The Nuclear Many-Body Problem

1 The Liquid Drop Model.- 1.1 Introduction.- 1.2 The Semi-empirical Mass Formula.- 1.3 Deformation Parameters.- 1.4 Surface Oscillations About a Spherical Shape.- 1.5 Rotations and Vibrations for Deformed Shapes.- 1.5.1 The Bohr Hamiltonian.- 1.5.2 The Axially Symmetric Case.- 1.5.3 The Asymmetric Rotor.- 1.6 Nuclear Fission.- 1.7 Stability of Rotating Liquid Drops.- 2 The Shell Model.- 2.1 Introduction and General Considerations.- 2.2 Experimental Evidence for Shell Effects.- 2.3 The Average Potential of the Nucleus.- 2.4 Spin Orbit Coupling.- 2.5 The Shell Model Approach to the Many-Body Problem.- 2.6 Symmetry Properties.- 2.6.1 Translational Symmetry.- 2.6.2 Rotational Symmetry.- 2.6.3 The Isotopic Spin.- 2.7 Comparison with Experiment.- 2.7.1 Experimental Evidence for Single-Particle (Hole) States.- 2.7.2 Electromagnetic Moments and Transitions.- 2.8 Deformed Shell Model.- 2.8.1 Experimental Evidence.- 2.8.2 General Deformed Potential.- 2.8.3 The Anisotropic Harmonic Oscillator.- 2.8.4 Nilsson Hamiltonian.- 2.8.5 Quantum Numbers of the Ground State in Odd Nuclei.- 2.8.6 Calculation of Deformation Energies.- 2.9 Shell Corrections to the Liquid Drop Model and the Strutinski Method.- 2.9.1 Introduction.- 2.9.2 Basic Ideas of the Strutinski Averaging Method.- 2.9.3 Determination of the Average Level Density.- 2.9.4 Strutinski's Shell Correction Energy.- 2.9.5 Shell Corrections and the Hartree-Fock Method.- 2.9.6 Some Applications.- 3 Rotation and Single-Particle Motion.- 3.1 Introduction.- 3.2 General Survey.- 3.2.1 Experimental Observation of High Spin States.- 3.2.2 The Structure of the Yrast Line.- 3.2.3 Phenomenological Classification of the Yrast Band.- 3.2.3 The Backbending Phenomenon.- 3.3 The Particle-plus-Rotor Model.- 3.3.1 The Case of Axial Symmetry.- 3.3.2 Some Applications of the Particle-plus-Rotor Model.- 3.3.3 The triaxial Particle-plus-Rotor Model.- 3.3.4 Electromagnetic Properties.- 3.4 The Cranking Model.- 3.4.1 Semiclassical Derivation of the Cranking Model.- 3.4.2 The Cranking Formula.- 3.4.3 The Rotating Anisotropic Harmonic Oscillator.- 3.4.4 The Rotating Nilsson Scheme.- 3.4.5 The Deformation Energy Surface at High Angular Momenta.- 3.4.6 Rotation about a Symmetry Axis.- 3.4.7 Yrast Traps.- 4 Nuclear Forces.- 4.1 Introduction.- 4.2 The Bare Nucleon-Nucleon Force.- 4.2.1 General Properties of a Two-Body Force.- 4.2.2 The Structure of the Nucleon-Nucleon Interaction.- 4.3 Microscopic Effective Interactions.- 4.3.1 Bruckner's G-Matrix and Bethe Goldstone Equation.- 4.3.2 Effective Interactions between Valence Nucleons.- 4.3.3 Effective Interactions between Particles and Holes.- 4.4 Phenomenological Effective Interactions.- 4.4.1 General Remarks.- 4.4.2 Simple Central Forces.- 4.4.3 The Skyrme Interaction.- 4.4.4 The Gogny Interaction.- 4.4.5 The Migdal Force.- 4.4.6 The Surface-Delta Interaction (SDI).- 4.4.7 Separable Forces and Multipole Expansions.- 4.4.8 Experimentally Determined Effective Interactions.- 4.5 Concluding Remarks.- 5 The Hartree-Fock Method.- 5.1 Introduction.- 5.2 The General Variational Principle.- 5.3 The Derivation of the Hartree-Fock Equation.- 5.3.1 The Choice of the Set of Trial Wave Functions.- 5.3.2 The Hartree-Fock Energy.- 5.3.3 Variation of the Energy.- 5.3.4 The Hartree-Fock Equations in Coordinate Space.- 5.4 The Hartree-Fock Method in a Simple Solvable Model.- 5.5 The Hartree-Fock Method and Symmetries.- 5.6 Hartree-Fock with Density Dependent Forces.- 5.6.1 Approach with Microscopic Effective Interactions.- 5.6.2 Hartree-Fock Calculations with the Skyrme Force.- 5.7 Concluding Remarks.- 6 Pairing Correlations and Superfluid Nuclei.- 6.1 Introduction and Experimental Survey.- 6.2 The Seniority Scheme.- 6.3 The BCS Model.- 6.3.1 The Wave Function.- 6.3.2 The BCS Equations.- 6.3.3 The Special Case of a Pure Pairing Force.- 6.3.4 Bogoliubov Quasi-particles-Excited States and Blocking.- 6.3.5 Discussion of the Gap Equation.- 6.3.6 Schematic Solution of the Gap Equation.- 7 The Generalized Single-Particle Model (HFB Theory).- 7.1 Introduction.- 7.2 The General Bogoliubov Transformation.- 7.2.1 Quasi-particle Operators.- 7.2.2 The Quasi-particle Vacuum.- 7.2.3 The Density Matrix and the Pairing Tensor.- 7.3 The Hartree-Fock-Bogoliubov Equations.- 7.3.1 Derivation of the HFB Equation.- 7.3.2 Properties of the HFB Equations.- 7.3.3 The Gradient Method.- 7.4 The Pairing-plus-Quadrupole Model.- 7.5 Applications of the HFB Theory for Ground State Properties.- 7.6 Constrained Hartree-Fock Theory (CHF).- 7.7 HFB Theory in the Rotating Frame (SCC).- 8 Harmonic Vibrations.- 8.1 Introduction.- 8.2 Tamm-Dancoff Method.- 8.2.1 Tamm-Dancoff Secular Equation.- 8.2.3 The Schematic Model.- 8.2.3 Particle-Particle (Hole-Hole) Tamm-Dancoff Method.- 8.3 General Considerations for Collective Modes.- 8.3.1 Vibrations in Quantum Mechanics.- 8.3.2 Classification of Collective Modes.- 8.3.3 Discussion of Some Collective ph-Vibrations.- 8.3.4 Analog Resonances.- 8.3.5 Pairing Vibrations.- 8.4 Particle-Hole Theory with Ground State Correlations (RPA).- 8.4.1 Derivation of the RPA Equations.- 8.4.2 Stability of the RPA.- 8.4.3 Normalization and Closure Relations.- 8.4.4 Numerical Solution of the RPA Equations.- 8.4.5 Representation by Boson Operators.- 8.4.6 Construction of the RPA Ground State.- 8.4.7 Invariances and Spurious Solutions.- 8.5 Linear Response Theory.- 8.5.1 Derivation of the Linear Response Equations.- 8.5.2 Calculation of Excitation Probabilities and Schematic Model.- 8.5.3 The Static Polarizability and the Moment of Inertia.- 8.5.4 RPA Equations in the Continuum.- 8.6 Applications and Comparison with Experiment.- 8.6.1 Particle-Hole Calculations in a Phenomenological Basis.- 8.6.2 Particle-Hole Calculations in a Self-Consistent Basis.- 8.7 Sum Rules.- 8.7.1 Sum Rules as Energy Weighted Moments of the Strength Functions.- 8.7.2 The S1-Sum Rule and the RPA Approach.- 8.7.3 Evaluation of the Sum Rules S1, S?1, and S3.- 8.7.4 Sum Rules and Polarizabilities.- 8.7.5 Calculation of Transition Currents and Densities.- 8.8 Particle-Particle RPA.- 8.8.1 The Formalism.- 8.8.2 Ground State Correlations Induced by Pairing Vibrations.- 8.9 Quasi-particle RPA.- 9 Boson Expansion Methods.- 9.1 Introduction.- 9.2 Boson Representations in Even-Even Nuclei.- 9.2.1 Boson Representations of the Angular Momentum Operators.- 9.2.2 Concepts for Boson Expansions.- 9.2.3 The Boson Expansion of Belyaev and Zelevinski.- 9.2.4 The Boson Expansion of Marumori.- 9.2.5 The Boson Expansion of Dyson.- 9.2.6 The Mathematical Background.- 9.2.7 Methods Based on pp-Bosons.- 9.2.8 Applications.- 9.3 Odd Mass Nuclei and Particle Vibration Coupling.- 9.3.1 Boson Expansion for Odd Mass Systems.- 9.3.2 Derivation of the Particle Vibration Coupling (Bohr) Hamiltonian.- 9.3.3 Particle Vibration Coupling (Perturbation Theory).- 9.3.4 The Nature of the Particle Vibration Coupling Vertex.- 9.3.5 Effective Charges.- 9.3.6 Intermediate Coupling and Dyson's Boson Expansion.- 9.3.7 Other Particle Vibration Coupling Calculations.- 9.3.8 Weak Coupling in Even Systems.- 10 The Generator Coordinate Method.- 10.1 Introduction.- 10.2 The General Concept.- 10.2.1 The GCM Ansatz for the Wave Function.- 10.2.2 The Determination of the Weight Function f(a).- 10.2.3 Methods of Numerical Solution of the HW Equation.- 10.3 The Lipkin Model as an Example.- 10.4 The Generator Coordinate Method and Boson Expansions.- 10.5 The One-Dimensional Harmonic Oscillator.- 10.6 Complex Generator Coordinates.- 10.6.1 The Bargman Space.- 10.6.2 The Schrödinger Equation.- 10.6.3 Gaussian Wave Packets in the Harmonic Oscillator.- 10.6.4 Double Projection.- 10.7 Derivation of a Collective Hamiltonian.- 10.7.1 General Considerations.- 10.7.2 The Symmetric Moment Expansion (SME).- 10.7.3 The Local Approximation (LA).- 10.7.4 The Gaussian Overlap Approximation (GOAL).- 10.7.5 The Lipkin Model.- 10.7.6 The Multidimensional Case.- 10.8 The Choice of the Collective Coordinate.- 10.9 Application of the Generator Coordinate Method for Bound States.- 10.9.1 Giant Resonances.- 10.9.2 Pairing Vibrations.- 11 Restoration of Broken Symmetries.- 11.1 Introduction.- 11.2 Symmetry Violation in the Mean Field Theory.- 11.3 Transformation to an Intrinsic System.- 11.3.1 General Concepts.- 11.3.2 Translational Motion.- 11.3.3 Rotational Motion.- 11.4 Projection Methods.- 11.4.1 Projection Operators.- 11.4.2 Projection Before and After the Variation.- 11.4.3 Particle Number Projection.- 11.4.4 Approximate Projection for Large Deformations.- 11.4.5 The Inertial Parameters.- 11.4.6 Angular Momentum Projection.- 11.4.7 The Structure of the Intrinsic Wave Functions.- 12 The Time Dependent Hartree-Fock Method (TDHF).- 12.1 Introduction.- 12.2 The Full Time-Dependent Hartree-Fock Theory.- 12.2.1 Derivation of the TDHF Equation.- 12.2.2 Properties of the TDHF Equation.- 12.2.3 Quasi-static Solutions.- 12.2.4 General Discussion of the TDHF Method.- 12.2.5 An Exactly Soluble Model.- 12.2.6 Applications of the TDHF Theory.- 12.3 Adiabatic Time-Dependent Hartree-Fock Theory (ATDHF).- 12.3.1 The ATDHF Equations.- 12.3.2 The Collective Hamiltonian.- 12.3.3 Reduction to a Few Collective Coordinates.- 12.3.4 The Choice of the Collective Coordinates.- 12.3.5 General Discussion of the Atdhf Methods.- 12.3.6 Applications of the ATDHF Method.- 12.3.7 Adiabatic Perturbation Theory and the Cranking Formula.- 13 Semiclassical Methods in Nuclear Physics.- 13.1 Introduction.- 13.2 The Static Case.- 13.2.1 The Thomas-Fermi Theory.- 13.2.2 Wigner-Kirkwood ?-Expansion.- 13.2.3 Partial Resummation of the ?-Expansion.- 13.2.4 The Saddle Point Method.- 13.2.5 Application to a Sperical Woods-Saxon Potential.- 13.2.6 Semiclassical Treatment of Pairing Properties.- 13.3 The Dynamic Case.- 13.3.1 The Boltzmann Equation.- 13.3.2 Fluid Dynamic Equations from the Boltzmann Equation.- 13.3.3 Application of Ordinary Fluid Dynamics to Nuclei.- 13.3.4 Variational Derivation of Fluid Dynamics.- 13.3.5 Momentum Distribution of the Density ?O.- 13.3.6 Imposed Fluid Dynamic Motion.- 13.3.7 An Illustrative Example.- Appendices.- A Angular Momentum Algebra in the Laboratory and the Body-Fixed System.- B Electromagnetic Moments and Transitions.- B.l The General Form of the Hamiltonian.- B.2 Static Multipole Moments.- B.3 The Multipole Expansion of the Radiation Field.- B.4 Multipole Transitions.- B.5 Single-Particle Matrix Elements in a Spherical Basis.- B.6 Translational Invariance and Electromagnetic Transitions.- B.7 The Cross Section for the Absorption of Dipole Radiation.- C Second Quantization.- C.1 Creation and Annihilation Operators.- C.2 Field Operators in the Coordinate Space.- C.3 Representation of Operators.- C.4 Wick's Theorem.- D Density Matrices.- D.l Normal Densities.- D.2 Densities of Slater Determinants.- D.3 Densities of BCS and HFB States.- D.4 The Wigner Transformation of the Density Matrix.- E Theorems Concerning Product Wave Functions.- E.l The Bloch-Messiah Theorem [BM 62].- E.2 Operators in the Quasi-particle Space.- E.3 Thouless' Theorem.- E.4 The Onishi Formula.- E.5 Bogoliubov Transformations for Bosons.- F Many-Body Green's Functions.- F.l Single-Particle Green's Function and Dyson's Equation.- F.2 Perturbation Theory.- F.3 Skeleton Expansion.- F.4 Factorization and Brückner-Hartree-Fock.- F.5 Hartree-Fock-Bogoliubov Equations.- F.6 The Bethe-Salpeter Equation and Effective Forces.- Author Index.