Nuclear Physics for Applications

Stanley C. Pruss'm Nuclear Physics for Applications A Model Approach BICENTENNIAL WILEY-VCH Verlag GmbH & Co. KGaA

VII Table of Contents Preface XIII 1 Introduction 1 1.1 Low-Energy Nuclear Physics for Applications 1 1.2 Some General Observations and Notations 3 1.3 Overview of Radioactive Decay Processes and Nuclear Reactions 4 1.3.1 Alpha Decay 4 1.3.2 Beta Decay 5 1.3.3 Spontaneous Fission 7 1.3.4 Gamma Decay 8 1.3.5 Nuclear Reactions 9 1.4 The Model-basedCharacterofthis Text 10 1.5 Sources of Nuclear Data 11 2 Nuclear Masses and Energetics of Radioactive Decay and Nuclear Reactions 13 2.1 Introduction 13 2.2 Review ofthe Special Theory of Relativity 13 2.3 Masses of Atoms and Particles 18 2.4 Comments Concerning "Nuclear Stability" and Energetics 21 2.4.1 Spontaneous Transformations and Nuclear Masses 21 2.4.2 Nuclear Stability 24 2.5 Bound and Unbound States and Their Energetics: Potential Wells 25 2.6 Nuclear and Atomic Masses and Binding Energies 29 2.6.1 ß" Decay 30 2.6.2 ß + Decay or Positron Emission 32 2.6.3 Electron Capture Decay 34 2.6.4 Competitive Decay Modes 35 2.6.5 a Decay 36 2.6.6 Spontaneous Fission 37 2.7 Nuclear Reactions 38 Nuclear Physics for Applications. Stanley G. Prussin Copyright 2007 WILEY-VCH Verlag GmbH & Co., Weinheim ISBN: 978-3-527-40700-2

VIIII Table of Contents 3 Phenomenology of Radioactive Decay and Nuclear Reactions 41 3.1 Introduction 41 3.1.1 The Phenomenology of Radioactive Decay 41 3.1.2 Units for Describing Radioactive Decay 45 3.1.3 Radioactive Growth and Decay 45 3.1.4 Simple Decay Scheines and Decay Chains 51 3.2 Statistical Considerations in Radioactive Decay 54 3.2.1 The Binomial Distribution 54 3.2.2 The Poisson Distribution 56 3.2.3 Application of Statistical Analysis to Common Experimental Conditions 61 3.2.4 Propagation of Errors 62 3.3 Reaction Cross Sections 66 4 Nuclear Binding Energies: Empirical Data and the Forces in Nuclei 75 4.1 Empirical Masses and Average Binding Energies of Nucleons 75 4.2 The Forces Acting Between Nucleons 77 4.3 The Average Nuclear Interaction Between Nucleons in the Nucleus and Nuclear Radii 83 4.4 Quantization of the Nucleus: Pairing of Identical Nucleons 92 4.5 Quantization of the Nucleus: Asymmetry Energy 100 5 The Semi-Empirical Mass Formula and Applications to Radioactive Decay 109 5.1 Introduction 109 5.2 The Semi-Empirical Mass Formula 110 5.3 The Nuclear Mass Surface 115 5.4 The Semi-Empirical Mass Formula and ß Decay 117 5.5 The Semi-Empirical Mass Formula and a Decay 123 5.6 The Semi-Empirical Mass Formula and Nuclear Fission 125 5.7 Discrepancies Between Experimental Masses and those Predicted by the Semi-Empirical Mass Formula 128 6 Elements of Quantum Mechanics 133 6.1 Introduction 133 6.2 Elements of Quantum Mechanics 134 6.2.1 The Schrödinger Equation and Conservation Laws 134 6.2.2 Elementary Properties of Operators 235 6.2.3 Elementary Properties of Wave Functions 137 6.2.4 Operators, Eigenfunctions and Conservation Laws 138 6.2.5 Parity 146 6.3 Angular Momentum in Quantum Mechanics 147 6.3.1 Operators for Orbital Angular Momentum 148 6.3.2 Angular Momentum and Magnetic Moments 150

Table of Contents IIX 6.4 The Vector Model for Angular Momentum 152 6.5 The Wave Functions of Many-Particle Systems 358 7 Nuclear Structure: The Spherical Shell Model 161 7.1 Introduction 161 7.2 The Independent Particle Model 161 7.2.1 The Angular Equations: Angular Momentum and Parity 164 7.2.2 Some Properties of the Wave Functions 170 7.2.3 The Radial Equation and the Centrifugal Potential 172 7.2.4 Models for the Average Central Potential in the Independent Particle Approximation 175 7.2.5 The Infinite Spherical Potential Well 178 7.2.6 The Isotropie Harmonie Oscillator 185 7.3 The Single-Particle Levels of Spherical Nuclei 189 7.4 Comparison of the Predictions of the Single-Particle Model with Experiment 195 8 Nuclear Shapes, Deformed Nuclei and Collective Effects 205 8.1 Introduction 205 8.2 Collective Excitations 212 8.3 Rotational Excitations in (Even,Even) Nuclei 213 8.4 Rotational Excitations in Odd-A Nuclei 219 8.5 Vibrational Excitations in Nuclei 222 8.6 Nuclear Structure in a Deformed Potential 229 8.7 The Nilsson Model 234 9 u Decay and Barrier Penetration 245 9.1 Introduction 245 9.2 Q a and a Decay Half-Lives 248 9.3 Binding of Valence Nucleons and the Potential for Interaction Between an a Particle and a Heavy Nucleus 253 9.4 The Wave Functions for Particles in Finite Potential Wells and Barrier Penetration 258 9.5 A Simple Model for a Decay 266 9.6 Application of the Model to the Decay of Even-Even Nuclei 268 9.7 Angular Momentum Effects in et Decay 272 9.8 Decay of Odd-A Nuclides and Structure Effects 274 10 ß Decay 281 10.1 Introduction 281 10.2 ß Decay and Conservation Laws: The Neutrino and the Weak Interaction 282 10.3 The Fermi Golden Rule No. 2 285 10.4 The Fermi Theory of Allowed ß Decay 287 10.5 ß Spectra 292

X I Table of Contents 10.6 Decay Probabilities for ß~ and ß* Decay 296 10.7 Some Implications of the Simple Theory of Allowed ß Decay 300 10.7.1 Angular Momentum Effects 300 10.7.2 Nuclear Matrix Elements: Fermi Transitions 302 10.7.3 Nuclear Matrix Elements: Gamow-Teller Transitions 305 10.8 Classification ofß Transitions and Experimental Log ]0 ft 306 10.9 Electron Caprure Decay 307 10.9.1 X-ray Emission 308 10.9.2 Auger Electron Ejection 311 10.10 Elementary Theory of Electron Caprure 314 10.11 Ratio of Electron Caprure to Positron Emission 318 10.12 ß-Decay Schemes 320 10.13 ß-Delayed Particle Emission 325 10.14 Comments on Fermi Transitions 327 11 y Decay and Internat Conversion 331 11.1 Introduction 331 11.2 The Angular Momentum of Photons and Conservation Laws 332 11.3 Introduction to the Theory of Photon Emission 334 11.3.1 The Radiation Field and Matrix Elements for Photon Emission 334 11.3.2 Matrix Elements and Transition Rates 341 11.4 Examples of Nuclear Isomerism 347 11.5 Some General Observations 350 11.5.1 El Transitions 350 11.5.2 E2 and Ml Transitions 351 11.5.3 Other Transitions 351 11.6 Internal Conversion 351 11.6.1 Elementary Theory of Internal Conversion 353 11.7 Decay Schemes 357 12 Nuclear Fission 373 12.1 Introduction 373 12.2 The Discovery of Nuclear Fission 374 12.3 The Liquid-Drop Model and Nuclear Fission: The Nuclear Potential Energy Surface 375 12.4 Empirical Data on Spontaneous and Neutron-Induced Fission 383 12.5 Energy Release in Fission 392 12.5.1 Fission Fragment Kinetic Energy 392 12.5.2 Kinetic Energy of Prompt Neutrons 395 12.5.3 The Spectrum of Prompt y-rays 399 12.5.4 Summary of the Sources of Energy Release in Fission 399 12.6 Fission Barriers and Fission Probabilities 401

Table of Contents XI 13 Low-Energy Nuclear Reactions 405 13.1 Introduction 405 13.2 Kinematics of Nonrelativistic Reactions 407 13.2.1 Kinematics of Elastic Scattering in the Laboratory Coordinate System 408 13.2.2 Kinematics of Elastic Scattering in the Center of Mass Coordinate System 412 13.2.3 Kinematics of General Nonrelativistic Nuclear Reactions 418 13.3 Cross Sections for Nuclear Reactions from First-Order Perturbation Theory 424 13.4 The Reciprocity Theorem 435 13.5 Qualitative Considerations of the Mechanisms of Low-Energy Nuclear Reactions 440 13.5.1 Potential Scattering 440 13.5.2 The Compound Nucleus 441 13.5.3 Direct Reactions 446 13.6 The Properties of Time-Dependent States 447 \3.7 A Physical Approach to the Form of Cross Sections for Compound Nucleus Reactions: The Breit-Wigner Single-Level Formula 451 13.8 Scattering in Quantum Mechanics: Partial Wave Analysis 457 13.9 Extension of the Partial Wave Analysis to Nuclear Reactions 468 13.10 S-Wave Scattering and Reactions in the Limit of the Spherical Potential Well Model 472 13.11 The Breit-Wigner Single-Level Formula and Experimental Cross Sections 478 13.12 About Fission Cross Sections 487 14 The Interaction of lonizing Radiation with Matter 493 14.1 Introduction 493 14.2 The Interaction of Photons with Matter 495 14.2.1 Elastic Scattering of Photons on Unbound Electrons 496 14.2.2 Compton Scattering 502 14.2.3 The Photoelectric Effect 536 14.2.4 Pair Production 519 14.2.5 Total Cross Sections and Attenuation Coefficients 520 14.3 The Interaction ofcharged Particles with Matter 524 14.3.1 The Stopping of Heavy Charged Particles in Matter 525 14.3.2 The Stopping of Electrons and Positrons in Matter 538 Appendix 1 Atomic Masses 545 Appendix 2 Nuclide Table 565

XII I Table of Contents Appendix 3 Physical Constants 607 Appendix 4 First-Order Time-Dependent Perturbation Theory 611 Index 619