Physics of Nuclei and Particles [I, 1 ed.]

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© 1969, BY ACADEMIC

PRESS, INC.

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PRINTED IN THE UNITED STATES OF AMERICA

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FOREWORD

The last decade or so has witnessed an increasing rate of progress in the attack on problems concerned with nuclei. Not only have new and sophisticated techniques been responsible for a vast improvement in the quality of experimental data but also new and penetrating theoretical ideas, supported by rapid electronic computational facilities, have gone far toward consolidating the experimental gains. Models of the static and dynamic properties of nuclei have been refined and polished to account for the improved data, but we still lack a complete and general model that incorporates all of the essential features of the individual models. Significant progress has been made in dealing with the many-body aspects of nuclei, but a practical general approach has not yet been devised. Great effort has been expended in the interpretation of proton-proton and neutron-proton scattering data, but we still do not have precise knowledge of the details of the fundamental nucleonnucleon interaction. Although there has been much progress, there are still many outstanding unsolved problems; nuclear physics is far from being a closed subject. With the current intensity of interest in nuclear problems, it is particularly appropriate that there should appear, at this time, a new textbook devoted to a comprehensive review of the status of the entire field. Professors Marmier and Sheldon have succeeded in this staggering task by producing, in these three volumes, a thorough and incisive discussion of nuclear physics today. These volumes, a prerequisite for which is a one-year course in quantum v

VI

Foreword

mechanics, were inspired by the success of the lecture notes which the authors developed for the course at E.T.H., Zurich. The text is unique in that the presentation attempts to unify the concepts of particle physics and nuclear physics. Each chapter starts from an elementary point of view and the subject matter is developed to a high degree of sophistication. The reader will find here, in this collaboration of experimentalist and theorist, an approach to the subject that reveals the authors' deep and active involvement in current research problems. Details have not been spared but neither have the authors lost sight of the unified structure of the subject. This is indeed a welcome and needed addition to the literature of the field. JERRY

University of Maryland March, 1969

B.

MARION

CONTENTS

Foreword.

v

Summary ofContents, Volumes II and III

XVll

1. Historical Developm.ent of Nuclear Physics. The Size and Constitution of the Atom.ic Nucleus A

SURVEY

OF

NUCLEAR

AND

PARTICLE

PHYSICS

WITHIN

THE

CONTEXT OF PHYSICS AS A WHOLE

1.1. The Present Status of Nuclear Physics 1.2. Brief History of the Development of Atomic, Nuclear, and Particle Physics. 1.3. The Domain of Nuclear Physics 1.4. The Size of the Nucleus and Nuclear Constitution Exercises .

2 7 8 19

2. Nuclear Radii and the Liquid Drop Model of the Nucleus STABILITY AND RADIUS OF NUCLEI

2.1. Energy Considerations 2.2. The Radius of Nuclei and the Liquid Drop Model vii

22

28

Contents

viii

2.3. The Liquid Drop Model and the Semiempirical Mass Formula 2.3.1. 2.3.2. 2.3.3. 2.3.4. 2.3.5. 2.3.6. 2.3.7. 2.3.8.

Von Weizsacker's Approach to Binding Energy Volume Energy Surface Energy Coulomb Energy Asymmetry Energy (Neutron Excess) Pairing Energy Summary Other Mass Formulae

2.4. Applications of the Mass Formula to Considerations of Stability 2.4.1. 2.4.2. 2.4.3. 2.4.4.

Coulomb Radius Constant Radius Constant from Binding Energy for Mirror Nuclei Uranium Fission The Behavior of Isobars in f3 Decay

Exercises .

31 31 32 32 33 34 36 37 39 42 42 43 44 45 51

3. Interactions and Nuclear Cross Sections INTERACTIONS, TRANSITION PROBABILITY, AND REACTION CROSS SECTION

53 55 55 55 55 57

3.1. Nuclear Force Characteristics 3.2. Classification of Interactions 3.2.1. 3.2.2. 3.2.3. 3.2.4.

Strong Interactions. Electromagnetic Interactions Weak Interactions . Gravitational Interactions.

3.3. Response of Particles to Strong, Electromagnetic, and Weak Interactions

62

3.4. Transition Probability 3.4.1. Time-Dependent Perturbation Theory 3.4.2. Transition Probability per Unit Time

3.5. Reaction Probability and Cross Section. 3.5.1. Definition of Reaction Cross Section 3.5.2. Partial and Total Cross Sections

3.5.3. 3.5.4. 3.5.5. 3.5.6. 3.5.7.

60 61 61

3.3.1. Strong Interactions. 3.3.2. Electromagnetic Interactions 3.3.3. Weak Interactions .

.

Differential and Total Cross Section (Angular Distribution) Double-Differential Cross Section (Angular Correlation) Geometrical and Absolute Cross Section . Cross Sections Referred to Electrons and Atoms. Classical Elastic Coulomb Scattering Cross Section (Rutherford Formula) 3.5.8. Coulomb Scattering Cross Section for Like Particles (Mott Formula)

64 64 67 68 71 77 77 78 79 82

83 88

ix

Contents 3.6. Transition Probability and Cross Section

89

3.6.1. Box Normalization and the Relationship of Transition Probability per Unit Time to the Differential Cross Section for Elastic Scattering 3.6.2. Elastic Scattering Matrix Element. 3.6.3. Level Density 3.6.4. Elastic Scattering Cross Section (Born Collision Formula)

Exercises .

89 91 92 93 95

4. Passage of Ionizing Radiation through Matter IONIZING EFFECTS OF ELECTROMAGNETIC RADIATION AND CHARGED PARTICLES

4.1. Survey of Electromagnetic Interaction Processes 4.2. Thomson and Compton Scattering of Gamma Radiation. 4.2.1. 4.2.2. 4.2.3. 4.2.4.

Thomson Cross Section Compton Effect Compton Cross Section (Klein-Nishina Formula) Atomic Compton Cross Section

4.3. Rayleigh Scattering 4.4. Photoelectric Effect

112 113 117 119 120

4.4.1. Energy and Atomic-Number Dependence 4.4.2. Angular Distributions 4.4.3. Attenuation Coefficients

4.5. Auger Effect 4.6. Pair Production. 4.6.1. 4.6.2. 4.6.3. 4.6.4. 4.6.5.

Dirac Electron Theory Electron-Positron Conjugation Feynman Graphs Differential and Total Pair Cross Sections Inverse Pair Production: Annihilation Bremsstrahlung

120 122 123 125 127 138 Radiation

and

4.7. Nuclear Scattering of Gamma Rays 4.8. Total Attenuation Coefficient for Electromagnetic Radiation Passing through Matter 4.9. Interaction of Charged Particles with Matter. 4.9.1. Energy Loss of Heavy Charged Particles (Stopping Power, Range, and Straggling) 4.9.2. Energy Loss of Electrons (Stopping Power, Range, and Straggling) . 4.9.3. Bremsstrahlung 4.9.4. Cerenkov Radiation

4.10. Energy Loss of Heavy Ions. Exercises .

98 98 100 103 106 110

144 147 148 155 156 170 176

190 199 204

Contents

x

5. Nuclei and Particles as Q.uantulD-Mechanical Systerns QUANTUM PROPERTIES OF NUCLEI AND PARTICLES

5.1. The Need to Treat Nuclei and Particles Quantum-Mechanically 5.2. Quantization of Angular Momentum 5.3. Quantum Numbers of Individual Particles 5.3.1. Orbital Quantum Number l 5.3.2. Magnetic Orbital Quantum Number m l 5.3.3. Spin Quantum Number s . 5.3.4. Magnetic Spin Quantum Number m. 5.3.5. Total Angular Momentum Quantum Number j . 5.3.6. Magnetic Total Angular Momentum Quantum Number mj 5.3.7. Radial (nr ) and Principal Quantum Number n 5.3.8. Isospin Quantum Numbers T and T, 5.3.9. Strangeness S 5.3.10. Parity 7T

5.4. Quantum Properties of Nuclear States 5.4.1. 5.4.2. 5.4.3. 5.4.4. 5.4.5. 5.4.6. 5.4.7. 5.4.8.

Nuclear Energy Levels Nuclear Angular Momentum (Spin and Coupling Schemes) Nuclear Parity Isospin Magnetic and Electric Moments of Particles and Nuclei. Anomalous Nucleon Spin Magnetic Dipole Moments Magnetic Dipole Moments of Nuclei Electric Moments .

5.5. Symmetries, Invariances, and Conservation Laws Exercises .

208 210 212 212 212 212 213 213 214 214 215 216 220 224 224 225 227 227 240 247 250 251 257 261

6. Radioactivity RADIOACTIVE DECAY

6.1. Mean Lifetime toward Radioactive Decay 6.1.1. Level Width and Decay Probability 6.1.2. Half-Life and Specific Activity

6.2. Branching Ratios (Partial Widths) 6.3. Radioactive Decay: Daughter Activity 6.3.1. Daughter Activity in Special Cases 6.3.2. Production of Radioactive Sources (Induced Radioactivity) 6.3.3. Mixture of Activities

6.4. Decay Schemes of Widely Used Radioactive Sources 6.5. Parent-Daughter Relationships in Radioactive Dating 6.6. Nuclear Stability Limits according to the Liquid Drop Model Exercises .

263 265 269 270 272 275

281 283 286 290 294 295

Contents

Xl

7. Alpha Decay ALPHA DECAY

7.1. Introduction

7.2. Semiempirical Mass Formula Applied to IX Decay 7.3. Relation between IX Energy and Decay Half-Life 7.4. Penetration of Potential Barriers 704.1. Rectangular Barrier. 704.2. Barrier of Arbitrary Shape. 7.4.3. Nuclear Potential Barrier .

7.5. Short- and Long-Range IX Radiation 7.6. Application of the Gamow Formula to

IX

Decay

7.6.1. IX Energy and Intensity 7.6.2. Nuclear Radius Constant . 7.6.3. Spontaneous Nuclear Disintegration

Exercises .

298 299

301 303 306 309 310 320 321 321 322 323 324

8. Beta Decay THE WEAK BETA-DECAY INTERACTION

8.1. Introduction 8.1.1. Decay Modes 8.1.2. Mass-Energy Balance 8.1.3. Beta-Energy Spectrum

8.2. The Neutrino 8.2.1. Neutrino Properties. 8.2.2. Neutrino Hunting

8.3. Beta -Decay Theory 8.3.1. Formulation. 8.3.2. Probability Function and the Beta-Momentum Spectrum 8.3.3. Statistical Factor (Final-State Density) 8.304. Interaction Matrix Element 8.3.5. Coulomb Correction Factor 8.3.6. Kurie Plot . 8.3.7. Neutrino Rest Mass. 8.3.8. Neutrinos and Cosmology 8.3.9. Beta-Decay Lifetime

8.4. Classification of Beta Transitions 804.1. Degrees of Forbiddenness 804.2. Superallowed Transitions 8.4.3. Allowed Transitions. 8.404. Forbidden Transitions

327 328

329 331 334

334 336 351 351 352 353 355 357 360 361 367 368 371

373 374 375 376

xii

Contents

8.5. Electron Capture 8.5.1. Discrete Neutrino-Energy Spectrum 8.5.2. Capture Lifetime andft Value 8.5.3. Electron Capture Ratios

8.6. Forms of Beta Interaction 8.6.1. Restrictions upon the Coupling Strengths. 8.6.2. Fermi and Gamow-Teller Transitions 8.6.3. ft Values and Nuclear Matrix Elements 8.6.4. Electron-Neutrino Angular Correlation

8.7. Parity Nonconservation in Beta Decay 8.7.1. Test of Parity Violation 8.7.2. Neutrino Helicity .

8.8. Beta Decay Coupling Strengths and Interaction Characteristics Exercises .

377 377 378 381 388 391 391 393 396 397 399 404 408 410

9. Radiative Transitions in Nuclei GAMMA DECAY

9.1. Multipole Character of Gamma Radiation 9.2. Multipole Transition Probability 9.2.1. 9.2.2. 9.2.3. 9.2.4.

Reduced Transition Probability "Forbidden" Transitions Nuclear Isomerism . Multipole Mixing .

9.3. Nuclear Level Scheme Compilation 9.4. Angular Distributions and Correlations. 9.5. Recoil-Free Gamma Spectroscopy 9.5.1. Nuclear Resonance Absorption and Fluorescence 9.5.2. Mossbauer Effect

Exercises .

414 418 425 427 427 428 433 440 446 446 450 460

10. Internal Conversion INTERNAL CONVERSION AND INTERNAL PAIR FORMATION

10.1. Conversion Coefficients 10.1.1. 10.1.2. 10.1.3. 10.1.4.

Partial and Total Conversion Coefficients Experimental Study of Conversion Evaluation ofInternal Conversion Coefficients Approximate Analytic Expressions for Conversion Coefficients and Mean Lifetime

467 467 468 473 478

Contents 10.2. Selection Rules 10.2.1. Mixed Multipo1arity

10.3. Conversion Distributions and Correlations (Particle Parameters) Exercises.

xiii 481 482 482 483

11. Fundamental Characteristics of Nuclear Reactions NUCLEAR REACTION CHARACTERISTICS

11.1. Reaction Energetics . 11.1.1. 11.1.2. 11.1.3. 11.1.4. 11.1.5.

Energy and Momentum Conservation in Nuclear Reactions Nonrelativistic Q Equation Relativistic Q Equation . Threshold Energetics Energy-Correlation Analysis

11.2. General Features of Reaction Cross Sections. 11.2.1. Probability Considerations 11.2.2. General Cross-Sectional Trend for Elastic Neutron Scattering. 11.2.3. Characteristic Cross Section for Exothermic Reactions Induced by Low-Energy Neutrons 11.2.4. Characteristic Cross Section for Inelastic Neutron Scattering. 11.2.5. Characteristic Cross Section for Endothermic NeutronInduced Reactions Leading to Emission of Charged Particles. 11.2.6. Cross-Sectional Trend for Exothermic Reactions involving Charged Incoming and Uncharged Outgoing Particles 11.2.7. Characteristic Cross Section for Exothermic Reactions with Charged Incident and Emergent Particles

11.3. Detailed Balance Predictions for Inverse Reaction Cross Sections 11.3.1. Experimental Investigation of Inverse Reactions

11.4. Resonance Reactions 11.4.1. 11.4.2. 11.4.3. 11.4:4. 11.4.5.

Resonance Anomalies in Excitation Functions. Breit-Wigner Formula Resonance Cross-Section Nomenclature Modified Breit-Wigner Theory for Elastic Scattering. Statistical Spin Factor

11.5. Formal Reaction Theory 11.5.1. 11.5.2. 11.5.3. 11.5.4. 11.5.5. 11.5.6.

Partial-Wave Approach to Scattering of Spinless Particles Phase Shifts Resonance Cross Sections Reaction Theory in Matrix Formalism. Transition Amplitude in S-Matrix Theory Basic Properties of the S Matrix (Probability Conservation and Time-Reversal Invariance). 11.5.7. Cross Sections in Matrix Formalism

Exercises.

485 485 487 488 488 490 505 505 507 507 509 509 510 511 511 514 519 519 520 523 524 530 531 532 538 539 541 541 542 544 555

Contents

xiv ApPENDIfC A.

Kinem.atics of Relativistic Particles

A.I. Lorentz Transformation A.I.I. A.I.2. A.I.3. A.l.4. A.l.5.

A.2. A.3. AA. A.5.

Lorentz Contraction and Time Dilation. Geometrical Representation of the Lorentz Transformation Composition of Collinear Velocities Relativistic Addition of Noncollinear Velocities. Doppler Effects (Frequency Shifts)

Relativistic Mass, Momentum, and Energy "Relativistic" Particles Lifetimes of Relativistic Particles. Speeds of Relativistic Charged Particles Exercises.

ApPENDIX

B,2.l. B,2.2. B,2.3. B,2.4.

Velocity Relations. Kinetic Energy Relations . Angular Relations . Relations between Velocities, Energies, and Scattering Angles in the Laboratory System . B.2.5. Solid Angle Relations B,2.6. Angular Intensity Relations

B.3. Relativistic Elastic Collision of a Fast-Moving Particle with a Stationary Target Velocity Relations . Expressions for Energy in the Center-of-Mass System Angular Relations . Solid Angle Relation

Exercises .

ApPENDIX

569 572 574 574 578

B. Transform.ation Relations between the Laboratory and Center-of-Mass Syseerns for Elastic Collisions

B.I. Characteristics of the Center-of-Mass System. B.2. Nonrelativistic Elastic Collision of a Moving Particle with a Stationary Target

B,3.l. B.3.2. B,3.3. B,3.4.

560 562 563 565 565 567

580 581 582 583 583 585 586 586 590 591 592 595 597 599

C. The Dynamics of Decay and Reaction Processes

C.I. Decay and Reaction Kinematics . C.2. Energetics and Kinematics for Two-Particle Decay. C.3. Scattering Kinematics C.3.1. Nonrelativistic Elastic Scattering Kinematics C.3.2. Relativistic Elastic Scattering Kinematics

601 601 604 605 605

Contents C.3.3. Graphical Treatment of Elastic Scattering C.3.4. Graphical Treatment of Inelastic Scattering C.3.S. Inelastic Scattering at High Incident Energies Leading to the Threshold of Particle Creation .

C.4. Nonrelativistic Reaction Kinematic Formulae Exercises. ApPENDIX

D.l. D.2. D.3. DA.

607 609 610 612 614

D. Wave Mechanics

Schrodinger Equations Probability Density and Electron Probability Distribution Heisenberg Uncertainty Relations Klein-Gordon Equation for Spin-O Particles. DA.I. Relativistically Invariant Notation

D.5. Dirac Equation for Spin-j Relativistic Particles D.S.I. Covariant Form of the Dirac Equation (Gamma Matrices) D.S.2. Properties of the Gamma Matrices

D.6. Dirac Electron-Positron Theory. D.7. Weyl Equation for Massless Particles (Two-Component Neutrino Theory) . D.8. Wave Equations for Bosons. Exercises .

ApPENDIX

xv

617 620 624 626 626 628 631 633 637 639 641 645

E. Angular Momentum in Q.uantulD Mechanics (Racah Algebra)

E.l. Angular Momentum Operators E.2. Composition of Angular Momentum Wave Functions (ClebschGordan Coefficients) . E.3. Properties of Clebsch-Gordan Coefficients and Wigner 3-j Symbols . EA. Values of Simple 3-j Symbols E.5. Examples of Wave-Function Coupling E.6. Recoupling of Angular Momenta (Racah Coefficients and Wigner 6-j Symbols) . E.7. Coupling of Four Angular Momenta (Wigner 9-j Symbols) E.8. Racah Functions in Angular Distribution and Correlation

646 647 649 651 651 658 661

Theory

664

E.8.1. Composition of Angular Distribution Functions. E.8.2. Composition of Angular Correlation Functions.

666 670 674

Exercises .

xvi

Contents

ApPENDIX

F. FeynD1an Interaction Theory

F.l. The Underlying Motivation Approach. F.2. Interaction Matrix Elements F.3. Feynman Graphs F.3.1. F.3.2. F.3.3. F.3.4.

ApPENDIX

behind

a

Field-Interaction

Relation to Matrix Elements Feynman Graphs in Momentum Space Second Quantization Formation of the S-Matrix Element

G. SOlDe Measueement Techniques in Nuclear Physics

G.!. Introduction G.2. Beta Spectrometry G.2.1. Principle of Magnetic Spectrometers G.2.2. Focusing Arrangements

G.3. Scintillation Counters G.3.1. Principle G.3.2. Energy Resolution. G.3.3. Pile-Up

G.4. Semiconductor Detectors G.5. Energy Scale in Low-Energy Nuclear Spectroscopy G.6. Coincidence Techniques Exercises. ApPENDIX

698 698 699 703 706 706 709 713 713 721 723 727

H. Radiation DosilDetry

H.l. Biological Effects of Radiation H.2. Dosimetry Units Exercises. ApPENDIX

675 676 680 680 686 687 693

731 732 734

I. Constants and Conversion Factors in AtoJDic, Nuclear, and Particle Physics

Text and Tables

738

References .

745

Solutions to Exercises

776

Subject Index

791

SUMMARY OF CONTENTS Volumes IT and m

12. Nuclear Particles and Their Interactions N eu trons; antinucleons ; deu terons and two-body nuclear forces; three-nucleon systems [3H, 3He]; four-nucleon systems [4He]; heavy ion physics

13. Nuclear Forces and Potentials, as Deduced froID Nuclear DynaJDics (Scattering and Polarization) Nuclear force characteristics: central, tensor, spin-orbit components; velocity-dependent forces; scattering formalism and effective-range theory; Wolfenstein parameters; phase-shift analysis

14. Scattering and Reaction Models in Nuclear Physics Optical model; compound-nucleus formalism; direct-interaction formalism; unified formalism; statistical fluctuations and strength functions xvii

Contents

XVlll

15. Nuclear Models Statistical models; liquid-drop model; Fermi-gas model; shell model; collective model; unified model; cluster model; nuclear matter

16. Certain Specialized Reaction Processes Spallation; fission; fusion; nuclear astrophysics; high-energy physics

17. Fundamental Particle Physics Physics of leptons, mesons, baryons, and resonances; conservation properties

18. Group-Theoretical Methods in Nuclear and Particle Physics Elements of group theory and unitary groups; group-theoretical treatment of angular momentum; the SU(3) group: algebra and irreducible representations; particle classification in the SU (3) scheme; quark states, mass formulae and broken symmetries; electromagnetic interactions and dynamic predictions of the SU(3) scheme; SU(6) and higher symmetry groups; principal linear groups; relativistic group theory for hadrons; grouptheoretical classification of nuclear states

ApPENDIX

J.

Rotation and Angular-MoDlentuDl Calculus

Rotation group, rotation matrices and application to the quantum mechanics of angular-momentum coupling

2.4. Applications of Mass Formula

45

case of2 38U fission, but the numerical estimate appliesjust as well to the more pertinent case of 235U symmetric fission treated here.) On the other hand, the energy of 173 MeV is set free during just the one initial fission step. The fission products are, however, extremely unstable because they have a large neutron excess (the neutron emission is indicated by the term xn in the reaction equation) and consequently undergo aseries of decay stages involving n, ß-, and y emission before finishing up as stable nudei. It is these subsequent stages wh ich are associated with the evolution of the remaining 30 to 40 MeV of energy. The two terms wh ich make up the value of Q originate from the Coulomb energy which is set free and the surface energy which has to be supplied. The energy balance in fission is accordingly a tug-of-war between electrostatic and surface tension forces from the viewpoint ofthe liquid drop model. From a dose examination of the behavior of BIA depicted in Fig. 2-3, one can infer that the fission process becomes exothermic for nudei wh ich are much lighter than thorium 01' uranium; in fact, it is al ready exothermic below about A ~ 150. Why, then, does the process not set in spontaneously for nudei which are much lighter than thorium 01' uranium? As we shall see in Section 7.6.3, the reason is to be found in the existence of nudear potential barriers whose inhibiting effect has to be overcome before fission can occur. Only for the heaviest nudides is there sufficient energy available to render the potential barrier ineffectual so far as fission is concerned. 2.4.4. THE BEHAVIOR OF ISOBARS IN ß DECAY We have al ready considered a special case of nudei with A = constant, namely pairs of mirror nudei. A more general situation, however, is represented by the parent and daughter nudides in ß decay, for these also are isobaric pairs. Most of the unstable nudides depicted in Fig. 2-5 are natural ß emitters, typical examples being 4oK, 87Rb, 115In, etc. The stability of ß-emitting isobars can be predicted from the mass formula and from an examination of the above plot of N vs Z, for when the latter is viewed diagonally-that is to say, along aseries of normals to the line N = Z-each set of isobars is located on a straight line. Closer scrutiny reveals that whereas each set of odd-A isobars normally indudes just one stable nuclide, the sets of even-A isobars often indude two, and sometimes even three, stable members. This characteristic property becomes even more evident if one conceives of a third coordinate axis as perpendicular to the (N,Z)-plane, to represent the values of a tomic mass M (Z,A). The ensuing three-dimensional figure has been termed a NUCLEAR ENERGY SURFACE, for the stable region lies along the trough to such a surface. The stable members lie in the vaIley, and the unstable along the sides and rim of the slopes, not unlike a dass of beginners in skiing. The

3.2. Classification of Interactions

59

formed by recombination at the emitter site would acquire in falling back down through the field iJ,p to the detector, as depicted schematically in Fig. 3-I(c). Of course, a direct way to check the sign of the gravitational mass of antimatter would be to study the gravitational deflection of a horizontal beam of positrons (beams of more massive antiparticles cannot as yet be produced with sufficient intensity). However, such an experiment is simpler in conception than in execution, for the elimination of extraneous electromagnetic fields which would perturb the positron motion in a highly evacuated tube of considerable length presents very pronounced technical difficulties. 'Y - Emitter (Energy E)

Height difference H

Gra\'itational potential difference !i¢= gH

Resonance Y- detector (able to measure energy change Il.E)

-- --- -----

In conclusion, it may be mentioned that the effect of gravity upon matter has been tested not only with massive bodies but also with beams of neutral free cesium and potassium atoms [Es 47], as well as with beams of highand low-energy neutrons [McR 51, Da 65a], and the results conclusively indicate that all matter is at all times subject to the same gravitational interaction.

3.3. Response ofParticles to lnteractions

61

that, of the appreciably more sluggish weak interaction in order to reduce its energy content. But the actual process is always the one which, without contravening various selection and conservation rules, is fastest. At the risk of anticipating some points discussed more fully in later sections, we present some examples to illustrate the various types of interaction, pointing out meanwhile that there may in fact be further as yet undiscovered interaction modes (such as "superweak" interaction, currently being studied in connection with K-meson ["kaon"] phenomena), the detection of which constitutes one of the aims of particle physics.

3.3.1.

STRONG INTERACTIONS

Clearly, internucleon bindingforces accounted for by the exchange of pions furnish an instance of a strong interaction. The pion-nucleon interaction that operates in pion scattering is also strong, as is that in the scattering of kaons by nucleons. A strong interaction is, moreover, responsible for kaon production: it operates in reactions such as

(3-20) However, the decay ofmesons, nucleons, and hyperons proceeds by an electromagnetic or weak interaction (or both) except when, as we shall see in examples which follow, it involves as intermediate step the (virtual) formation of a nucleon-antinucleon pair, this being an instance of a strong interaction.

3.3.2.

ELECTROMAGNETIC INTERACTIONS

In most instances, processes which proceed by the electromagnetic interaction involve one or more photons. Photon exchange is, of course, responsible for e1ectromagnetic bindingforces, and photon capture can effect the production of mesons or hyperons by an electromagnetic interaction, for example : (photopion production) (associated photoproduction of a

(3-21 )

A hyperon and K meson) An examp1e of a radiative capture reaction is provided by the process

(3-22)

Decay processes illustrate a particularly important point which at first sight may appear paradoxical, namely that uncharged particles can also seem to be subject to electromagnetic interactions, even though this would appear to be fundamentally impossible. For example, 7T ü decay into two photons,

(3-23) occurs VIa an electromagnetic interaction, with a mean lifetime of about

3.4. Transition Probability

67

This can be integrated to obtain a~l) if we assume the perturbation to be describable by a step function which has a constant finite value between times 0 and t, viz.

a?) =;/i Hf~ Jt exp(iwfi t) dt = l

0

_1_ Hf~[l - exp(iwfi t)]

/iwfI

(3-61)

Ifwe neglect higher orders of approximation, we obtain the probability that a transition from state i tofhas taken place within a time tunder the influence of a step-function perturbation, (3-62) 3.4.2.

TRANSITION PROBABILITY PER UNIT TIME

Ifwe have a group offinal statesfwhose energy is nearly the same as that of the initial state i, we can treat them as a continuum and express the number of such states per unit energy interval around Er in the form of an energy LEVEL DENSITY p(Er ) , whose explicit value we shall derive in Section 3.6.3. Here, we avail ourse1ves of this concept which represents a measurable quantity in order to derive the final expression for the transition probability per unit time in a simple wide1y used form. /

Ifwe assume that Hf~ varies but slowly withfwithin the group ofstates in the region around E f , the quantity IHf~12 can be taken to denote a mean value, independent of the actual energy E, and treated as essentially constant in the calculation which folIows. The evaluation of the transition probability per unit time to all accessible states f according to first-order perturbation theory then consists of replacing the summation in the expression (3-63)

by an integral over all energy intervals dEf' (3-64)

noting that dEf = /i dWfi. Thus, if we confine the consideration to final states within an energy band of width LlE ~ we may bring IHf ~ 12 p (E f ) outside the integral and write

s;

(3-65)

in which the integral is simple to evaluate since it is of the form

J:oo x-

2

sin 2 (kx) dx = 7Tk

(3-66)

-Delbrück scattering Nuc1ear scattering -coherent scattering -incoherent scattering

Interaction with a Coulomb field -pair production

Nuc1ear photoeffeet

a

;;:: 100 MeV

>I MeV, especially at 5-10 MeV >2 MeV (t as E t)

A more extensive tabulation is given by Siegbahn [Si 65, pp. 38 and 39].

Mössbauer, resonance, Thomson Compton, with individual nucleons

with e1ectron Coulomb field (inelastic pair production and triplet production) in nuclear Coulomb fie1d

with nuclear Coulomb field (elastic pair production)

;;:: 10 MeV

incoherent scattering

with "free electrons" with bound atomic electrons, causing ejection of e1ectrons with nucleus as a whole, causing emission of photons, particles, and (above threshold) mesons

with bound atomic e1ectrons

- Thomson scattering

-Compton scattering Photoe1ectric effect