The Nuclear Many-Body Problem

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