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The Nuclear Many-Body Problem

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Table of Contents

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 Schroedinger 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 Bruckner-Hartree-Fock.- F.5 Hartree-Fock-Bogoliubov Equations.- F.6 The Bethe-Salpeter Equation and Effective Forces.- Author Index.


From the reviews: "The monography by Peter Ring and Peter Schuck covers the techniques used to solve the nuclear many-body problem ! . is recognized as a reference by the nuclear physics community. Theoretical developments are explained pedagogically, with a constant rigour, are well documented and are illustrated with suitably chosen examples. The book contains a lot of references ! . It is served by a concise style. By its scope and rigour, it has no real rival and will expectedly remain a familiar introductory text in nuclear structure theory for many years." (Joseph Cugnon, Physicalia, Vol. 57 (3), 2005) "In many ways, the 1950s through to the 1970s may be seen as a golden period for the development of nuclear physics, both experimental and theoretical. ! The book contains an excellent description of many basic theoretical methods, which continue to be relevant today, it is still of value to specialist students of nuclear theory." (J. P. Elliott, Contemporary Physics, Vol. 46 (6), 2005)

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