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BOOK

THE ATOMIC NUCLEUS

Robley D. Evans, Ph.D.

PROFESSOR OF PHYSICS MASSACHUSETTS INSTITUTE OF TECHNOLOGY

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

THE ATOMIC NUCLEUS

1955 by McGraw-Hill, Inc.

All Rights Reserved

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

T M H Edition

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

New York.

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

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

Preface

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

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

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

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

vi Preface

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

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

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

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

Preface vii

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

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

Keferencr tables of many of the reasonably well-established nuclear properties accompany the corresponding text. For more comprehensive tables, explicit references arc made to the voluminous and valuable standard compilations. For the latest data, thcsn compilations must be augmented by the1 Miminarics of new nuclear data published quarterly in NurJcar iS'ci'rwr Abstracts.

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

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

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

viii Preface

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

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

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

ROBLEY D. EVANS

Contents

Preface V

INTRODUCTION

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

CHAPTER 1 CHARGE or ATOMIC NUCLEI

Introduction 6

1. Chemical Origin of Atomic Number 6

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

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

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

5. The Displacement Law 25

CHAPTER 2 RADIUS or NUCLEI

Introduction 28

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

2. Coulomb-energy Difference between Isobars . 31

3. Coulomb Potential inside a Nucleus 38

4. The Nuclear Potential Barrier .... 45

5. Wave Mechanics and the Penetration of Potential Barriers 49

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

7. Anomalous Scattering of a Particles ... 81

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

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

CHAPTER 3 MASS OF NUCLEI AND OF NEUTRAL ATOMS

Introduction 96

1. The Discovery of Isotopes and Isobars . 96

2. Nomenclature of Nuclei 98

3. Mass Spectroscopy 101

4. Atomic Mass from Nuclear Disintegration Energies 117

5. Tables of Atomic Mass 135

CHAPTER 4 NUCLEAR MOMENTS, PARITY, AND STATISTICS

Introduction 140

1. Nuclear Angular Momentum 141

x Contents

2. Nuclear Magnetic Dipole Moment 148

3. Anomalous Magnetic Moments of Free Nucleons 151

4. Relationships between / and M 155

5. Electric Quadrupole Moment 163

6. Parity . . 174

7. The Statistics of Nuclear Particles 177

CHAPTER 5

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

Introduction. ... 1S1

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

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

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

CHAPTER 6 EFFECTS or NUCLEAR MOMENTS AVD PARITY ON

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

Introduction . .... 202

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

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

3. Lifetime in tf Decay . - 20f>

4. Radiative Transitions in Nuclei . 211

5. Internal Conversion 218

6. Nutlfur Ipomers 22*. *

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

and 7- Transit inn Probabilities , 2H2

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

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

CHAPTER 7 ISOTOPIC ABUNDANCE RATIOS

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

1. Ratios from Mass Spectrosoopy . . 250

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

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

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

5. Chemical and Physical Scales of Atomic Weight . 202

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

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

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

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

CHAPTER 8 SYSTEMATIC^ OF STABLE NUCLEI

Introduction 276

1. Constituents of Atomic Nuclei 276

2. Relative Abundance of the Chemical Elements 279

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

Contents xi

CHAPTER 9 BINDING ENERGY OF NUCLEI

Introduction. ... 294

1. Packing Fraction . . 294

2. Total Binding Energy .... 295

3. Average Binding Energy . 297

4. Separation Energy for One Nucleon 302

CHAPTER 10 FORCES BETWEEN NUCLEONS

Introduction. ... 309

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

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

3. Neutron-Proton Scattering at 0 to 10 Mev . 317

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

5. The Proton-Proton Force at 0 to 10 Mev . . . 338 G. Equivalence of (/in) and (pp) Forces . . . 344

7. Summary of Central Forces . . 345

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

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

CHAPTER 11 MODELS OF NUCLEI

Introduction 357

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

2. The Nuclear Shell Model 358

3. The Liquid-drop Model . . 365

4. Statistical Model of Excited Levels 397

CHAPTER 12 CONSERVATION LAWS FOR NUCLEAR REACTIONS

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

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

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

CHAPTER 13

NUCLEAR REACTIONS, ILLUSTRATED BY Bl°(«,p) AND TTS ASSOCIATES

Introduction .... . 422

1. Energy Distribution of Protons from BH\rt,p)C113 423

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

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

5. Energy Loss in Inelastic Scattering . . . 438

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

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

Introduction. . . 441

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

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

xii Contents

CHAPTER 15 RADIOACTIVE-SERIES DECAY

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

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

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

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

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

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

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

7. Growih of a Granddaughter Product . . 486

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

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

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

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

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

CHAPTER 16 SPECTRA

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

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

2. Genealogy of Nuclides Which Emit a Ra}rs ...... 517

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

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

CHAPTER 17 0-RAY SPECTRA

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

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

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

3. Fermi Theory of 0 Decay . . ............ 548

CHAPTER 18 lONIZATION OF MATTER BY ClIARQED PARTICLES

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

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

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

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

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

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

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

CHAPTER 19 ELASTIC SCATTERING OF ELECTRONS AND POSITRONS

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

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

CHAPTER 20

RADIATIVE COLLISIONS OF ELECTRONS WITH ATOMIC NUCLEI

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

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

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

Contents ziii

CHAFTEB 21 STOPPING OF ELECTRONS BY THICK ABSORBERS

Introduction 611

1. Path Length and Range of Electrons 611

2. Thick-target Bromsstrahlung . 614

3. Range-Energy Relations for Electrons 621

4. Annihilation Radiation 629

CHAPTER 22 PASSAGE OF HEAVY CHARGED PARTICLES THROUGH MATTER

Introduction. . 632

1. Capture and Loss of Electrons . 633

2. Energy Loss per Unit Path Length 637

3. Range- Energy Relationships 647

4. lonization of Gases. 654

5. Straggling . . 660

6. Range of Fission Fragments 668

CHAPTER 23 THE INTERACTION OF ELECTROMAGNETIC

RADIATIONS WITH MATTER. COMPTON SCATTERING AND ABSORPTION

Introduction. . . . 672

1. Compton Collision and the Conservation Laws 674

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

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

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

5. Energy Distribution of Compton Electrons and Photons 692

CHAPTER 24 PHOTOELECTRIC EFFECT AND PAIR PRODUCTION

1. Photoelectric Effect. . . 695

2. Pair Production by Photons 701

CHAPTER 25 ATTENUATION AND ABSORPTION OF ELECTROMAGNETIC RADIATION

Introduction. . 711

1. Attenuation Coefficients 711

2. Energy Absorption 719

3. Multiple Scattering of Photons 728

4. Distributed y-Ray Sources 736

CHAPTER 26 STATISTICAL FLUCTUATIONS IN NUCLEAR PROCESSES

Introduction 746

1. Frequency Distributions 747

2. Statistical Characterization of Data 757

3. Composite Distributions . 766

CHAPTER 27 STATISTICAL TESTS FOR GOODNESS OF FIT

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

1. Lexis' Divergence Coefficient 774

xiv Contents

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

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

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

CHAPTER 28

APPLICATIONS OF POISSON STATISTICS TO SOME INSTRUMENTS USED IN NUCLEAR PHYSICS

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

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

2. Scaling Circuits. . . ... 794

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

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

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

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

APPENDIX A

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

Introduction. . . . 819

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

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

APPENDIX B

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

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

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

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

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

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

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

APPENDIX C THE WAVE MECHANICS OF NUCLEAR POTENTIAL BARRIERS

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

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

lar Barrier .... . 852

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

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

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

AlTJfiNIlIX D

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

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

Contents xv

APPENDIX F

Table of the Elements 900

APPENDIX G

Somo Useful Inofficiont Statistics 902

Bibliography 905

Glossary of Principal tfymhols 930

Index 953

INTRODUCTION

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

The earliest speculations on the atomic hypothesis of the ultimate structure of mattci are ascribed to the Ionian philosophers of the fifth century B.C. Anaxagoras, Leucippus, and Democritus postulated that all matter is made up of a set of particles which were called atoms to denote their presumed indivisibility. Their concept of a world made up of invisible, incompressible, eternal atoms in motion is best known now through the writings of the Latin poet Lucretius (98 to 55 B.C.), especially through his six-book scientific poem " Concerning the Nature of Things1' (De Rerum Natura) (Dl).f

Bodies of things are safe 'till they receive

A force which may their proper thread unweave,

Nought then returns to nought, but parted falls

To Bodies. of their prime Originals.

. . . Then nothing sure its being quite forsakes,

Since Nature one thing, from another makes;

. . . LUCRETIUS

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

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

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

1

2 The Atomic Nucleus

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

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

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

By this time a rays from radioactive substances were under intensive study. Following Rutherford's (R42) semiquantitativc observation of the scattering of a rays by air or by a thin foil of mica, ( JeigiT (() 10) found the most probable angular deflection suffered by a rays in pas.sing through 0.0005-mm gold foils to be of the order of 1°. (loiger and Marsden (G13) had shown that 1 a ray in 8,000 is deflected more than 00° by a thin platinum film. The Thomson model had predicted only small deflec- tions for single scattering and an extremely minute probability for large deflections resulting from multiple scattering. The predict ioius of the Thomson model fell short of these experimental results by at least a factor of 10in. 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. Rutherford1^ theory did not predict the sign of the nuclear chtirRi*, but the electronic mass and the X-ray and spectral data indicated thai it must, be positive, with the negative electrons distributed about it to form the neutral atom.

The quantitative dependence of the1 intensity of «-ray scattering on

Introduction 3

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

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

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

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

The Atomic Nucleus

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

Advance	Date	By whom	Where
First experimental basis for the atomic hypothesis. Chemical combining weights Atoms and molecules distinguished. Gas laws unified	1808 1811	Dalton Avogadro	England Italy
Precursor of mass number. Hydrogen as a basic unit in structure of heavy atoms. Periodic chemical classification of the elements Discovery of continuous X rays Discovery of radioactivity of urn-ilium. • •	1815 1868 1895 1896	Prout Mendeleev Rontgen Becquerel	England Russia Germany France
Discovery of electron as constituent of all atoms . . ... ... ....	1897	J. J. Thomson	England
Charge of electron measured by cloud method. Avopadro's number estimated . . Identification of <* particle as a helium nucleus Equivalence of mass and energy Number of electrons per atom estimated from X-ray scattering Isotopes, isobars identified Discovery of stable isotopes of Ne20-22. . . Atomic nucleus discovered by interpretation of a-ray -scattering results Nuclear atom model "completed" by expla- nation of origin of spectra. Quantization of atomic states . Assignment of atomic numbers, from X-ray spectra Nuclear transmutation induced; proton iden- tified	1903 1909 1905 1904- 1911 1911 1912 1911- 1913 1913 1913 1919	H. A. Wilson Rutherford Einstein Barkla Soddy Thomson Rutherford, Geiger, and Marsden Bohr Moseley Rutherford	England England Switzerland England England England England Denmark England England
Compton effect	1923	A. H. Compton	U.S.A.
Wavelength proposed for corpuscles	1924	dc Broglie	France
The wave equation . .	1926	Schrodingcr	Germany
Uncertainty principle	1927	Hoi sen berg	Germany
De Broglie wavelength observed when elec- trons diffracted by crystals a-ray decay explained as wave penetration of a nuclear barrier Discovery of deuterium	1927 1928 1932	Davisdon and Germer Gamow, Condon, and Gurney Urey	U.S.A. Germany, U.S.A. U.S.A.
Discovery of the neutron . . .	1932	Chad wick	England
Nuclear transmutation by artificially acceler- ated particles Positron discovered	1932 1932	Cockcroft and Walton Anderson	England U.S.A.
Anomalous magnetic dipole moment of proton discovered /Neutrino hypothesis	1933 1933	R. Frisch and 0. Stern Pauli	Germany Switzerland
1 Theory of /9 decay. ...	1934	Fermi	Italv

Introduction

CHBONOLOGICAL REVIEW OF SOME MAJOR STEPS IN THE ACCRETION or

EXPERIMENTAL KNOWLEDGE CONCERNING THE ATOMIC NUCLEUS

(Continued)

Advance	Date	By whom	Where
/Radioactive light nu elides discovered	1934	I. Curie and F.	France
1		Joliot
| Radioactive nuclides produced by acceler-	1934	Lawrence et al.,	U.S.A.
V ated particles		Lauritsen et al.
Transformation of nuclei by neutron capture	1934	Fermi	Italy
Anomalous proton-proton scattering	1936	White, Tuve,	U.S.A.
		Hafstad, Herb,
		Breit, etc.
± p meson discovered ...	1936	Anderson and	U.S.A.
		Ncddermeyer
Precise measurements of nuclear moments
by molecular-beam magnetic-resonance
methods	1938	Rabi	U.S.A.
Nuclear fission discovered	1939	O. Hahn and F.	Germany
		Strassmann
Measurement of magnetic moment of the	1940	Alvarez and	U.S.A.
neutron		Bloch
+ v meson discovered . .	1947	Powell et al.	England
Artificial production of v mesons ...	1948	Gardner and	U.S.A.
		Lattes

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

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

CHAPTER 1

Charge of Atomic Nuclei

The number 7. of positive elementary charges (r = 4.8 X 10~in esu, or 1.0 X 10~lu coulomb) carried by the nuclei of all i.solopcw of an ele- ment Is called the atomic number of Miai element. At least five* different experimental approaches have been needed for the ultimate1 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 error1' for all nuclei. Of course, the actual charyr Zc contains (he experimental uncertainty of the best determinations of the elementary charge c. Thus the absolute nuclear charge, like everything else in physics, is known only within expen mental accuracy.

1. Chemical Origin of Atomic Number

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

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

6

§2]

Charge of Atomic Nuclei

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

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

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

OF POTASSIUM

Element	Atomic number	Miiws numbers arid their relative abundance	Average atomic weight
:3(i	37	38	39	40	41
A	18	0.3		0.06		99.6		39.9
K	19			. L .	93.4	0.01	6.6	39.1

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

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

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

8 The Atomic Nucleus [CH. 1

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

From classical electromagnetic theory, Thomson showed that each electron should radiate, or "scatter,11 a definite fraction of the energy flux which is incident on the electron. In Thomson's theory, the fraction of the incident radiation scattered by each electron is independent rf the wavelength of the X ray. It is now known that this is true only for electromagnetic radiation whose quantum energy hv is large compared with the binding energy of the atomic electrons, yet small compared with the rest energy, m0c2 = 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 tQ ergs when it is trav- ersed for a time t sec by a plane wave of X rays whose intensity is I ergs/ (cm2) (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 m0 = rest mass of electron

c = velocity of light

ez/m0c2 = "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 =cm2 ergs/ (cm2) (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 ea is called the Thomson electronic cross section. When the most probable values of the fundamental physical constants (see Appendix E) e} ra0, 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 cm2

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/cm3, there arc (NZ) elec- trons/cm3 and (NZ AT) electrons/cm2 of absorber area as seen by an incident beam of X rays. If each electron has an effective cross sectior of t<T cma/clectron, then the total effective scattering area in 1 cm2 of area of absorbing foil is (NZ AJT) rcr "cm2 of electrons "/cm2 of foil. Thus (NZ AT) ,<r is ike fraction of the superficial area of the foil which appears to be "opaque" to the incident X rays.

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

A/ = - JNZ a Ax

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

~=-adx (2.3)

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

— = era* = 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/cm3 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/cm2, and the mass attenuation coefficient is (cr/p) cm2/g.

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

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

m 9 in carbon. These are the excitation

of fluorescence radiation following • photoelectric absorption of the X

• O • rays by K and L electrons, and the

* coherent and diffuse scattering from

m m the crystal planes in graphite. The

crystal effects were unknown at the Thomson Rutherford-Bohr tjme rf Barkla>g WQrkj but thcy ap_

Fig. 2.1 The atom model of J. J. *>™ to hfve been fortuitously aver- Thomson (T23) distributed the doc- a6ed out bv the combined effects of trons, shown as black dots, inside a wavelength inhomogeneity in the in- large sphere of uniform positive clectri- cident X rays and wide-angle geome- fication. The Rutherford-Bohr model try in the detection system. Barkla compressed all the positive charge, and recognized the influence of photoc- its associated large mass, into a small lectric absorption, which is strongly central nucleus, with thr electrons per- dependent on wavelength, and un- forming Copernicanlike orbits at dis- dertook to extrapolate this effect out tanres of the order of 10- to W time. fc comparing <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 e2/mnc2 and »iuiice on measured values of both e and e/m$. The numer- ical values of e and e/mQ were known only approximately in Barkla's time. They were sufficiently accurate to show unambiguously that the X-ray scattering would be produced by the atomic electrons, because of their small mass, and not by the positively charged parts of the atom.

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

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

§3] Charge of Atomic Nuclei 11

niquc, and better values of e and r/m0, led Barkla in 1911 (B12) to con- clude that, the mass attenuation coefficient cr/p for Thomson scattering by carbon is about 0.2 cm2/g, which corresponds to the currently accept- able value of six electrons per atom of carbon. For other light elements, Barkla concluded correctly that "the number of scattering electrons per atom is about half the atomic weight of the element."

It should be remarked that Barkla's results would have been incorrect if he had applied the Thomson theory to atoms of such large, Z that the (then unknown) electron binding energies were comparable with the relatively s'nall quantum energy [~40 kev (kiloelectron volts)] of his X rays. Secondly, if Barkla't* X-ray quantum energy had been suffi- ciently large HO that it was comparable with wnr2 = 0.51 Mev, the Thomson formula would also have been invalid because it omits consider- ation of the recoil of the electron, which was discovered much later by Oompton. At the time of Barkla's work, many phenomena now regarded as fundamental in atomic physics were unknown. The atomic nucleus had not yet been discovered, and Thomson's model of the atom was still fashionable. Bohr's explanation of atomic spectra and of the binding energy of atomic electrons and Compton's explanation of the interaction of higher-energy photons with electrons were yet unborn. The prin- ciples of Bragg diffraction of X rays by crystal planes were unknown.

With all these factors in view, Hewlett, in 1922 (H49), found a/p = 0.2 cm2/g for 17.5-kev X rays in carbon, and consequently six electrons per atom of carbon, and this result is acceptable from all viewpoints.

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

Problems

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

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

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

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

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

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

12 The Atomic Nucleus [en. 1

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

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

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

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

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

§3]

Charge of Atomic Nuclei

13

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

The use of C coordinates has a profound mathematical advantage. In both classical mechanics and wave mechanics, the use of C coordinates reduces any two-body collision prob- lem to a one-body problem, namely, the interaction of one particle having the reduced mass M 0 and velocity V with a potential field which can always be considered as centered at the origin of the C coordinates. The reduced mass M0 of a system of two particles having masses MI and M2 is

J_=_L+J_

M0 ~ Mi Mt

or

+ M,

(3.1)

Fig. 3.1 Coulomb elastic scattering of an a ray (Mi) by an oxygen nucleus (Af2), seen in the laboratory coordi- nates. The center of mass, marked C, moves through the laboratory with a constant velocity Vc which is one-fifth the initial velocity of the a ray. The impact parameter x is the separation at which the particles would pass if Lhore were no interaction between them. Note that the initial direction of Af2 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 Mz 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 Vc = 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 Tc which the center of mass possesses in

the L coordinates.

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

r

Fig. 3.2 The same collision as Fip. 3.1 but now seen as the particles actually experience it, in the center-of-mass coor- dinates. The center of mas.s, marked C, is now at rest. The total linear momen- tum is zero. Each particle traverses a true hjrperbolic 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 Mz 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 0 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 Mt is always 180° in the C coordinates but in the L coordinates has the values

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

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

for J»/t « Jl/j, *> + * ^ ][ + ?

^ ^

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

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

b . B /0 px

-r = - cot -- (3.0)

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

where ze = charge on incident particle

Zc = charge on target particle V — mutual velocity of approach

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

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

16 The Atomic Nucleus [CH. 1

attractive and for repulsive forces. For the special case of repulsive forces, as in the nuclear scattering of a rays, the collision diameter b is also equal to the closest possible distance of approach, i.e., to the minimum separation between the particles during a head-on collision. At this minimum separation the particles are stationary with respect to one another, and therefore their initial kinetic energy ^MQV2 is just equal to their mutual electrostatic potential energy Zzez/b.

d. Cross Section for Rutherford Scattering. In all collisions for which the minimum distance of approach b is significantly greater than the radius of the nucleus, the only force acting will be the inverse-square coulomb force, and Eq. (3.6) will be valid. All collisions for which the impact parameter lies between 0 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 0 is the area of a disk of radius x, or

•w

*(> 0) =7rz2 = ~62cot2| (3.8)

Thus the cross section for backscattering (0 > 90°) is simply ir62/4, which is the area of a disk whose radius equals the collision radius 6/2.

For 0 = 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 0 and 0 + d0 is the area of a ring of radius x and width dx, or

da = \2vx dx\ = *- 62 cot f esc2 f d© (3.9)

4 22

The solid angle d!2, into which particles scattered between 0 and 0 + d0 are deflected, is

dfi = 2ir sin 0 d©

A ' ® *M

= \if sin — cos — d© J 4

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

, __&_'[ !_'

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

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

§3] Charge of Atomic Nuclei 17

centimeter to normally incident a rays. If the cross section of each scattering center is a cm2/atom, then the scattering centers comprise the fraction crN As of the total area of the foil. Then if no a rays are incident normally on the foil, n a. rays will be scattered in the directions repre- sented by the particular value of the cross section a being used. The fraction so scattered is simply

- = <r(N As) or — = AT (N Aa) (3.11)

7l(j Tlo

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

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

f. Experimental Verification of Rutherford's Nuclear Atom Model. In proposing that all the (positive) charge in the atom should be regarded as concentrated in a small central nucleus, Rutherford made use of experimental results which had been obtained by Geiger on the'angular distribution of the a rays scattered by a thin gold foil. These results were in sharp contrast with the predictions of the Thomson model of the atom, but they were in substantial agreement with the I/sin4 (0/2) distribution of Eq. (.3.10) predicted by the nuclear model in which the central positive charge has such small dimensions that it is not reached by swift a rays even in head-on collisions (R43).

Geiger and Marsden subsequently completed a beautiful series of experiments which completely verified Eqs. (3.10) and (3.11) point by point. Their original paper (G14) warrants reading by every serious student. The angle of deflection # in the laboratory coordinates was varied in small steps from 5 to 150°; this brings about a variation in sin4 (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 1/74 term which enters all the cross sec- tions through 62 was varied in seven steps over a factor of about 9 to 1. Secondly, the nuclear charge Ze was varied by studying the scattering from gold, silver, copper, and aluminum foils. It was found that the intensity of the scattering per atom was approximately proportional to the square of the atomic weight.

18

The Atomic Nucleus

[CH. 1

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

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

g. The Equivalence of Nuclear Charge and Atomic Number. It fell to van den Broek (B125) in 1913 to collect the various types of evidence then available and to make the fertile suggestion that the charge on the atomic nucleus ?s actually equal to the atomic number. Bohr adopted this suggestion and developed his quantum theory of the structure of atoms and the ori- gin of spectra. This theory pre- dicted that the frequency of the X-ray lines in the K scries should increase with the square of the charge on the atomic nucleus, i.e., with Z2. Moscley's observations of these X-ray lines showed instead that the frequency v is substantially proportional to (Z — I)2, if it be assumed that the charge Z on the nucleus equals the atomic number and that the atomic number of aluminum is 13. Mosclcy sug- gested correctly that the effective charge on the atomic nucleus, for K -series X rays, is about one unit less than the actual charge Z on the nucleus because of screening of the nuclear charge, especially by the one /v-shell electron which is pres- ent in the initial atomic state of any A"-series transition. Any doubt which may have persisted about this interpretation was later re- moved by Chadwick's direct measurement of the nuclear charge of Cu, Ag, and Ft by the a-ray- scattering method.

30° 60° 90° 120° 150° 180° Mean angle of scattering tf

Fig. 3.3 Differential cross section for the single scattering of a rays by a, thm foil of gold. The vertical scale represents the relative number of n rays scattered into n constant element of solid angle at the mean scattering angles tf which arc shown on the horizontal scale. The curve is proportional to I/sin4 (d/2), as predicted hy the classical theory, and is fitted to the arbitrary vertical scale at tf = 135°. The closed and open circles are the experi- mental data of Goiger and Marsden (G14) in two overlapping series of observations, one at small and one at large scattering angles. The agreement at all angles shows that, under the conditions of these experiments, the only force acting be- tween the incident a ra-ys and the gold nuclei is the inverse-square coulomb re- pulsion. The closest distance of ap- proach in these experiments was 30 X 1(T13 cm (for 150° scattering of the 7.68- Mev a rays from RaC'), and so the posi- tive charge in the gold atom is confined to a small central region which is defi- nitely smaller than this, or about JO"4 of the atomic radius.

§3] Charge of Atomic Nuclei 19

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

Scintillation detector

Baffle to stop \Or/X^ for a rays scattered

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 iJ2, as used for his direct measurement of the nuclear charge on Cu, Ag, and Pt. This annular geometry for the scattering body has subsequently been widely adapted to a variety of other scat- tering problems, e.g., the shadow scattering of fast neutrons by lead (Chap. 14).

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

Problems

1. A thin gold foil of thickness As cm has N atoms of gold per cubic centi- meter. Each atom has a nuclear cross section a cm2 for scattering of incident a rays through more than some arbitrary angle @. The fraction of normally inci- dent a rays scattered through more than 0 is n/n0 = <rN As. Show clearly what

20 The Atomic Nucleus [CH. 1

fraction of the incident a rays is scattered through more than 0 if the a rays are incident at an angle \l/ with the normal to the foil.

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

(r(backscatter) = TT

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

3. In an a-ray-scattering experiment, a collirnated beam of polonium a rays (5.30 Mev) strikes a thin foil of nickel, at normal incidence. The number of a rays scattered through a laboratory angle greater than 90° (i.e., reflected by the foil) is measured. Then the nickel foil is replaced by a chromium foil, and the measurements are repeated. It is found that the chromium foil reflects 0.83 times as many a rays as the nickel foil. The foils are of such thickness that each weighs 0.4 mg/cm2.

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

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

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

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

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

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

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

(e) Scattering angle in C coordinates.

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

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

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

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

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

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

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

(m) The fraction of 1.02-Mev a rays, incident normally on an Al foil 0.01 mg/ cm2 thick, which are deflected through more than the angles found in (e) or (/).

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

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

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

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

§4] Charge of Alomic Nuclei 21

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

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

(4-2)

where ?M and n2 are the principal quantum numbers for the initial and final electron vacancies (HI = 1, nz = 2, for the Ka series; n\ = 2, nz = 3, for the La series), a is the fine-structure constant (a ^TTT; «2wi0c2/2 = 13.6 ev), and all other symbols have their customary meaning and the numer- ical values given in Appendix E.

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

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

Motley's data are shown in Fig. 4.1. The plot of atomic number against v* for the Ka 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 Ka series have the form

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

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

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

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

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

22

The Atomic Nucleus

[CH. 1

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

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

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

Moseley examined the Ka X rays of 21 elements from i3Al to 4?Ag, and also the La X rays of 24 elements from 4i»Zr to ygAu. The overlap, be- tween 40Zr and 4?Ag, oriented the L series and permitted its use for bridging over the rare-earth group of elements in order to establish for the first time the atomic- numbers in the upper part of the periodic table.

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

£J 20 2 10 0			/
	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 Ka and La X-ray lines of all available elements and (fie uniform variation of v^ with integers % assignable as atomic num- bers to the 38 elements tested. Each A"« and La 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 H1 and He3.

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

§4] Charge of Atomic Nuclei 23

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

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

Electron Capture. The capture of an atomic, electron by a nucleus is an important mode of radioactive decay, which generally competes with all cases of positron £-ray decay. Several radioactive substances are known in which the transition energy is insufficient to allow positron jS-ray emission, and in which all radioactive transitions proceed by elec- tron capture (for example, 4Be7, 24Cr51, a,Ga67, 49Inul). It is generally more probable that a K electron will be in the vicinity of the nucleus and will be captured than that an L, M, . . . electron will be captured. The majority of the vacancies are therefore produced in the K shell. If Z is the atomic number of the parent radioactive substance, then (Z — 1) is the atomic number of the daughter substance in which the electron vacancy exists and from which the X rays are emitted. The existence of the electron-capture mode of radioactive decay was first established by Alvarez's observation (A22) of relatively intense Ka X rays of titanium (Z = 22) among the radiations emitted in the radioactive decay of the 16-day isotope of vanadium, 2.iV48. More rigorous experimental proof was subsequently obtained from absorption curves (A23) and from Abel- son's bent-crystal spectrometer studies (Al) of the X rays of zinc (Z = 30) which are emitted in the pure electron-capture decay of 3iGa67. Several isotopes of technetium (Z = 43) decay predominantly by electron cap- ture, and the early identification of element 43 was aided by the obser- vation of the molybdenum (Z = 42) X rays which are emitted in the decay of these technetium isotopes.

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

24

The Atomic Nucleus

[CH. 1

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

Internal-conversion transitions are therefore accompanied by X-ray- emission spectra. Neither internal conversion nor 7-ray emission involves any change in the nuclear charge, so that the X-ray spectra are characteristic of the element in which the actual nuclear transition took place. For example, the 0-ray decay of 79Au198 results in the production of the daughter nucleus 8oHg198 in an excited level which is 0.41 Mev above the ground level of Hg198. About 95 per cent of these excited nuclei go to ground level by emitting a 0.41-Mev 7 ray. The others go to ground level by internal conversion, 3 per cent in the K shell, 1 per cent in the L shell, and 0.3 per cent in the M shell. The X-ray-emission spectra are characteristic of mercury (Z = 80), not gold.

The chemical identification of a number of radioactive nuclides among the transuranium elements has been made or confirmed by obser- vations of the L-series X rays of »0Th, 9iPa, 92!!, 9aNp, 94Pu, gBAm, and 96Cm (B18).

Nuclear Isomers. Nuclear isomers are long-lived excited levels of nuclei, in which the decay by internal conversion and 7-ray emission to the ground level is measurably delayed (Chap. 6, Sec. 6). Many nuclear isomers are sufficiently long-lived to permit them to be isolated chem- ically and to be dealt with as a parent radioactive substance. The iso- meric transition to the ground level involves, no change in Z. Conse- quently, the X rays which are associated with the isomeric transition by internal conversion will be characteristic of the Z of the parent radio- active element, even if its ground level is a /3-ray emitter (for example, siSb122). This X-ray-emission property is useful in identifying nuclear isomers, especially in those cases in which isomeric transitions are in competition with 0-ray emission from the excited level (for example,

Problems

1. The wavelengths of the Ka\ 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	i»Al	2,Cu	4 2 Mo	73Ta	«u
Kah A ....	44 54	8.3205	1 5Ii74	0.7078	0 2149	0.12640
A.nd|e, A . . . .	43.5	7.9356	1 . 3774	0.6197	0 1836	0.10658

(a) Make a new table, expressing Kai and K9d€9 energy in kev.

(6) Test the simple Bohr theory: (hv)K.^ = m&*(a*/2)Z* = 0.0136Z2 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 K9df9 to Kai 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 La X rays of Ag, I, and Pt are 4.1456, 3.1417, and 1.3103 A. Taking the atomic numbers of Ag and I as known (47 and 53), deter- mine the atomic number of Pt.

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

8j64 kev

£= energy of conversion electrons

The X-ray energies for various lines of zsCu and 32Ge are

Element	Z	Ka, kev	La, kev
Cu	29	8.06	0.93
Ge	32	9 89	1 19

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

5. The Displacement Law

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

goTh"2 A BaMsTh?28 A 89MsThr8 -^ 9oRdTh»8 A

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

26

T/tc Atomic Nucleus

[CH. 1

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

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

Type of radioactive transformation	Usual ay mbol	Atomic number of initial state, or parent	Atomic number of final state, or daughter
Alpha decay	a	Z	Z - 2
Positron beta decay	0+	z	Z - 1
Electron rapture . ...	EC	Z	Z - 1
Gamma ray . . Internal conversion .	y e~	z z	Z Z
Isomeric transition Neutron emission . ...	IT n	z z	Z Z
Negatron beta decay . . .	ft-	z	2+1

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

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

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

Type of nuclear	Atomic number	Atomic number
reaction	of target	of product
(«,n)	Z	Z + 2
'«,?) W,n)	Z	Z + l
fd,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 (93Np) and plutoniuin (94Pu) were first assigned their atomic numbers from studies of the negatron ft decay of &2U239 which was formed in the reaction U288(^,T)U239. A part of this series is

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

§5] Charge of Atomic Nuclei 27

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

Problems

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

Element	Principal radiations	Half -period
1 2 3	a, 7, X ray ftr, 7, X ray X ray (no 0~, /3+, or 7 ray)	490 yr 6.6d 40 d

Each of these three elements emits the same line spectrum of X rays, which is characteristic of a certain atomic number Z. The LaZ line of this spectrum has a quantum energy of 13.79 kev. It is known that the LaZ line of ai.C'ru (curium) has a quantum energy of 14.78 kev, while the La2 line of 9oTh (thorium) has an energy of 12.84 kev.

(a) From the X-ray data, determine the atomic number Z of the atoms which emit the 13.79-kev La-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 g2U235 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 Kal line of 92U23B is 0.1267 A, calculate the wave- length to be expected for the Kai line of the stable atoms formed in (a).

(c) The observed value for the Kai 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 92U23B to that of the nucleus of the nuclide formed in (a).

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

CHAPTER 2

Radius of Nuclei

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

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

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

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

1. The Growth of Concepts Concerning the Size of Nuclei

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

28

§1] Radius of Nuclei 29

We shall discuss the contemporary interpretation of the experiments on anomalous scattering later, in Sec. 7. Here it is worth noting that the early efforts to interpret these results, in terms of collisions which could be described by classical mechanics, led to the introduction of a number of ad hoc, if not bizarre, models of the inner structure of atomic nuclei. Some of these models had to stay in vogue for over a decade because no more acceptable model could then be found. These included Chad- wick's (CIS) "platelike a particle'7 and Rutherford's (R49) "core-and- neutral-satcllite " nucleus which contained a small positively charged core, surrounded by other nuclear matter in the form of heavy but uncharged satellites moving in quantized orbits, under a central 1/r6 law of attraction which was attributed to polarization of the neutral satellites. The early speculations on the idea of neutrons are visible in this model.

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

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

The existing experimental evidence also supports the view that within the nucleus the spatial distribution of positive charge tends to be sub- stantially uniform; thus the protons are not appreciably concentrated at the center, the surface, the poles, or the equator of the nucleus. Small asymmetries of the distribution of positive charge are present in some nuclei, as is known from the fact that many nuclei have measurable electric quadrupole moments. These charge asymmetries are discussed in Chap. 4; here we note that, if the positive charge in a nucleus is regarded as uniformly distributed within an ellipsoid of revolution, then the largest known nuclear quadrupole moment (of Lu176) corresponds to a major axis which is only 20 per cent greater than the minor axis of the assumed ellipsoid. In most nuclei the corresponding ellipticity is only of the order of 1 per cent. Therefore we may regard most nuclei as having nearly uniform and spherical internal distributions of positive charge.

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

30 The Atomic Nucleus [CH. 2

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

R = fl0A* (1-1)

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

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

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

#o = (1.5 ± 0.1) X 10-13 cm (1.2)

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

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

RQ = (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 cm3 = 3.1 X 10~87 cm3. 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 cm3. Thus the nuclei occupy only about 2 parts in 1014 of the volume of the solid material. The density of nuclear matter is then of the order of 1014 g/cm8.

§2]

Radius of Nuclei

31

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

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

Experimental phenomenon which depends on nuclear radius	Basic physical principles on which the method rests	Type of mechanics which can provide an interpretation of the observations
1. Energy of radioactive jS-ray decay (coulomb-energy difference between isobars)	Coulomb energy of a sphere of charge	Classical
2. Isotope shift in line spectra 3. Elastic scattering of fast electrons by nuclei 4. Characteristic electromagnetic radi- ations from /u-mesonic atoms 5. Fine-structure splitting of ordinary electronic X-ray levels in heavy atoms	Coulomb potential in- side a sphere of charge	Wave
6. Lifetime of «-ray emitters 7. Anomalous scattering of « rays 8. Cross section for nuclear reactions produced by charged particles, such as («,n), (a,2n), (p,7i), etc.	Penetration of nuclear potential barriers by charged particles	Wave
9. Elastic scattering of fast neutrons by nuclei	Diffraction of un- charged matter waves	WTave

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

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

2. Coulomb-energy Difference between Isobars

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

w - W-* -

(2.1)

32 The Atomic Nucleus [GH. 2

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

*--!£* (2-2)

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

TF«,-|^Z(Z-l) (2.3)

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

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

In ft decay, the mass number A does not change, and therefore R does not change, at least not within the domain of validity of the constant- density model R = RoA*. In positron 0 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 Emu[ of the positron 0-ray spectrum) plus the rest energy of the positron (m0c2) 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^ + m0c2 (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 Mp and the product neutron Mn. We shall derive general expressions for all these contributions in Chap. 11, but we need not await those generalizations in order to estab- lish the one special case with which we are concerned here.

Fowler et al. (F61, B130) first drew attention to a group of nuclei, which undergo ft decay, in which the binding energy due to short-range nuclear forces between the nucleons is substantially the same in both parent and product. These nuclei constitute a series of so-called mirror nuclei, one example of which is the isobaric pair, OJB and N1B, in which O16 undergoes positron ft decay to stable N1B, according to

8O16 -» 7N15 + 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 6Cn to ziSc41 have just one more proton than the number of neutrons. Their stable decay products each contain just one more neutron than the number of protons; hence each of these particular pairs of parent and product are mirror nuclei. In each of these nuclei, the mass number A is

A = 2Z - 1 (2.6)

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

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

As an example, consider the isobaric pair 801B and 7N1B 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 yN14. Adding one proton to this structure gives us O1B, whereas adding one neutron gives us N1B. We can express the mass of the O1B and N16 nuclei as the mass of their constituent protons and neutrons, diminished by the net binding energy resulting from the short-range attractive nucleon forces and from repulsive coulomb forces. Then the mass of the nuclei of O1B and of N1B can be expressed as

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

1B)

[(7M p + 7Mn) + Mn] - [(nucleon binding energy) - Wml] (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

Mp - Mn - ATPnuc + AH^ (2.9)

in which AlFcolli has been defined and evaluated in Eq. (2.4), while ATVrnuo 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

- (Mn - Mp) - AJFnllc (2.10)

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

Em = AJFeou! - 1.80 Mev - ATFnuo (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 R0

Emn = |-£ A* - 1.80 Mev - ATFnuo (2.13)

5 /to

Table 2.1 lists the current (H61) experimental values of the maximum positron energy -Bmax 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 = 0 axis at -1.80 Mev. Thus in Eq. (2.13) we find AJFnuo = 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 RQ and corresponds in Fig. 2.1 to about

RQ ~ 1.45 X 10~13 cm (2.14)

Dotted lines for R0 = 1.4 and 1.6 X 10~18 cm are shown; they clearly bracket the probable value of /?0 for the coulomb-energy unit radius.

§2]

Radius of Nuclei

35

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

ENERGY Em^ OF THE POSITRON fl DECAY SPECTRA IN THE MIRROR NUCLEI

FOR WHICH A = 2Z - ]

Z Element A	#max, MCV	Z Element A	Em**, Mev
5B 9		14 Si 27	3.48
6C 11	0.99	15 P 29	3.94
7 N 13	1.24	16 S 31	3.9
8O 15	1.68	17 Cl 33	4.1
9F 17	1.72	ISA 35	4 4
10 Nc 19	2.18	19 K 37	4.6
]1 Na 21	2.50	20 Ca 39	5.1
12 Mg23	2.99	21 Sc 41	4.94
J3A1 25		22 Ti 43

-2

14

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

36 The Atomic Nucleus [CH. 2

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

Atf— - ~ (2Z - 2) (2.15)

5 K

- 1.80 Mev - ATT... (2.16)

5 /to

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

The coulomb-energy unit radius B0 — 1.45 X 10~ia cm obtained from Fig. 2.1 is in good agreement with the nuclear radii obtained for these same nuclei by other methods. We may note that this constitutes some degree of verification of the factor £ in Eq. (2.2). Physically, the factor 7 is due to the assumed uniform distribution of charge throughout the volume of the nucleus. For example, if all the charge were on the nuclear surface, this factor would be £ instead of f , and the coulomb unit radius would be only about 1.2 X 10~13 cm.

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

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

2. The coulomb-energy radii of nuclei having A < 41 follow the constant-density model Rmu^ = R0A* and have a unit radius of RQ ~ 1.45 X 10~18 cm.

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

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

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

Wml =* \£ (Z(Z - 1) - 0.77Z*] (2.17)

o ri

instead of the classical expression of Eq. (2.2). The correction term — 0.77Z*(3e2/5#) arises from the antisymmetry of the proton wave func-

§2] Radios of Nuclei 37

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

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

flo~ 1.2 X 10~13 cm (2.18)

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

Problems

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

2. Calculate a predicted value for the maximum kinetic energy of the positron ft rays emitted by (a) 12Mg" -> ft+ + nNa23 and (6) 13A1» -> ft+ + 12Mg» using the constant-density nuclear model, with R = 1.45 X 10~13A*.

(c) Compare with observed values found in tables.

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

38 The Atomic Nucleus [CH. 2

3. Coulomb Potential inside a Nucleus

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

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

e JT" iP(r)Z7(r)4irr2 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 UA(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 Uv(r) at distance r from the center can be shown to be

§3]

Radius of Nuclei

39

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

= e

t2(r)(Vv - U)4*r* dr

(3-5)

where V = Ze/r for the point nucleus and the integration extends only throughout the nuclear volume 0 < r < R.

The three potentials, V(r) for a point nurleus, Us(r) for a surface- charged nucleus, and U1t(r) for a uniform volume-charged nucleus, are compared graphically in Fig. 3.1, from which a qualitative idea of the

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

o

o

I

Ze

^Extranuclear region

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

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

It is foiind experimentally that there are a number of elements having two or more stable isotopes which differ in mass number by two units. Examples include H2Pb2"4, Pb2"6, Pb208; 8,>Hg'!'fi, Hg19«, Ilg200, Hg202, Hg204, etc. Under high resolution, certain lines in the emission spectra of these elements will be found to consist of a number of closely spaced com- ponents, one for each isotope of even mass number. These are the isotope-shifted components in which we are interested here. Each of these components is itself single, i.e., it is not further split into a group of hyperfinc-structure components, because the nuclear moments are zero-valued. [The actual spectral "line" will generally contain other components which arise from one or more stable isotopes whose mass numbers are odd, for example, Pb207, Hg201, 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 5dn6p — » 5dn6s. 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 Rz arc different for the two isotopes. More exactly, the "electromagnetic radius," which is the true meaning of 7? in Eq. (3.5), is found to be larger in the heavier isotope. Of course, the nuclei of both isotopes contain the same number of protons and have the same total charge Zc. If the heavier isotope were formed from the lighter isotope by merely adding two extra neutrons to the outside of the lighter nucleus, and not also increasing the proton-occupied volume, there would be no isotope shift. Thus the very existence of the isotope shift shows that the protons in the nuclei of both isotopes move in regions of different size. The penetrating s electron serves as a useful probe because it spends a part of its time actually within the nuclear volume, and its noncoulomb interactions with protons and neutrons are negligible.

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

Figure 3.2 summarizes (B123, F59, B61) the present experimental data on 19 elements in a form which allows comparison with the predic- tions of existing theory. It will be rioted that the observed isotope shifts are about one-half as large as the calculated shifts if the nuclear unit radius is taken as R0 = 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!0 = 1.1 X 10~13 cm. Important improvements in the use of isotope shift as a means of studying the inner structure of nuclei (C52) can be expected as spectroscopic investigations are extended to enriched or separated isotopes and as advances are made in the theory (B83), especially with regard to the evaluation of the elec- tronic wave functions.

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

§3]

Radius of Nuclei

41

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

100

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

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

42 The Atomic Nucleus [CH. 2

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

tfo -^ (1.1 ± 0.1) X 10-13 cm (3.6)

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

c. Characteristic Electromagnetic Radiations from ^-Me sonic Atoms. The properties and behavior of w mesons and /i mesons are now rather well understood (M14, M15, P29, Bf>3, T25). Bombardment of nuclei by high-energy particles or photons (*> 150 Mev) (Bll) can evoke the emission of positive or negative TT mesons from the target nuclei. Because of their positive charge, the Tf mesons are repelled by nuclei. They decay with a mean life of about 0 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 ra0 of the electron or positron, the rest masses of the TT meson and \i meson are close to Mr ~ 273m0, 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 orbit7* 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?2m0, will be about 200 times smaller than the corresponding radius for an electron, because of the larger rest mass of the i±~ meson. As all the mesonic " atomic states" are unoccupied, the \r meson will fall to states of lower energy, the transitions being accompanied by the emission of character- istic electromagnetic radiation, or of Auger electrons, and taking place within a time of the order of 10~13 sec. In the " K shell," the /*" meson will be some 200 times nearer the nucleus than is a 7if -shell electron, and the p~ meson will therefore spend an appreciable fraction of the time within the nucleus itself. The life of the individual uT meson mav

§3] Radius of Nuclei 43

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

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

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

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

In Pb, the radius of the ^-mesonic K shell for a point nucleus (RQ — 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 R2Z*, where R — R^A* is the radius of an assumed uiuform distribution of charge within the finite volume of the nucleus (F52, C42, W33), The shift is greatest for the Is level, much less for the 2pj level (L\\ level in X-ray notation), and still smaller for the 2^,. level (Lu\ 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 2p3 — > Is transition in nine elements. Comparison with the calculated values for a point nucleus (7?0 — 0) and for a homo- geneously charged nucleus having a unit radius of RQ = 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, RQ = R/A^ which fall in the domain

RQ = (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 105 tons/mm3. 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 ^lO17 g/cm2. This circumstance

44

The Atomic Nucleus

[CH. 2

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

The fine-structure splitting of the ^-mesonic "X-ray11 spectra appears to have been resolved in the experiments by Fitch and Rainwater (F52), Tn electronic X-ray spectra, the electron spin of T gives rise to the two fine-structure levels 2p± (or Ln) and 2pa (or Lin). Transitions from these to the ls$ (or K) level constitute the Ka* and Kai lines, respectively, and the Kai energy slightly exceeds the Kaz energy. For the /z-mesonic levels in Pb, a fine-structure splitting of about 0.2 Mev is observed for the

0.2,

20

50 60 70 80 100

30 40 Atomic number Z

Fig. 3.8 Energies of the ji-mesonic transition 2pj — > Is, which corresponds to the Ka\ line in electronic X-ray spectra. Calculated values for point nuclei (flo = 0) and for homogeneously charged nuclei with /20 =* 1.3 X 10~n cm are shown by the two'curves. The experimental values obtained by Fitch and Rainwater (F52) are shown as open circles.

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

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

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

§4] Radius of Nuclei 45

is minute when compared with the shifts in /i--mesonic atoms. Schawlow and Townes (87) have summarized the pertinent theoretical and experi- mental material, showing that a homogeneously charged nucleus, whose electromagnetic unit radius is fi0 ^ 1.5 X 10~18 cm, should produce a change of only 0.3 per cent in the fine-structure separation of the 2p» and the 2p} levels for Z = 90. The effect diminishes rapidly for smaller Z. Schawlow and Townes found that the existing data on the Kai-Ka2 X-ray fine-structure separations for Z = 70 to 90 appear to be in agreement with an electromagnetic unit radius of R0 ~ 1.5 X 10~18 cm. Further improvements in X-ray energy measurements and in the theory of X-ray fine structure would be required in order to improve the accuracy of this estimate of R0.

Problems

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

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

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

4. The Nuclear Potential Barrier

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

As we decrease r, we finally come to some small distance which is of the order of the nuclear radius of mercury. Here the short-range attractive nuclear force begins to be felt, and as we continue to decrease r this attractive force increases until it just equals the coulomb repulsive force, leaving zero net force between the two particles. On decreasing r still further, the attractive force dominates, and the two nuclei coalesce. If the original nucleus was eoHg204, then the addition of an a particle (2He4) forms MPb208. Now BaPb208 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 Hg204 and He4 nuclei.

Figure 4.1 is the usual schematic illustration of the potential energy

46

The Atomic Nucleus

(en. 2

U as a function of distance r for such a system. The simplest model is the so-called square-well model, in which the potential energy of the bound system is taken as constant and equal to — f/0 for r = 0 to r =* R} while at r — R the potential energy increases discontinuously to the coulomb value (Ze)(ze)/R. For r > R the potential energy consists only of the coulomb energy (Ze)(ze)/r. In the square-well model, R is called the nuclear radius.

Zzv2

Barrier height, B= ~-

K

-B

R 0 u

r

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

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

(4.1)

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

b. Modifications Due to Short-range Forces. Clearly the discon- tinuities in this square-well model are unrealistic. The simplest refine- ment is to replace the infinite potential slope at r = R by a finite but very steep slope and to round off the bottom and top of the potential well, as indicated by the dotted potential curve in Fig. 4.1. When this is done, the definition of nuclear radius requires reconsideration; it will generally be some parameter entering the analytical functions which are chosen to describe the new potential well. In some such models the nuclear radius remains defined as the position of the top of the rounded-off barrier, i.e., the distance for zero force. In other models the nuclear radius may signify the point of maximum slope within the potential well. Moreover, the distance r signifies only the separation between the centers of the two particles, Zc and ze. Each particle has an assignable radius of its own, and the radius of the ze particle will obviously depend on whether ze is a proton, an a particle, or even some larger nucleus such as, say, O16. If Ze and ze are regarded as uniformly charged spheres, it is well known that their external electrostatic fields are the same as though

§4] Radius of Nuclei 47

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

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

This marked difference in behavior between short-range and long- range forces has to be recognized in those models in which nuclear radius signifies some particular point along the mutual potential energy curves of Fig. 4.1, such us the top of the barrier. For example, if Ze and ze are spheres having radii Rz and Rz, then their surfaces first make contact when the centers are separated by r = K7i + Rz. 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 < (Rz + Rz) the nuclear attractive forces will just balance the coulomb repulsion. This is the "top of the barrier," and it corresponds to some separation r lying between the radius of the huger nucleus and the sum of the radii. When the joint action of long-range and short-range forces is included in the model, a more realistic definition of barrier height #, in terms of nuclear radii Rz and RZJ would be

where Rz < r < (Rz + Rz) (4.3)

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

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

In the particular case of the radioactive decay of 92U238, 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 90Th234.

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

30

I"

£ 20

15

10

5

^-B-28 Mev

Coulomb potential demonstrated by scattering of a rays by uranium N

tr

10 20

50

60 70 80

30 40

r in 10"13cm

Fig. 4.2 The coulomb barrier to « particles (z = 2) for Z about 90 or 92. The region of the solid curve beyond r = 30 X 10~13 cm is verified by direct a-ray-scatter- ing experiments. If the 4.2-Mev a rays of U238 were emitted classically, i.e., over the top of the barrier, the coulomb potential would have to atop at about r — 60 X 10~13 cm. Classical mechanics is therefore unable to provide a simple, single model which can account for both observations.

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

The subsequent development of the wave-mechanical treatment of the interaction of charged particles with potential barriers provided a satisfactory description of a wave mechanism whereby particles can pene- trate through potential barriers, instead of being required to surmount them as they must in classical mechanics. In the case of the uranium decay, the evidence is now that the radius of the residual nucleus is about 9.3 X 10~1! cm and the barrier height about 28 Mev. The 4.2-Mev uranium a ray has a probability of only 10~19 of penetrating this barrier in a single collision, either from outside or inside the nucleus, but this is sufficient to account for the known radioactive half-period of U28B. We

§5] Radius of Nuclei 49

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

Problems

1. Consider the details of the collision of a 5.3-Mev a particle with a nucleus of chromium (24Cr62). Calculate the following parameters, and locate them on a plot of the coulomb barrier.

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

(c) The initial kinetic energy in C coordinates.

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

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

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

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

3. The a rays emitted by U23H 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 U23M 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 U23B nucleus and an incident 8.57-Mev a ray for the case of 160° scattering in the laboratory coordinates?

5. Wave Mechanics and the Penetration of Potential Barriers

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

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

a. Particles and Waves. The original quantum concepts of Planck (1901) introduced the quantum of action h into the theory of electro- magnetic waves. Thus the frequency of oscillation v when multiplied by h was recognized as representing the quantum of energy hv in electro- magnetic radiation. A number of physical phenomena involving light were soon found to be best understood by descriptions in terms of these photons. The simplest classical properties of electromagnetic waves in free space are the frequency vy 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/c2 = hv/c2, into the description of the physical behavior of these waves. This "dual" approach has been fruit- ful in the theoretical description of black-body radiation, of the photo- electric effect, of the Comptoii effect, and of many other phenomena. Corpuscular properties arc conferred on waves by the introduction of h.

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

X = - (5.1)

P

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

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

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

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

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

§5] Radius of Nuclei 51

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

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

wave velocity in free space

wave velocity in medium

(5.2)

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

wavelength in free space

wavelength in medium

(5.3)

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

w = \P (5.4)

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

% dw dv ., ..

( }

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

X = - de Broglie (5.6)

P

W

v = — Schrodinger (5.7)

h

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

W = iJlf r» + U __ _ (5.8)

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

52 The, Atomic Nucleus [CH. 2

where M = mass of particle

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

hW W V , U W

w = \v =__=_=_+

. --- .

p h p 2 MV V2M(W - U)

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

d /V2	\M(hv - U)'	) -£ <5'n) (5.12)
dv\	h _ M P

m_m ^ d_ jV2M(W - U) g dv

or g

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

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

,,,_EJ> E „ wg^W ,__

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

M = ^ (5.14)

W

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

M = ^ (5.15)

00

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

V p \0

M = pr = - = — 15.16;

V0 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) = Zzez/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)

dz2 w2 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 * = -4c±»i<-«±«/w (5.20)

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

(-0

const (5-21)

54 The Atomic Nucleus [CH. 2

represents a surface of constant phase. Differentiation gives

^ = i>\ = w (5.22)

at

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

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

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

cr** = cos v? — i sin <p (5.24)

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

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

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

£-£'«

uZ " OS"

and Eq. (5.18) becomes

Separating the variables, we have two differential functions which must be equal to each other for all values of z and t and which therefore must be equal to some constant. We will call this separation constant — fc2. Then Eq. (5.18) can be written

Sc ions of these two separated differential equations include

*« = ^ (fcz) and ^(2) = e±a" (5'27)

L>Uo

™ (wfcO and »(0 = c-fc-*' (5.28)

C Oo

§5] Radius of Nuclei 55

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

k = — (5.29)

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

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

* = iKsJc-2"" (5.30)

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

A = 0 (5.31)

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

W = ~M + r (5-32)

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

and substitution of the do Broglic wavelength X = hip gives

Then Eq. (5.31) can be written

av + srw^.-n

dz2 /r

which is known as "Schrodinger's amplitude equation," or simply as tichrodingcr1 s equation. Tn three dimensions, Schroclinger's etjiiation becomes

W + ^/c^_rJO , = o (5.35J

where V2 is the Laplaciaii operator aiid, in the cartesian coordina x, y, z, has the value

v^-32 +

dx'2 dyz' ' 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 27- (\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^'dc2 to obtain a term proportional to p2, 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 A2 = -7?2, or .4 = ±iB. The choice of sign here is arbitrary, and most commonly the minus sign is chosen, so that A = — iB. Then the wave function of Eq. (5.36) becomes

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

* (539)

v

ih dt dz2 hz

which is Schrodinger1 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* + g2 = A* (5.40)

58 The Atomic Nucleus [CH. 2

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

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

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

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

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

z = 1 (5.42)

then \t\*dxdydz (5.43)

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

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

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

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

§5] Radius of Nuclei 59

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

Ap Arc > A s A (5.44)

^7T

Equation (5.44) can be expressed in 1111 equivalent form which is con- venient for numerical applications. For a particle having mass M, velocity V = fie, momentum p = pMCj and rest energy Me1

A(pc) AJT ~ he A(pMc2) Ax ~ he = J.97 X 10~n Mev-rm (5.44a)

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

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

2?r

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

Al A^ ~ = I radian (5.45a)

The uncertainty in kinetic energy AT1, and in the time A/ during which the energy is measured, are related by

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

2?r

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

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

60

The Atomic Nucleus

[CH. 2

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

Within this domain we must therefore forgo the possibility of experi- mental knowledge of the intimate details of the interaction. If two theories of the interaction specify two different models or mechanisms for the interaction within this domain, we have no way of experimentally deter- mining which, if either, actually occurs. This is a domain of "blackout1' which prevents our observing the mechanism of the collision too inti- mately. Within this domain, whose boundaries are set quantitatively by the uncertainty principle, we cannot reject a particular model merely

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

Incident particles

Reflected particles

C7, = 0-

©

Transmitted particles

CD

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

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

In Fig. 5.1, a particle which has mass M, velocity V, and kinetic energy T = ?MV2 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 Uz = U occurs and that this continues for a distance z = a, where the potential again drops to zero. In classical mechanics, all particles whose incident kinetic energy is smaller than U would be thrown back by the barrier, while all particles of greater energy would pass the barrier. In the wave mechan- ics neither of these statements is exactly true. A fraction of the incident particles, when represented as waves, will be reflected, and the remainder will pass the barrier when T = U. The fraction transmitted will increase when T > T7, and it will decrease when T < U. The classical values will be approached most closely when the thickness of the bar- rier is large compared with the de Broglie wavelength of the incident particles.

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

§5] Radius of Nuclei 61

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

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

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

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

Ap ~ - (5.48)

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

T, =

2M

" "*"

2M "" M "" 2M

T + -vy Ap whenever Ap « p

" * "*" M a

= T + — * -; (5.49)

This can be written as

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

a/ V

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

62 The Atomic Nucleus [CH. 2

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

hV 1 (fc/W)/F\

AT = - - = — --- I — I (m0c2)

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 — Tr} = 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,,i«i = 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 |^,ncident|2 multiplied by the group velocity Fi of the particles in region 1 ; thus

Incident flux = ^M\tVl = I^ITi (5.54)

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

§5] Radius of Nuclei 63

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

" (5-55)

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

li = .4 !<"•*•" + Bic-*i* (5.50)

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

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

k*'z (5.57)

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

n

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

^/2M (U — T) . f

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

n

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

The disturbance under the barrier is then represented by

1^2 = Azc~kzZ + B2rk*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 ft3 = 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 = 0

(5'62)

These boundary conditions give us four linear equations, from which the amplitudes A2, A3, BI, B2 can be obtained in terms of the incident amplitude .-li.

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

\Ai\*V* (5.IB)

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

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

""

4 '' I r

* i 3 " r

3

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

P = {£!-! (5-65)

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

The penetration factor will arise later in connection with our dis- cussions of nuclear barriers and nuclear reactions [e.g., Eq. (8.fm)]. In general, Tt = P(V^/Vi). It happens in the present problem that I's = Vi, because U = 0 f or regions 1 and 3. In such special cases the transmission coefficient and the penetration factor are equal.

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

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

§5]

Radius of Nuclei

65

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

while

V2M(U - T)

Hence kza = 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:2a ^ 1 ) in thickness. For barriers which are thicker than about a ~ Ai/2 (or Ar2a^2), log Tz 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

0 1 2 3 4 5 6 Barrier thickness aA,

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

The probability of reflection, (5.67)

Reflectance = 1 — L

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

or

(5.68)

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

66

The Atomic Nucleus

[CH. 2

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

Graphical Representation. It is helpful to visualize the boundary

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

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

Fig. 6.3 Graphical representation of the total wave functions ^\t ^2, and ^3 in the three regions of Fig. 5.1. There are two vertical .scales. One is an energy scale, with respect to whieh the horizontal lines show the total energy W (equal in this case to the initial kinetic energy T) and the potential energies. The second verti- cal scale is the real part (or, alternatively, the imaginary part ) of the wave functions, which are plotted with respect to the en- ergy line W as a horizontal axis. This schematic representation of ^ can be re- garded as applying to some arbitrary value of time t, because the time factor e 2ir"f is common to all three regions. The bound- ary conditions are satisfied by making ^ go straight across the discontinuities of potential at z = 0 and z = a.

(5.09)

In any region of positive kinetic energy (H' — T), it is necessary that dz\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/dz2 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 = 0 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 X2 = 27r/fc2 of a particle whose energy is (C7 — T). This condition corresponds to fc2a ^> 1 in Eq. (5.66). Such barriers could be described as either "thick" (large a) or as "high" (large /c2). 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 0 or 1 , the coefficient of the exponential term is of the order of unity. The domi- nant term it- the exponential. A plot of Eq. (5.70) for T = 0.817 is shown as the dotted line in Fig. 5.2, where the domain of validity of the approximation can be seen clearly.

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

I,-*- (5.71)

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

7 = — V2M h

[U(z) -

(5.72)

dz

Here Zi and z2 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) = 2Jfc2o (5.73)

Ft

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

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

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

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

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

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

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

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

(5.75)

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

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

X = r R(r) (5.76)

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

§5] Radios of Nuclei 69

equation, which is

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

/(/ + j)

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

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

'

The second term, therefore, has the dimensions of energy. In its denom- inator, the quantity Mr" will be recognized as the classical moment of inertia for two particles whose reduced mass is M and whose separation is r. In classical mechanics, rotational kinetic, energy can be written as J2/2/, 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 J2/2I shows that a portion of Eq. (5.79) can be identified as the angular momentum J. Thus the magnitude of the angular momentum of the wave-mechanical system of

70 The Atomic Nucleus [CH. 2

two particles is taken as _

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

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

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

Plane Wave in Polar Coordinates. In collision problems, we express a collimated beam of monoenergetic particles as the usual plane wave elk*. 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 / = 0 to I = •», must equal the plane wave eikz = el*rcoB*.

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

(21 + \)ilji(kr) P,(cos tf) (5.81)

j-o

Here ji(kr) are the spherical Bessel functions of order I, and Pz(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 = 0 to at least x ~ \ and thus including classical angular momenta between 0

§5]

Radius of Nuclei

71

and at least J = MVx ~ ft. These correspond to the ''central collisions" I = 0 of the wave theory. Similarly, collisions whose classical angular momentum lies between MVx ~ Ih and MVx ~ (/ + 1)A correspond to the Z-wave collisions of the wave theory, for which the quantized value of the angular momentum is the geometric mean between Ih and (I + l)ft,

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

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

I	0	1	2	3	4	5
Letter designation	s	V	d	f	0	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 = 0 or s-wave collisions. The s wave from Eq. (5.81) has the simple form

sin AT

— — kr

(5.82)

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

Xo = r #0(r) = -

sin kr

k

(5.83)

Transmission through a Nuclear Coulomb Barrier by s Wave.. For the I = 0 partial wave, the modified radial wave equation becomes

[W - t^(r)]xo = 0 (5.84)

14' III

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

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

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

t/W - o

r < R r> R

(5.85)

72

The Atomic Nucleus

[CH. 2

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

SrZze2

T " hV

where B = Zze*/R = coulomb barrier height

T = ?MV2 = 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 — 0 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

si0-6

V

|£ 04

S

.5 0.2

I

0 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 (isAl27), tin LoSn118), and uranium (gall288), on the assumption that these nuclei have radii R = 1.5 X 10-" A* cm.

R = 1.5 X 10-»A» cm

The transmission exponent 7 takes on a simpler form for the physically important case of a 1 ' high " barrier. When the kinetic energy T is small compared with the barrier height B, the transmis- sion coefficient T0 = 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

= 2ire2/hc = fine-structure constant

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

velocity of light

= reduced mass in terms of rest mass ra0 of electron /j/ro = nuclear radius in terms of classical electron radius r0 = = 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 ~