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Physics of Atomic Nuclei

ISBN: 978-3-527-41350-8
688 pages
June 2017
Physics of Atomic Nuclei (3527413502) cover image

Description

This advanced textbook presents an extensive and diverse study of low-energy nuclear physics considering the nucleus as a quantum system of strongly interacting constituents.
The contents guide students from the basic facts and ideas to more modern topics including important developments over the last 20 years, resulting in a comprehensive collection of major modern-day nuclear models otherwise unavailable in the current literature. The book emphasizes the common features of the nucleus and other many-body mesoscopic systems currently in the center of interest in physics. The authors have also included full problem sets that can be selected by lecturers and adjusted to specific interests for more advanced students, with many chapters containing links to freely available computer code. As a result, readers are equipped for scientific work in mesoscopic physics.
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Table of Contents

Dedication xiii

Preface xv

1 Building Blocks and Interactions 1

1.1 What Are the Nuclei Made Of? 1

1.2 Proton and Neutron 3

1.3 Strong Interactions 4

1.4 Electromagnetic Interactions and Charge Distribution 5

1.5 Magnetic Properties 10

1.6 Weak Interactions 11

1.7 Neutron Decay 13

1.8 NuclearWorld 15

References 19

2 Isospin 21

2.1 Quantum Numbers in the Two-Body Problem 21

2.2 Introducing Isospin 23

2.3 Isospin Invariance 25

2.4 Space–Spin Symmetry and Isospin Invariance 26

2.5 Glimpse of a More General Picture 30

2.6 Relations between Cross Sections 31

2.7 Selection Rules 35

2.8 Isobaric Mass Formulae 38

References 41

3 Two-Body Dynamics and the Deuteron 43

3.1 Low-Energy Nuclear Forces 43

3.2 Example: Argonne Potential 45

3.3 Meson Exchange 48

3.4 Deuteron: Central Forces and s-Wave 51

3.5 Tensor Forces and d-Wave 55

3.6 Magnetic Dipole Moment 58

3.7 Electric Quadrupole Moment 59

References 65

4 Two-Body Scattering 67

4.1 Scattering Problem 67

4.2 Phase Shifts 69

4.3 Scattering Length 71

4.4 Sign of the Scattering Length 78

4.5 Resonance Scattering at Low Energies 80

4.6 Effective Radius 82

4.7 Scattering of Identical Particles 83

4.8 Coulomb Scattering 86

4.9 Coulomb-Nuclear Interference 87

References 89

5 Liquid Drop Model 91

5.1 Binding Energies 91

5.2 Shape Variables 95

5.3 Microscopic Variables 97

5.4 Multipole Moments 98

5.5 Kinetic Energy and Inertial Parameters 100

5.6 Shape Vibrations 102

5.7 Stability of the Charged Spherical Liquid Drop 104

References 111

6 Vibrations of a Spherical Nucleus 113

6.1 SoundWaves 113

6.2 Isovector Modes 117

6.3 Giant Resonance and Linear Response 119

6.4 Classification of Normal Modes 121

6.5 Quantization of Nuclear VibrationalModes 125

6.6 Multiphonon Excitations 128

6.7 Angular Momentum Classification 132

References 134

7 Fermi Gas Model 135

7.1 Mean Field and Quasiparticles 135

7.2 Perfect Fermi Gas 137

7.3 Ground State 138

7.4 Correlation Between Particles 142

7.5 Asymmetric Systems and Chemical Equilibrium 143

7.6 Pressure and Speed of Sound 146

7.7 Gravitational Equilibrium 148

7.8 Nuclear Matter Equation of State 150

References 151

8 Spherical Mean Field 153

8.1 Introduction 153

8.2 Magic Numbers 153

8.3 Separation Energy 155

8.4 Periodicity of Nuclear Spectra 156

8.5 Harmonic Oscillator Potential 157

8.6 Orbital Momentum Representation 160

8.7 SquareWell Potential 162

8.8 Spin–Orbit Coupling 163

8.9 Realistic Level Scheme 165

8.10 Semiclassical Origins of Shell Structure 166

References 168

9 Independent Particle Shell Model 169

9.1 Shell Model Configurations 169

9.2 Particle–Hole Symmetry 171

9.3 MagneticMoment 172

9.4 Quadrupole Moment 174

9.5 Recoil Corrections 177

9.6 Introduction to Group Theory of Multiparticle Configurations 178

References 183

10 Light Nuclei 185

10.1 A ShortWalk along the Beginning of the Nuclear Chart 185

10.2 Halo in Quantum Systems 190

10.3 Nuclear Halos 192

10.4 One-Body Halo 193

10.5 Two-Body Halos 195

10.6 Efimov States 199

References 202

11 Many-Body Operator Formalism 203

11.1 Secondary Quantization 203

11.2 Physical Observables: One-Body Operators 208

11.3 Two-Body Operators 209

11.4 Interparticle Interaction 210

11.5 Interaction in a Spherical Basis 213

11.6 Recoupling of Angular Momentum 215

References 222

12 Nuclear Deformation 223

12.1 Idea of Nuclear Deformation 223

12.2 Collective Model 224

12.3 Adiabatic Approximation 226

12.4 Onset of Deformation 228

12.5 Quadrupole Deformation in the Body-Fixed Frame 230

12.6 Quadrupole Shape Variables 232

12.7 Variety of Quadrupole Shapes 233

12.8 Empirical Deformation 235

12.9 Single-Particle Quantum Numbers 239

12.10 Anisotropic Harmonic Oscillator 240

12.11 Asymptotic Quantum Numbers 245

12.12 Nilsson Potential 246

12.13 More Examples 247

References 250

13 Pairing Correlations 251

13.1 Physical Evidence 251

13.2 Seniority Scheme 256

13.3 Multipole Moments in the Seniority Scheme 260

13.4 Degenerate Model 261

13.5 Canonical Transformation 265

13.6 BCS Theory: TrialWave Function 269

13.7 Energy Minimization 271

13.8 Solution for the Energy Gap 273

13.9 Excitation Spectrum 276

13.10 Condensation Energy 278

13.11 Transition Amplitudes 279

References 281

14 Gamma-Radiation 283

14.1 Introduction 283

14.2 Electromagnetic Field and Gauge Invariance 283

14.3 Photons 285

14.4 Interaction of Radiation with Matter 288

14.5 Radiation Probability 291

14.6 Electric Dipole Radiation 292

14.7 Electric Quadrupole Radiation 295

14.8 Magnetic Dipole Radiation 296

14.9 Photoabsorption 298

14.10 Multipole Expansion 299

References 303

15 Nuclear Gamma-Transitions and Related Electromagnetic Processes 305

15.1 Single-Particle Transitions 305

15.2 Collective Transitions 308

15.3 Nuclear Isomerism 310

15.4 Isospin 312

15.5 Structural Selection Rules 315

15.6 Monopole Transitions 318

15.7 Internal Electron Conversion 320

15.8 Coulomb Excitation 322

15.9 Nuclear Photoeffect 326

15.10 Electron Scattering 330

References 335

16 Nuclear Rotation 337

16.1 Introduction: Rotational Bands 337

16.2 Finite Rotations 345

16.3 Rotation Matrices as Functions on the Group 346

16.4 Euler Angles 347

16.5 Angular Momentum in Euler Angles 351

16.6 Eigenfunctions of Angular Momentum 354

16.7 Rigid Rotor 355

16.8 Symmetry Properties 357

16.9 Simplest Solutions 358

16.10 Ground-State Band 359

16.11 Intensity Rules 360

16.12 Electric Quadrupole Moment 363

16.13 MagneticMoment 366

16.14 Symmetry Properties Revisited 367

16.15 Coriolis Mixing and Decoupling Parameter 368

16.16 Classical Rotation and Routhian 370

16.17 Cranked Rotation 372

16.18 Moment of Inertia 375

16.19 Adiabatic Expansion 377

16.20 Rotation of a Perfect Fermi Gas 379

16.21 Perfect Bose Gas and Ideal Liquid 381

16.22 Pairing Effects 384

16.23 Band Crossing 385

References 388

17 Self-Consistent Field 391

17.1 Exchange Interaction 391

17.2 Hartree–Fock Equations 395

17.3 Operator Formulation 397

17.4 Single-Particle Density Matrix 398

17.5 Hartree–Fock–Bogoliubov Approximation 400

17.6 General Canonical Transformation 402

17.7 Solutions 404

17.8 Generalized Density Matrix 407

17.9 Pairing and Particle Number Conservation 409

17.10 Effective Interaction 411

17.11 Skyrme Functionals 413

17.12 Generalization to Nonzero Temperature 418

References 419

18 Collective Modes 421

18.1 Schematic Model 421

18.2 Random Phase Approximation 426

18.3 Canonical Form of the RPA 427

18.4 Model with Factorized Forces 430

18.5 Collective Modes as Bosons 432

18.6 Mapping of Dynamics 433

18.7 Normalization and the Mass Parameter 435

18.8 Symmetry Breaking 438

18.9 Generator Coordinate Method 444

References 446

19 Bosons, Symmetries and Group Models 447

19.1 Introduction 447

19.2 Low-Lying Quadrupole Excitations as Interacting Bosons 448

19.3 Algebra of Boson Operators 450

19.4 Subgroups and Casimir Operators 452

19.5 s–d Model 455

19.6 Irreducible Representations and Quantum Numbers 458

19.7 Vibrational Limit 461

19.8 óG(6) Limit 466

19.9 óKóM(3) Limit 468

19.10 Shapes and Phase Transitions in the IBM 470

References 473

20 Statistical Properties 475

20.1 Introduction 475

20.2 Level Density: General Properties 478

20.3 Darwin–FowlerMethod 480

20.4 Relation to Statistical Thermodynamics 482

20.5 Thermodynamics of a Nuclear Fermi Gas 483

20.6 Statistics of Angular Momentum 486

20.7 Shell Model Monte Carlo Approach 488

20.8 Thermodynamics of Compound Reactions 490

20.9 Statistical Description of Resonances 492

References 497

21 Nuclear Fission 499

21.1 Introduction 499

21.2 Alpha-Decay 502

21.3 Neutron Fission 505

21.4 Photofission 509

21.5 Fission as a Large-Amplitude Collective Motion 510

21.6 Nonadiabatic Effects and Dissipation 512

21.7 Fission Isomers 514

21.8 Parity Violation in Fission 518

References 522

22 Heavy-ion Reactions: Selected Topics 525

22.1 Introduction 525

22.2 Experimental Indications 526

22.3 Macroscopic Description 530

22.4 Equilibration as a Diffusion Process 534

22.5 Toward a Microscopic Description 540

22.6 Sketch of a More General Approach 541

22.7 A Simple Model 545

22.8 Nuclear Multifragmentation 547

22.9 More about Fusion Reactions 550

References 553

23 Configuration Interaction Approach 555

23.1 Center-of-Mass Problem 555

23.2 Matrix Elements of Two-Body Interactions 558

23.3 Ab initio Approach 559

23.4 Three-Body Forces 564

23.5 Semiempirical Effective Interactions 565

23.6 Shell-Model Hamiltonian, Properties and Solutions 570

23.7 Effective Non-Hermitian Hamiltonian 571

23.8 From Isolated to Overlapping Resonances 576

23.9 Realistic Nuclear Calculations 581

References 583

24 Weak Interactions 585

24.1 Introduction 585

24.2 Beta-Spectrum in the Simplest Case 587

24.3 Nuclear Transitions 590

24.4 Dirac Formalism 595

24.5 Four-Fermion Theory 599

24.6 Nuclear Structure Effects 601

24.7 Parity Violation 604

24.8 Electric Dipole Moment 607

24.9 Nuclear Enhancement 609

24.10 On theWay to ElectroweakTheory 612

24.11 Higgs Mechanism 616

24.12 Neutrino: Oscillations 618

24.13 Neutrino:Majorana or Dirac? 620

References 623

25 Nucleus as a Chaotic System 627

25.1 Introduction 627

25.2 Strength Function 628

25.3 Level Density Revisited 633

25.4 Complexity ofWave Functions 636

25.5 Correlations between Classes of States 639

25.6 Invariant Entropy 643

25.7 Random Matrix Ensembles 646

25.8 Thermalization 650

References 652

General Nuclear Data Resources 655

Index 657

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Author Information

Vladimir Zelevinsky is professor at the Department of Physics and Astronomy and at the National Superconducting Cyclotron Laboratory at Michigan State University, USA. In the 1980s he was Head of the Theory Division at the Budker Institute and Head of Theoretical Physics at Novosibirsk University, Russia. He spent three years as visiting professor at the Niels Bohr Institute in Copenhagen, Denmark. He is the author of over 250 scientific publications, deputy editor of the EPL journal and associate editor of the journal Nuclear Physics. He is also the author of Quantum Physics, 2 Volume Set, published with Wiley VCH in 2010.

Alexander Volya is professor of Physics at the Florida State University, USA. His education includes diploma from Tallinn Tynismae Science School, Estonia; bachelor's degree from St. Petersburg State University, Russia; doctoral degree in theoretical nuclear physics from Michigan State University; and postgraduate research work at the Argonne National Laboratory. In the fall of 2003, he joined the faculty at Florida State University where he currently leads a research program in theoretical nuclear physics and mesoscopic physics. He has published over 100 publications and has been regularly teaching nuclear physics courses at Florida State University.

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