTeller selection rules on A/ for allowed
transitions but having A/ = 2 on the singleparticle shell model. These
/forbidden transitions (A/ = 1, no, A = 2) have log/ ~ 5 to 9, hence
halfperiods 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 overall 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 firstforbidden
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 magneticangularmomenturn 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 evenZ evenJV 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), 7ray 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 7Ray 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 7ray photons
are those combinations of I and parity which give nonvanishing values
of the transition probability. These are summarized in Table 4.1.
d. 7Ray 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 yray 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 singleparticle shell model of nuclei. In the single
particle theor}', a yray 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 yray 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)
(49)
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 liquiddrop
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 1Mev 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 pre1951 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 singleparticle 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 pre1951
theories ; especially for low energies).
TABLE 4.2. THE TYPE OF MULTIPOLE RADIATION EMITTED IN TRANSITIONS
BETWEEN NUCLEAR LEVELS WHOSE ANGULARMOMENTUM 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 parityfavored transitions (no "spin flip")
and parityunfavored transitions for which, in the singleparticle 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 " parityfavored,"
whereas /j >/ s would involve a "spin flip" (AS = 1) and would be
"parityunfavored."
218 The Atomic Nucleus [CH. 6
e. Mixed Transitions. Admixtures of the predominant multipole
with the weaker competing multipole shown in the righthand 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 7ra'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
socalled 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
longerlived 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 7ray 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 392kev
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 U92kev 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
singleparticle 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 aray emitters, which possess no continuous
/3ray 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 Xray 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 7ray
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 "internalconversion 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. Internalconversion 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 "directcoupling"
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
internalconversion 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 internalconversion coefficient depends on
W, the energy of the transition
Z, the atomic number of the transforming nucleus
I, the multipole order of the transition
parityfavored (electric multipole) or parityunfavored (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 "internalphoto
electriceffect " model of internal conversion is the occasional reappear
ance of the pre1933 definition of internalconversion 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 shellconversion 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
shellconversion 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 (parityunfavored, as shown in Table 4.2), the TiCshell internal
conversion coefficient for raoc 2 )> W y> B K reduces to
(56)
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 angularmomentum 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 internalconversion 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 /3ray emitters in 1934. The
strong Z dependence (~ Z 3 ) made the observation of internal conversion
in the new lowZ 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 electroncapture 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 firstrank 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. Lshell 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 socalled
" 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 Lconversion 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 angularmomentum 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 internalconversion coefficient,
Eq. (5.2), and of the shellconversion 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 singleparticle 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 positronnegatron
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 states is the experimentally observed
positron, while the electron in a positive energy state is the negatron
member of the observed positronnegatron 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 internalconversion 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 highenergy
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 7ray 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 pairconversion coefficient is almost independent of Z;
indeed, it decreases slightly with increasing Z. Convenient graphs of the
pair internalconversion 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 internalconversion coefficient.
Experimentally, the radioactive
nuclides provide only a few 7ray
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 1.7fiMov and 2.2Mev transi
tions, and that the 2.02Mev 7ray
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 internalconver
sion coefficients have been measured by Slatis and Siegbahn (S4(>) for the
following wellknown 7ray transitions:
TABLE 5.1. PAIR INTERN ALCONVJCHSJON COEFFICIENTS
l
Fig. 5.5
2 3
Transition energy W, in Mev
Representative theoretical
curves and experimental points? for the
pair internalconversion coefficient.
Note the strong contrasts with the ordi
nary internalconversion 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 2.76Mcv and 1.38Mev 7ray 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 internalconversion 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
higherenergy 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 174 Mev
Fig. 5.6 Energy spectra of the positrons produced by pair internal conversion, in
competition with the 2.76Mev and 1.38Mev 7ray 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 internalconversion coefficients 7.1 X 10~ 4 for the 2.76Mev transition and
0.6 X 10~ 4 for the 1.38Mev 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 Kshell internalconversion coefficients, pair internalconversion coefficients,
and TT angular correlation); the ft transition is allowed (log .ft = 6.1). A competing
feeble (0.003 per cent) 4. 17 Mev, secondforbidden ft transition is omitted in the
figure, as is also a 0.04per cent crossover E4 yray transition.
in > transitions, ordinary single quantum yray 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 7ray emission. Thus the internal
conversion coefficient is infinite.
228 The Atomic Nucleus [CH. 6
Emission of a Single Nuclear Internalconversion 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 Kshell
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
1.412Mev 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 longrange 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 1.412Mev
level in RaC' emits more than twice as many longrange a rays as all the
other 11 excited levels put together.
A second experimental example of the > 0, no, transition appears
to be the 0.7Mev 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 evenZ evenN composition
of this nucleus. Because of the absence of 7 rays, the 0.7Mev 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 /STelectron
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 6.04Mevexcited level in O 16 . Because of the
low Z and high W, no singleelectron "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 positronnegatron 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 6.04Mev > 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 gausscm
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 (sameZ, sameA), with clearly dis
tinguishable properties such as different radioactive halfperiods, 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 longlived excited levels was first made on
sound theoretical grounds by Weizsiicker (W27), who pointed out in 1936
that 7ray decay of levels whose excitation energy W is small should be
delayed by an easily measurable amount if the angularmomentum
change A/ is large.
a. Longlived 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 longlived excited level of UZ (T^ = 6.7 hr).
AU
i i i i i i i i i
8

.... QddJV
"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 oddA isomeric pairs displays "islands oi
isomerism" (F18) in which the oddnucleon number is less than 50, or less than 82.
The solid bars represent the number of oddZ evenAT cases, while the cvenZ oddJV
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 socalled 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 halfperiod 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
angularmomentum values with the singleparticle 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 7ray emission and of the internalconversion 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 pre1951 theories were
valid only in a few limiting cases.
b. Shortlived 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 halfperiods
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 shorterlived group is made up of Ml,
M2, and E2 transitions. Experimental techniques for exploration of the
millisecond domain await development and systematic utilization.
c. Halfperiod for Isomeric Transitions. The halfperiod would be
that of 7 decay if the excited nucleus were stripped of its atomic electrons.
Internal conversion shortens the halfperiod by providing an alternative
mode of deexcitation. Then, from Eq. (5.3), the halfperiod 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
2.6min 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 halfperiod (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
highenergy 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 halfperiod 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 secondfor
bidden transition, for which the selec
tion rule involves no parity change.
The shape (L5) of the 0ray 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 lowenergy ft transition of Cs 187 has a measured decay energy of
0.51 Mev and a partial halfperiod of 33 yr/0.92 ~ 36 yr. The shape
(O6, L8) of the /3ray spectrum corresponds in the theory of forbidden
ft decay [Chap. 17, Eq. (3.22)] to an angularmomentum change of 2,
accompanied by a change in parity (A7 = 2, yes). This is a first
forbidden transition according to GamowTeller selection rules and is in
quantitative agreement with the long halfperiod, 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 0.66Mev 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 wellrecognized
isomeric level whose halfperiod is 2.6 min. This implies that there is an
angularmomentum difference of several units between the ground level
and the 0.66Mev level in Ba 137 . Among the choices available for the
0.66Mev level we must elect 7 = TT This decision is independently
established by measurements of the internalconversion coefficients for
Stable
Fig. 7.1 Decay scheme of Cs 137 , with
angularmomentum and parity assign
ments for all levels.
7]
Nuclear Effects of Nuclear Moments
233
the transition to the ground level. For a 0.66Mev 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
0.66Mev 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 0.66Mev 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 angularmomentum
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 magneticresonance method (D9).
c. /37 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 10.7min isomeric level in Co 60 is
nearly the same as the cross section for formation of the 5.2yr ground
level of Co 60 , both through the reaction Co b9 (71,7) Co 60 . The 10.7min
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 1.33Mev 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 2.50Mev "crossover"
7 ray is not observed and has an abundance less than 2.5 X 10~ 7 of that
of the 1.17 plus 1.33Mev cascade (F55).
Problems
1. On a basis of the singleparticle 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 2.50Mev 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 7ray transitions will be isotropic in the labora
tory coordinates. There is no preferred direction of emission for the
7ray photon from the individual transition I A * IB, because the atoms
and nuclei are oriented at random. The same is true for aray, /3ray,
and conversionelection 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 twostep cascade transition, such as I A I B > /r,
there is often an angular correlation between the directions of emission
of two successive 7ray 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 ay, 07, /3e~ (where e~ means a conver
sion electron), 7e~, .... 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 magneticangularmomentum 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. DipoleDipole Angular Correlation. Any pure 77 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
/C0 *
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 dipoledipole, 7179
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 7172 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 dipoledipole, y7
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 foreandaft symmetry. The
angle # is actually in centerofmass co
ordinates, but the nuclear recoil from
Tray 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 foreandaft
symmetry which arises from even
powers of cos # is evident. This
foreandaft symmetry, in centerofmass 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 77 cascades which have been measured thus far are 4(2)2(2)0,
or quadrupolequadrupole cascades, in evenZ evenJV nuclides such as
Mg 24 , Ni 60 , etc.
b. General Case for 77 Angular Correlation. The principles which
we have just outlined can be applied to any 77 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 77 cascade /A(ZI)/B(/S) Jc the
238 The Atomic Nucleus fen. 6
angularcorrelation function W(&) for the angle d between the successive
7 rays can be shown to be (Yl, F5, B67)
iZ,
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 overall index of the complexity of the angular distribution
is given by the socalled 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 "backtoback" 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, ay, py, 77, 76', 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 halfperiod 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 halfperiod 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 243kev
In 111
49 in 62
2.8 day
'EC
"* Jr0.177
i Ml
Mev
y0.243
E2
Mev
Fig. 8.5 The 77 cascade in Cd 111 , follow
ing the electroncapture transition In 111
Cd 111 . The influence of an external mag
netic field on the 77 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. 77 Angularcorrelation Coefficients. Hamilton (H14), Falkoff
(F4), and others have deduced the angularcorrelation coefficients o 2 , a 4 ,
... for most of the multipoles expected in 77 cascades. As illustra
tions, we give in Table 8.1 the coefficients (H14) for some of the possible
dipole and quadrupole 77 cascades for the important case I c = 0, as
met in evenZ evenTV nuclei.
TABLE 8.1. ANGULARCORRELATION COEFFICIENTS FOR SOME DIPOLE AND
QUADBUPOLE 77 CASCADE TRANSITIONS
[When the angularmomentum quantum number Ic for the final level, such
as the ground level of evenZ evenTV nu elides (H14, B107).]
W^(#) ^^ ~ (1 H~ *a cos 2 i? ~\" o>4 cos 4 i?) dn
77 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 77 cascades
which follow the 0ray 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 77 cascade in Ni 80 , and is in agreement with a
quadrupolequadrupole 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 77 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 77 cascade appears to be 1(1)1(2)4, while in Cd 111 (Fig. 8.5)
the 77 cascade 0j(l)d 9 (2)sj seems well established (F66).
e. Parity of Excited Levels. 77 Polarizationdirection Correlation.
The 77 angularcorrelation 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 yy 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 1.33Mev and 2.50Mev
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
7ray triple coincidence polarimeter, shown in Fig. 8.7, and measured the
polarizationdirection correlation of the 77 cascades in Ti 46 , Ni 60 , Pd 106 ,
and Ba 134 .
Hamilton's (H14) theory of the polarizationdirection 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 quadrupolequadrupole cascade, the polarizationcorrelation when
= 90 is
L > 1 f or E2E2 (I A and I c have same parity)
= 1 for E2M2, or M2E2 (I A and 7 C have opposite parity)
^L < 1 for M2M2
x
and I c have same parity)
242
The Atomic Nucleus
[CH. 6
Fig. 8.7 Tho 7ray triplecoincidence 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 KlcinNishina 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 77 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, E2E2, the plane of the B vector tends to he
parallel to the plane of the two y rays. For two magnetic quadrupoles
M2M2, 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 polarizationdirecti _>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 evenZ
TABLE 8.2. Py ANGULAR CORRELATION (F5)
Note that the angular distribution of 187 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
Firstforbidden .
Any
Any
1
1 + Oz COS 2 tf
Secondforbidden . . .
Dipole
1 + a 2 cos 2 &
Secondforbidden
>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
evenTV 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. 07 Angular correlation Coef
ficients. Angularcorrelation coeffi
cients for the successive emission of
any two nuclear radiations, such as
a7, 07, 7e~, . . . , 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 07 angular correlations,
irrespective of the character of the
interaction, the generalizations
shown in Table 8.2 apply to the com
plexity of the angulardistribution
function W(d) of Eq. (8.5). Near
the lowenergy 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 07 angular correlations
Fig. 8.8 Polarizationdirection 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 Iriplecoinridenrc 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 JVj/JVj_ < 1 at
= 90, hence /fiL > ]  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 7ray quanta, in the 0(1)1(1)0 cascade is about 55.
2. In a 77 angularcorrelation experiment show that the directional correla
tion will be disturbed if the halfperiod T of the intermediate level does not
satisfy the inequality
where AP is the hyperfinestructure 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 halfperiod 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 angularmomentum 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 inelasticresonancescattering 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 twostep 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 centerofmass 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 singlevalued 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 centerofmass 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 angulardis
tribution data suffices to determine
7 = 2+ for the 19.9Mev 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 entrancechannel 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 angulardis
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 ccnterofmass 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
pwave 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 7ray 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 antiparallelspm 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 eenterofmass 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 7ray
threshold at 2.22 Mev and falls as the 7ray 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
andaft 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 centerofmass cooi>
dinates. [From Butler (B146).]
Problems
1. In the nuclear reaction A + a+ B >c + C, show that, in centerofmass
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 foreandaft symmetry in centerofmass 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 centerofmass 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 wellknown 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 physicalchemical equilibrium constants are tem
peraturedependent. 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 wellcollimated beam
of highenergy 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 singlefocusing 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 vacuumtube 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 iiuiiradiogiiiic 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 massspectroscopic 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 massspectroscopic
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
threeelectron 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 FabryPe*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 isotopeshifted 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 hyperfincstructure 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 4,244A line where U 238 U 236 = 0.251 A
(3145).
Isotope Shift in Samarium. 8
That the isotope shift in heavy jgS
elements is not due to nuclear mass J
alone is well illustrated by com
parison of the massspectroscopic
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 aradioactive, and for a time this activity was
thought to be associated with the isotopeshift anomaly and to be due to
Sm 162 . However, further studies, using enriched isotopes, have now
proved that Sm 147 is the aactive isotope (R7, W13). The implications
of the anomalously large isotope shift between fl 2SmJg and fl 2SmJS 2 seem
to be related to the neutronshell 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 nuclearvolume 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 isotopeshifted pattern. Some intermediateweight 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 nuclearvolume 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 Pomcmiue 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 rotationalangularmomentum 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 vibrationrotation 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 VibrationRotation 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 vibrationrotation bands occur in the near infrared, in the neighbor
hood of 2.5 to 10j* 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 massspect.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, MoOawHffl 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 massspectroscopic 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 massspectroscopic 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 chemicalatomicweight 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
atomicweight (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
atomicweight 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 (5D
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 atmosphericoxygen
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
atomicweight 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 gramequivalent 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 freshwater 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. Massspectrographic 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 aray 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 longlived
0rayemitting 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 irm/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 fissionproduct 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 halfperiod 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
industrialplant scale. As an enormously useful byproduct 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 highintensity 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 beamcurrent 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 halfperiod 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 oddZ have one or, at most, two stable isotopes.
The elements of evenZ often have many stable isotopes. Thus there
are many cases in which one or two pure isotopes of elements having
evenZ 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 halfperiod 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. Molecularbeam 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 0ray emission, if E is the maximum energy of the 0ray 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 0.049Mev y ray?
(b) If the 0.049Mev 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
overall 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. Overall 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 onehalf 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 multiplestage poroustube 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 overall 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
largescale 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 isotopediscriminating 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
overall 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 singlestage 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 cyjnder
(~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 overal)
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 = gUCAfrJfO/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, selfdiffusion, 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 10fold enrichment over the normal C"/C 12 ratio.
4
tl
Heavy
Light
 Heavy fraction 
Fig. 8.3 Schematic explanation of the
ClusiusDickcl thermaldiffusion 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 overall enrichment fartor 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, singlestage 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
100fold 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. SzilardChalmers 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 internalconversion
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 protonelectron model, all oddZ nuclei
would contain an odd total number of protons plus electrons.
Both the proton and electron are known from direct evidence to be
FerrniDirac 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 halfinteger nuclear angular momentum, / (Chap.
4).
Nuclear Angular Momentum of N 14 . The earliest and soundest experi
mental contradiction to the protonelectron 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 EinsteinBose. 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 oddZ 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.
ElectronNeutrino 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
electronneutrino pair (Chap. 17) gives a satisfactory overall account
of the emission of both positron and negatron ft rays. The mechanism
by which the electronneutrino pair arises during the nucleon transition
is as yet obscure. It is clear, however, that the electronneutrino 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 oddZ
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 1Mev 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 neutronproton 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 neutronproton model of nuclear constitution.
Quantitatively, however, the detailed course of such reactions is in accord
with theoretical deductions based on the neutronproton 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 nuclearelectron 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) electronneutrino pains = (34 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 neutronproton 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 EinsteinBose 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 pionnucleon 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 evenZ
evenJV 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
clearcut generalization from these data is that by weight more than 85
per cent of the sampled earth consists of cvenZ evenN 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 evenZ evenN nuclides (Chap. 4, Table 4.1) and that of the
remainder all but four have either evenZ or evenAT.
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).]
EvenZ
OddZ
Element . . .
uO
i 4 Si
26 Fe 20 Ca
i 2 Mg
W A1 n Na iK
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 elementbuilding 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 halfperiods 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 "freezingin" 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 (halfperiod ~ 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 Radiativecapture cross sections for 1Me.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 radiativecapture (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 halfperiods are comparable with the age of
the universe.
Halfperiods. The nuclides which occur in nature can therefore be
defined in terms of their halfperiod T\ as
~ 10 9 yr < T* < * (3.1)
In addition, there are in nature some 40 shorterlived 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 longlived 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 . Betaray transitions have yet to be found
from V 60 , although this nuclide satisfies other criteria for instability.
Mass data, obtained from massspectroscopic 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 evenZ evenA r , hence probably 7 = 0. The ft transi
tions against A/ = 6 may well have such a long halfperiod 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 /3ray 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
oddA 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 massenergy valley is called a nuclear energy surface.
Relative Mass of OddA Isobars. Cross sections of the valley, in
planes of constant A, have a characteristic appearance. For oddA, the
relationship between atomic mass M and nuclear charge Z is as shown in
Fig. 3.2. The lowest isobar in the mastenergy 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 evenZ oddJV or oddZ evenA 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
} Jfperiod 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 massenergy valley
is more complicated. Successive
isobars no longer fall on a single
parabola. The isobars of evenZ
evenJV fall on a lower parabola and
therefore have more tightly bound
nuclear structures than the alter
nating oddZ 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 eveneven and odd
odd character of the protonneutron
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 evenZ, 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 oddZ oddN isobar on the upper parabola. When this is ener
getically impossible, both evenZ evenJV isobars are stable. Among the
known stable nuclides there are 54 pairs of stable evenZ even AT isobars
and four cases in which three evenZ evenN isobars are stable (A = 96,
124, 130, 136).
Double ft Decay. The only alternative transitions between a pair of
"stable" evenZ evenN isobars would be by the simultaneous emission
of two ft rays, or double ft decay. The theoretical half period for the
emission of two electronneutrino 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 radiationdetection techniques, arranged for the observation of
ftft coincidences, can at best explore down to a halfperiod 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 OddAT
/
.
V
rEv
,
nZ EvenW
/
/
/
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 evenA isobars, showing the charac
teristic doublevalued 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 halfperiod > 10 20 yr is presently the
only positive experimental evidence which distinguishes between two
formulations of 0decay 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 ncgatronantincutrino pair is
emitted:
p+n
Double 0~ decay requires the emission of two Dirae antineutrinos, and
the calculated halfperiods 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 halfperiods, ^ 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 "electronneutrino 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 evenZ [exceptions: (a) H 2 , Li 6 , B JO , N n , which have oddZ
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 evenZ: 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 evenJV (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 oddZ: There are never more than two stable isotopes. The
element is usually simple, and if so its only stable isotope is invariably
evenJV. In 10 cases there are two stable isotopes, both evenTV (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 evenJNT: 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 evenZ (up
to five for N = 82). There may also be stable oddZ isotones, numbering
0, 1, or 2 (only for N = 20 and N = 82).
2. For oddiV: There are never more than two stable isotones. Usually
there is only one stable isotone (example: aOj 7 ), and if so this is invariably
evenZ (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 oddZ 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
oddZ 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
oddZ oddN 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 excessneutron 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 oddZ elements to be
simple and evcnJV is the analogue of the tendency for the oddTV isotone,s
to be simple and to have even/?. The large number of stable isotopes
found for evcnZ elements is the analogue of the large number of stable
evenjV isotones. These large frequencies for evcnZ, and analogously
but independently for evenAT, 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 closedshell 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
excessneutron 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 liquiddrop 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 liquiddrop 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 wellfounded 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 shortrange 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 shortrange 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 socalled 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 massspectroscopic 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 massspectrosropic 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 ProtonNeutron 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 Accurding 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 righthand 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 protonneutron 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 ziX 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 0activc 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 neutronhydrogen mass difference (M n Mn) is about
0.78 Mov [Eq. (4A2), Chap. 3], all ft transitions for which the total /3ray
and 7ray 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 neutronhydrogen 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 evenZ evenJV, and N = Z.
They correspond to a sequence of completed ''fourshells" 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 fourshell 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 singlevalued 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 fourshells 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 twobody
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
liardball 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 evenZ evenJV 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 selfconsistent 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 evenZ
evenN 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 evenZ evenA 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
closedshell effects which may be displayed by your graph.
4. (a) Show that for a nucleus A\ to be energetically unstable against aray
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
heavyparticle 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, Nl)+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 lefthand 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 righthand 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 evenZ cvenJV nuclides are at or neai
the top and the oddZ or odd# nuclides form the lower envelope.
< 24, as computed from the selfconsistent 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 evenAT nuclides, the last neutron is more tightly bound; for
example, in 3 LiI, S n = 7.2 Mev. When, in addition, the nuclide has
eveiiZ, then S n > 15 Mev. These are the even2 evenAT 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 evenA 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 onehalf arises because each n represents the difference between
an evenAT and an oddJV nucleus.
3. There is a complete parallelism between S n for oddN and evenJV
nuclides and the behavior of the proton separation energies S p in oddZ
and evenZ 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. Independentparticle, central force
(Hartree) models, whose wave functions do not correspond to preformed
a particles but whose firstorder energies may show a marked fourshell
structure in LS coupling (F23), are favored over pure a models (112).
For a closer match with experimental results, independentparticle 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 fourshell 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
Lcoupling model, these four lowest levels correspond to the orbital
quantum numbers I = 0, and I = 1, m z = 1,0, +1.
Pointbypoint 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 fourshells are
offered, as in Be 8 , there is insufficient binding force between subshells to
form a stable nucleus. Three fourshells are required, as in C 12 , before
the forces between subshells are sufficient to form a stable evenZ eveiiAT
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 atomicmass values are unknown, by using nuclearreaction
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 groundtoground 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 neutronhydrogen 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 evenAT
than for oddA", 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
evenJV 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 groundtoground 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 halfperiod of 2.7 days. Au 198
emits a simple negatron spectrum with a maximum energy of 0.963 Mev followed
by one 0.412Mev 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 oddZ have so few stable isotopes (and
none of even A unless A = 2Z), while neighboring elements of evenZ 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 twobody
nucleonnucleon 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 nucleonforce field). Third, we must apply the
nucleonstructure theory to the general problem of calculating nuclear
energy levels.
The main features of nucleonnucleon interactions have become clear
as a result of much experimental and theoretical work, but the twobody
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 twobody 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 quantummechanical effects associated with
spin. The atom possesses a predominant central particle which is the
origin of a longrange 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 manybody
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 Twobody Forces between Nucleons.
The number of important twobody 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 socalled 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 evenparity, Sstate 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 evenZ and to triplet interactions for oddZ.
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 protonpairing energy and an approxi
mately equal neutronpairing 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 shortrange 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 TWOBODY SYSTEM
Sti
ate
Force
Operator
Ev<
mt
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 Sstate 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 shortrange. 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 nucleonnucleon interactions can be visual
ized from simple considerations outlined below, using a centralforce,
nonexchange (Wignerforce) approximation. The boundstate '(np)
interaction is obtained most simply from the theory of the deuteron, while
the continuum of unbound states is explored in np and pp scattering
experiments.
2. Ground Level of the Deuteron
a. Wave Function for the Rectangularwell 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 righthand side of Eq. (2.15) is small compared with
unity, and the lefthand side can be roughly represented by cot (x/2).
Then, approximately,
Db
AA 2 A 2
~: ( I =
\2/ 2M
1.0 X 10 24 Mevcm 2
(2.16)
c. Shape of the Deuteron Groundlevel 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 onequarter 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 rectangularpotentialwell 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 wavemechanical 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 Centralforce Potential Wells. A number of poten
tials U(r) besides the rectangular well have been studied extensively.
As long as these correspond to shortrange 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 "welldepth
parameter" and "intrinsicrange 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 rectangularwell model, is
given approximately by
D
3]
Forces between Nucleons
317
3. NeutronProton 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 np 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 centerofmass ,
coordinates (A19). This means that only Sstate interactions (I = 0)
are involved in this energy range. On the reasonable assumption that
some Pstate 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 neutronproton
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 np
scattering cross section. The cross section for the competing radiative
capture reaction H 1 ^,?)!! 2 is only 0.05 barn per proton for 1ev 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 protoncontaining 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
np 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 "zeroenergy 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. Phaseshift Analysis of np Scattering. The theoretical descrip
tion of 5wave, elastic np 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 swave (I = 0) interac
tions, the scattering amplitude /(#) is isotropic and for any shortrange
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 swave scattering.
The swave totalelasticscattering 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
(30)
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 zeroenergy 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 zeroenergy scattering cross section
becomes
ao = 4ira 2 (3.9)
which is the same as the zeroenergy 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. np 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 np cross section a
in terms of the remaining parameters. The result, after making use of
the approximations D B, D T7, and for lowenergy neutrons such
that kb 1, is
A_ is i / r. \
(315)
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 np 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 "zeroenergy 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 swave 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 threefourths of the collisions and
to zero in onefourth 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
"zeroenergy 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 centralforce 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 rectangularwell 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 pp scattering
1 D = 11.9 Mev, to match <T O ~ 20 barns for np scattering
The "zeroenergy 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), 00
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 "effectiverange theory
322 The Atomic Nucleus [CH. 10
of nuclear forces," or "shapeindependent 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 = swave 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 "shapeindependent approximation."
g. Ground Level of the Deuteron in the Shapeindependent 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 shapeindependent 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. np Scattering Cross Section in the Shape independent Approxi
mation. For the swave, np 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 secondorder range correction
term in (V /2p) 2 appears in the final bracket of Eq. (3.23).
Equation (3.23) and also the rectangularwell 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 np 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 np 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 np 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 neutronhydrogen
molecule collision where M = Af , as compared with the reduced mass
in the neutronfreeproton 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 squarebracket 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 clearcut predictions follow at once.
3] Forces between Nucleons 325
1. If the total (up) forces are spinindependent, 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 shapeindependent 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 spindependent, '(np) ^ l (np).
2. The singlet np 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 np 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 overall 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 (groundstate molecularrotation quantum num
ber L = 1) and parahydrogcn (groundstate molecularrotation 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) radiativecapture 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 0.1ev 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 Xray 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 Xray scattering by different types of atoms in a crystal. For example,
in crystalline sodium hydride NaH the Xray 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).
Neutrondiffraction 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
(340)
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 lowtemperature 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 1A 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. Nuclearforce Parameters in the Shape independent Centralforce
Approximation. We now have enough experimental data and intercon
necting theory to permit evaluation of the four parameters which enter
the "shapeindependent approximation" or "effectiverange theory" of
nuclear forces. We summarize the theoretical and experimental rela
tionships, with their original equation numbers for ease of reference.
1. np 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 np 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 np 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 centralforce, shapeindependent 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 angularmomentum
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 centerofmass
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 np scattering cross section <r as
given by Eq. (3.15), using the phaseshift relationship of Eq. (3.14) as starting
point.
, 4, Using the parameters of the shapeindependent 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 np System
The 3 (np) and l (np) forces also govern two other ba.sic processes in the
neutronproton 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
0o.p for the radiativecapture process, Eq. (4.1), is only significant for
verylowenergy neutrons. Even then, it is very small compared
with the elastic np 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 radiativecapture 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 finestrurture
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 np 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 np 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 7ray transition between them has no change in parity.
This restricts the important radiativecapture 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 np 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 " spinflip" magnetic dipole transi
tion can be shown to be proportional to the net magnetic dipole moment
in the singlet state (B68). For the centralforce 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 photomagneticcapture 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 radiativecap
ture cross section are finite because the nuclear force is spindependent,
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
centralforce zerorange 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 0.025ev 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 0.025ev
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 lowenergy 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 neutroncapture 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
firstorder 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 centralforce
shapeindependent 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 photomagneticcapture 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,
(4Y I >'"
\2Mc/ {
3 137
These cross sections, like o c . p , correspond to the centralforce zerorange
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 tip system, in the zerorange, centralforce 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 finestructure 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
centralforce shapeindependent 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 centralforce shapeindependent
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 shapeindependent (effectiverange) 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 photoelectriccapture cross section o oap (E])
and evaluate o p (El) in barns per proton for thermal neutrons. Compare with
the photomagneticcapture 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 0.025ev
neutrons.
2. Numerically evaluate o oap for the photomagnetic capture of thermal neutrons
(0.025 ev in L coordinates) by protons, using the centralforce zerorange model,
Eq. (4.6). Ans.: 0.30 barn per proton.
5. The ProtonProton 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
Sstate (I = 0) interactions will be overwhelmingly stronger than the
(pp) force for Pstate 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 oddl (P, F, . . .).
With primary interest thus centered in only one mode of interaction,
the singlet S state, detailed information on the (pp) twobody 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 np scattering experiments. However, the theoretical
interpretation of the lowenergy pp scattering is much more complicated
than for np scattering. This is due in part to wave interference effects
produced by the joint action of the shortrange (pp) force and the
"infiniterange" coulomb force.
Studies of pp 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 pp 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 swave 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 2.4Mev 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
foreandaft 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 headon 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 swave 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 pp Scattering from to 10 Mev. The
existence of a shortrange attraction between protons was demonstrated
in 1936 when White (W40) and Tuve, Heydenburg, and Hafstad (T32)
first observed the anomalous (nonMott) 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 centralforce 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 pp 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 swave 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 pp 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
shapeindependent 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 rr
0.10
30 40 50 60
70 80 90
* + (C coordinates)
Fig. 5.2 Angular distribution of pp 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 pp 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 prp 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 pp system is unbound. Therefore the diproton, 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 pp Repulsion. In Fig. 5.2, the observed pp 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 swave effect.
344
The Atomic Nucleus
[CH. 10
Upon analysis (H7), these effects are interpreted as evidence for a small
negative pwave 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 veryhigh
energy protonproton 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 pp 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 psp 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 twobody evidence on the force between two neutrons is
extremely limited. Scattering experiments, analogous to the np and
pp 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 Dineutron. There is one piece of negative evidence, the
nonexistence of a stable dineutron (C29). This might be expected by
analogy with the nonexistence of the diproton He 2 , because, even with
the coulomb field turned off, the pp 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
twobody 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 Sstate 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 bindingenergy 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 bindingenergy 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 pp and nn 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, clearcut quanti
tative conclusions have yet to emerge from the theoretical and experi
mental work on elastic nd 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 shapeindependent approximation, which characterizes the effec
tiverange 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 twobody phenomena, including:
1. Small binding energy of the deuteron
2. Large size of the deuteron
3. Large cpithermal np cross section OQ 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 pp scattering
11. Instability of the diproton and dineutron
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 shortrange 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 np 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 eveni but
changes sign and is repulsive for P states and all other states of oddZ.
To the extent that the lowenergy 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 lowenergy predictions,
and it is in accord with the small Pstate repulsion seen in the pp 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 *jks()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) shortrange, (b) nonelectric, (c) nongravitational,
(d) nonmagnetic, and (e) spindependent.
2. What experimental evidence shows that the (nn) and (pp) forces are sub
stantially equal?
8. Effects of Tensor Forces
Among the lowenergy 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 lowenergy 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 lowenergy
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 lowenergy 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 threebody problem, H 3 and He 3 , and the fourbody
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 lowenergy 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 lowenergy 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 evenZ (S, D, . . .)
and only change the sign of the force to a repulsion for states of odd/
(P 9 Fj . . .). The lowenergy data involve a P state only in the photo
electric disintegration of the deuteron (*S > 3 P), where the Pstate inter
action was taken as negligible in Eq. (4.20), and in the small (< 1 per
cent) anomaly in the pp 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 lowenergy 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. Highenergy np and pp Scattering
The (np) and (pp) twobody 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 np 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 highenergy scattering and
conditions within an average nucleus.
Both np and pp 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 lowenergy data. We shall indicate
here the general character of the experimental findings. The interpreta
tion of these highenergy 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. np Scattering above 10 Mev. Incident neutrons up to 20Mev
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 260Mev 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
higherangularmomentum states. At 90 Mev, for example, some 90 per
cent of the total cross section is still attributable to /Sstate scattering
(C21).
Exchange Forces in Highenergy np Scattering. The influence of
exchange forces and of ordinary forces on highenergy 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 260Mev 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
lowenergy collision data that the depth of the np 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 centerofmass 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 halfandhalf mixture
of ordinary and exchange force. This mixture is the socalled 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 np 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 np scattering below ~10 Mev, the effective range
for highenergy scattering is dependent on the assumed shape of the
potential well. Analytically (p. 501 of R4) this corresponds to the addi
tion of a shapedependent term Pk*r\ to the righthand 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 centralforce 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 np scattering distribution, it fails completely when applied
to highenergy pp scattering.
The fundamentally important result of the highenergy np scattering
studies has been the most suggestive experimental demonstration of exchange
forces in a twobody interaction.
b. pp Scattering above 10 Mev. In pp scattering the identity of
the two particles produces three effects which are not present in np
scattering.
1. For each proton scattered at any angle tf (C coordinates) the
partner is scattered at TT #. Therefore the experimental results always
display foreandaft 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 pp cross
sections inherently four times larger than the corresponding np cross
sections.
3. The Pauli principle excludes all oddZ singlet states and all evenZ
triplet states. Therefore the pp 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 pp scattering up to 340 Mev differ
sharply from what would be expected on a basis of the np 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 Sstate 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
PP
^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 pp 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
pp 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
pp 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 pp scatter
ing at 90 cannot be explained (C22) by potentials of the types
used by Christian and Hart (C21) for rap 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 pp and np
scattering data at all energies.
59]
Forces between Nucleons
355
c. Shortrange 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 shortrange
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 spinindependent 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 spinindependent shortrange 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 hardcore repulsion, and a triplet potential
U + *U with a negligible hardcore 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 highenergy np and pp 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
375Mev singlet depth
69Mev triplet depth
0.60 X 10~ 13 cm hardcore 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 Highenergy pp Scattering.
At sufficiently high energies, the singlet Swave 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 pp 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 repulsivecore model best predicts the observed singlescattering cross
sections, it does not appear possible at present to match the experiments
on pp double scattering with this same potential (O8).
Qualitative Effects on the Repulsive Core on Highenergy np Scattering.
In contrast with pp scattering, the np 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 overall effect, because the np
scattering is dominated by the triplet interaction.
Effects of the Repulsive Core on the Lowenergy Data. Some of the
effects of a hardcore singlet interaction on lowenergy 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 repulsivecore model
cannot eventually be found which will simultaneously satisfy the low
energy and highenergy 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
independentparticle models emerge. We shall discuss the shell model
as an example from the broad group of independentparticle models.
The extreme opposite view is that of very strong interaction between
all the nucleons in a nucleus. As representatives of the stronginteraction
models, some aspects of the liquiddrop 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 evenZ evenN nuclides, 7 = 0.
For oddZ oddN 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 twovalued 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 socalled "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 oddJV 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 lowlying excited levels in nuclei, in contrast with
the close spacing of highly excited levels (Chap. 11, Fig. 4.1)
17. Existence of resonancerapture reactions, such as (n,,y). Con
stancy of the fastneu troncapture 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
pendentparticle shell model. The liquiddrop 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 stronginteraction
models (liquiddrop 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 individualparticle
models (F54) involving closed shells of 2(21 + 1) neutrons or protons,
where I is the angularmomentum 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 individualparticle models arid centered around the development of
Bohr's idea of a liquiddrop nucleus and around Wigner's (W47) uniform
model, supermultiplet, and isobaricspin 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 liquiddrop model and the uniform model are inherently
incapable of predicting such discontinuities. Attention swung sharply
back to the individualparticle 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 spinorbit forces for individual nucleons,
a sequence of independentparticle states emerges which matches the
experimentally known "magic numbers." The justification for intro
ducing a strong spinorbit 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 shortrange potential well. Secondly, this potential is assumed
to be the same for all values of L
The Weakinteraction Paradox for Lowlying States. In the assumed
central potential, each nucleon is imagined to be capable of describing
an orbit of welldefined 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 welldemonstrated 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 weakinteraction 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 independentparticle 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 shortrange 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 Spinorbitcoupling 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 spinorbit 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 " spiuorbitcoupling model," or "^"coupling
model," involves the following assumptions (M25, M26), in addition to
those which are inherent in every independentparticle 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 lowlying 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 spinorbit
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 shortrange potentials, such as a threedimensional harmonic oscilla
tor potential U = UQ + ar z (B48, H24), or hybrids of oscillator and rec
tangularwell 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
spinorbit
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 independentparticle model using a deep rectangular
potential well (left). Center, the empirical splitting of j I + J and j = I $
states, which is attributed to spinorbit 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 spinorbit 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 spinorbit 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 jjcoupling 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 groundlevel 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 lowlying excited levels
Ml <[ 0.159 Mev of odd A nuclei correspond to occu
8 W J stabe pation of higherenergy 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 (104day 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 lowlying 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 groundlevel 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 groundlevel 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 spinorbit 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 7ray 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 1Mev neutrons, as compared with an ' 'average 7 ' heavy nuclide (^0,10 barn)?
3. The Liquiddrop Model
The liquiddrop 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 liquiddrop jnodel is the antithesis of the independentparticle
models. JThe interactions between nuclepns are assumed tp^ be. strong
instead of weak. Nuclear ieyels__are i jepresratedjs_quantized states
of the~M^earBytienras a whole and jiotjis states of a single particle in
an average fielcT TKeTrqurddr^~^^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 stronginteraction, liquiddrop 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 liquiddrop 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) socalled
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 spinindependent
as well as chargeindependent. If the coulomb force is turned off,
(np) = (nn) = (pp)
3. These nuclear forces have a shortrange 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 longrange 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 constantdensity 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 quantummecEamcaT 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 liquiddrop 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 lowestenergy 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 deepestlying 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 higherenergy 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 asymmetryenergy 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 liquiddrop model.
In Wigner's theory, the isobaricspin quantum number T, when pro
jected onto the f axis in a hypothetical isobaricspin space, has the value
T{ = v(N ~~ Z), or onehalf 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 onequarter 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 oddA 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=dropinodel
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 spinindependent forces. To correct
for this omission, we add the pairing_eneryJ .as .another correction term
B 4 for the totallunfeuQii^rgy., Conventionally, the pairingenergy cor
rection is usually taken as zero for odd 4 nuclides (F38, B96, K35).
Then for even^4 nuclides
for evenZ evenAf /* 10^
aforoddZodclAT 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 evenZ even.V \ (3.13)
for odd7 oddTV )
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 liquiddrop model, the average pairing energy should be
expressed as some smooth function of A, Fermi's (F41) empirical value
5a^ (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 oddZ and those which have oddJV. 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 oddZ evenAT nuclides, and by 5 = c/2 for
evenZ oddTV nuclides. In the present discussion we shall omit this
interesting (C46) refinement.
Closed Shells. The Ijauiddrpp _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 liquiddrop 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 coulombenergy unit radius, Eq. (2.14) of Chap. 2, which is
fto = (145 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 OddA 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 ASYMMETRYENERGY COEFFICIENT a 0 FROM
Z AND A OF STABLE ODDA 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 asymmetryenergy coefficient a a to
the coulombenergy 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 (326)
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# oddN:evenA
for oddZ evenAT : odd A ,
Oforeven2oddJV:oddA t 3 ' 3 ^
6 for evenZ evenJV : 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 = 2yZo, and is
M(Z*,A) = aA  2yZoZo + 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 oddA
(3.39)
Equation (3.30) is the parabolic mass relationship for oddA 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 oddA (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 electroncapture 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 oddA (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 EvenA 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 evenA (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 evenZ evenAT, 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 evenA mass parabolas
M (Z,A)  M (Z ,A) = 7(Z  Z ) 2
, (25 for oddZ \ A /0 eox
+ I } evenA (3.53)
1 for evenZ) '
The reaction energy Q, for Z (Z 1) at constant evenA, becomes
= M(Z 9 A)  M(Z 1, A)
' I O* *. ^JJ *7 1
evenA (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) oddA (3.55)
For even A, the oddZ oddJV nuclides are unstable, so that we need
study only the evenZ evenJV 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
evenZ evenAT nuclides is contained implicitly in Eq. (3.54). For all
0stable nuclides, Q & < 0. Then from Eq. (3.54),
Z*  Z\ (3.56)
a
<T/^ 2
for stable evenZ even2V nuclides, while
1 . ,
Z " ~ Z (3>57)
for unstable evenZ evenJV 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.
(evenZ evenJV) 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 > pi 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 0decay 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 0decay 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 mussis 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 liquiddropmodel treatment of the average
bindingenergy 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 coulombenergy difference
of light mirror nuclei, Eq. (3.16), and determined all four energy coef
ficients from a leastsquares adjustment to the mass data for 0stable
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 evenN evenZ. 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 leastsquares 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
AW = 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
08)
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
31
*:.i. x r i nr !
ti?
50 82 126
III U I I 1 LJl
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
groundlevel energetics are quite well represented by Eq. (3.15). How
ever, the liquiddrop 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 surfaceenergy 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 lowlying excited levels in nuclei. The
liquiddrop 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 nuclearfission 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 shortrange forces, as represented by surface tension, and the
longrange 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 longrange coulomb force a greater
advantage over the shortrange 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
aray emission, with a halfperiod 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 halfperiod 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 halfperiod 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 liquiddrop 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

oOddA
EvenA
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 evenZ nuclides. The partial
halfperiods 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 halfperiods
for the even4 isotopes of evenZ elements is a linear function of the fissionability
parameter Z*/A. The partial halfperiods arc several orders of magnitude greater
for the odd A isotopes than for the evenyl 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 evenZ nuclides from Z = 90 to 98. This
systematic relationship was first pointed out by Seaborg (S24, G20) and
is useful in predicting the spontaneousfission rates for undiscovered
nuclides. In general, the odd A nuclides have longer partial halfperiods
than the evenZ 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 7ray emission,
then fission may be a principal consequence of the neutron capture.
We have seen that S n is appreciably greater for evenJV' than for
oddAT 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 oddN target (say ozU 236 ) compared with an evenAT
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 evenN than in one having oddN. 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 Lowenergy 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 lowto
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 7ray 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 socalled " 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 "neutronrich," because N/Z is appreciably
greater for /3stable heavy nuclei than for 0stable middleweight nuclei.
TABLE 3.5. ASYMMETRY CHARACTERISTICS OF THE MASSYIELD CURVES FOR
LOWENERGY 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
peakto
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 lowenergy fission. It is also observed in the massyield
curves for the spontaneous fission of U 288 and Th 282 , which have been
evaluated by massspectroscopic 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 lowenergy fission of U 28 ', U m , and Pu*" by thermal neutrons, and
for the lowenergy 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 massyield curves for lowenergy
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 massyield distribution for the fission of
excited TJ 218 compound nuclei which are produced by bombardment of
Th 282 by 37Mev 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 lowenergy 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 " +
37Mev 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 peaktotrough ratio
from 600 to about 2. For veryhighenergy fission, e.g., that of Bi 209
bombarded by 400Mev 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 lowenergy fission.
These have been summarized and their inadequacies discussed by Hill
and Wheeler (H53), in connection with their discussion of the socalled
394
The Atomic Nucleus
[CH. 11
collective model of nuclei, which blends somewhat the individualparticle
and the liquiddrop 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 sGe 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
"magicnumber 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 5.8min 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 electroncapture 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. 7ray photons of sufficiently large quantum energy can be reliably detected
without interference from lowenergy radiations by the use of socalled "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 liquiddrop model to predict (say, 0.5 Mev)
the minimum photon energy which can produce the 12hr 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 timeofflight 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 evenZ evenJV nuclide, such as U 236 , than it is in a nearby evenZ oddJV
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 halfperiod (!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 halfperiod 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 /Jstable 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 lowestlying
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 oddnucleon,
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 lowlying 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 independentparticle 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 yray 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 7ray
emission or its everpresent 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 7ray emission or internal con
version to any lowerlying 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
quantummechanical 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 welldefined 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 + 2nh/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 lowestlying 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 lowlying
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 8Mev
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 welldefined quasistation
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 welldefined 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 overall 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 Gamowtype 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 socalled 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 Mevcentimeters 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 swave
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 welldefined 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 lowenergy 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 lowenergy 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
16Mev 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 16Mev 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 (412)
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 onelevel 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 7ray 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 swave 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 swave 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 7radiation 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" nuclides, i.e., nurlides which do not
involve a closed shell of neutrons. Within the accuracy of these values,
there is no marked dependence of D Q on the angular momentum / of the
target nucleus, nor on whether Z is odd or rven (so long as Z is not a
closed shell).
All the target mirlides studied have evenJV (except TiLuiof). Those
which have N = 50, 82, or 126 exhibit abnormally small capture cross
sections cr(n,7). This is to be correlated with the relatively small binding
4] Models of Nuclei 407
of the 51st, 83rd, or 127th neutron, whose separation energy we have
formerly found to be abnormally small. The compound nucleus (Z,
N + 1) therefore has a smaller excitation energy E when N = 50, 82, or
126. Consequently its level separation is expected to be greater and
its <r(n,y) smaller than for nonmagic target nuclides. The experimental
data suggest that, in addition to this expected effect, the