The Nuclear Many-Body Problem

It is the aim of this book to describe in concise form our present theoretical understanding of the nuclear many-body problem. The presen tation of the enormous amount of material that has accumulated in this field over the last few decades may be divided into two broad categories: One can either concentrate on the physical phenomena, such as the single-particle excitations, rotations, vibrations, or large-amplitude collec tive motion, and treat each of them using a variety of theoretical methods; or one may stress the methodology and technical aspects of the different theories that have been used to describe the nucleus. We have chosen the second avenue. The structure of this book is thus dictated by the different methods used-Hartree-Fock theory, time-dependent Hartree-Fock the ory, generator coordinates, boson expansions, etc. -rather than by the physical subjects. Many of the present theories have, of course, already been presented in other textbooks. In order to be able to give a more rounded picture, we shall either briefly review such topics (as in the case of the liquid drop or the shell model) or try to give more updated versions (as in the cases of rotations or the random phase approximation). Our essential aim, however, is to present the more modern theories-such as boson expansions, genera tor coordinates, time-dependent Hartree-Fock, semiclassical theories, etc. -which have either never been seen, or at best had little detailed treat ment in, book form.

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. 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. 7 Comparison with Experiment. - 2. 8 Deformed Shell Model. - 2. 9 Shell Corrections to the Liquid Drop Model and the Strutinski Method. - 3 Rotation and Single-Particle Motion. - 3. 1 Introduction. - 3. 2 General Survey. - 3. 3 The Particle-plus-Rotor Model. - 3. 4 The Cranking Model. - 4 Nuclear Forces. - 4. 1 Introduction. - 4. 2 The Bare Nucleon-Nucleon Force. - 4. 3 Microscopic Effective Interactions. - 4. 4 Phenomenological 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. 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. 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. - 7 The Generalized Single-Particle Model (HFB Theory). - 7. 1 Introduction. - 7. 2 The General Bogoliubov Transformation. - 7. 3 The Hartree-Fock-Bogoliubov Equations. - 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. 3 General Considerations for Collective Modes. - 8. 4 Particle-Hole Theory with Ground State Correlations (RPA). - 8. 5 Linear Response Theory. - 8. 6 Applications and Comparison with Experiment. - 8. 7 Sum Rules. - 8. 8 Particle-Particle RPA. - 8. 9 Quasi-particle RPA. - 9 Boson Expansion Methods. - 9. 1 Introduction. - 9. 2 Boson Representations in Even-Even Nuclei. - 9. 3 Odd Mass Nuclei and Particle Vibration Coupling. - 10 The Generator Coordinate Method. - 10. 1 Introduction. - 10. 2 The General Concept. - 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. 7 Derivation of a Collective Hamiltonian. - 10. 8 The Choice of the Collective Coordinate. - 10. 9 Application of the Generator Coordinate Method for Bound States. - 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. 4 Projection Methods. - 12 The Time Dependent Hartree-Fock Method (TDHF). - 12. 1 Introduction. - 12. 2 The Full Time-Dependent Hartree-Fock Theory. - 12. 3 Adiabatic Time-Dependent Hartree-Fock Theory (ATDHF). - 13 Semiclassical Methods in Nuclear Physics. - 13. 1 Introduction. - 13. 2 The Static Case. - 13. 3 The Dynamic Case. - 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.