The Nuclear Shell Model

1. Angular Momentum in Quantum Mechanics.- 1.1 Central Force Problem and Orbital Angular Momentum.- 1.2 General Definitions of Angular Momentum.- 1.2.1 Matrix Representations.- 1.2.2 Example for Spin 1/2 Particles.- 1.3 Total Angular Momentum for a Spin 1/2 Particle.- 1.4 Coupling of Two Angular Momenta: Clebsch-Gordan Coefficients.- 1.5 Properties of Clebsch-Gordan Coefficients.- 1.6 Racah Recoupling Coefficients: Coupling of Three Angular Momenta.- 1.7 Symmetry Properties of 6j-Symbols.- 1.8 Wigner 9j-Symbols: Coupling and Recoupling of Four Angular Momenta.- 1.9 Classical Limit of Wigner 3j-Symbols.- Short Overview of Angular Momentum Coupling Formulas.- 2. Rotations in Quantum Mechanics.- 2.1 Rotation of a Scalar Field-Rotation Group O(3).- 2.2 General Groups of Transformations.- 2.3 Representations of the Rotation Operator.- 2.3.1 The Wigner D-Matrices.- 2.3.2 The Group SU(2)-Relation with SO(3).- 2.3.3 Application: Geometric Interpretation of Intrinsic Spin 1/2.- 2.4 Product Representations and Irreducibility.- 2.5 Cartesian Tensors, Spherical Tensors, Irreducible Tensors.- 2.6 Tensor Product.- 2.7 Spherical Tensor Operators: The Wigner-Eckart Theorem.- 2.8 Calculation of Matrix Elements.- 2.8.1 Reduction Rule I.- 2.8.2 Reduction Rule II.- Short Overview of Rotation Properties, Tensor Operators, Matrix Elements.- 3. The Nuclear Shell Model.- 3.1 One-particle Excitations.- 3.1.1 Introduction.- 3.1.2 The Radial Equation and the Single-particle Spectrum: the Harmonic Oscillator in the Shell Model.- 3.1.3 Illustrative Examples of Energy Spectra.- 3.1.4 Hartree-Fock Methods: A Simple Approach.- 3.2 Two-particle Systems: Identical Nucleons.- 3.2.1 Two-particle Wavefunctions.- 3.2.2 Two-particle Residual Interaction.- 3.2.3 Calculation of Two-Body Matrix Elements.- 3.2.4 Configuration Mixing: Model Space and Model Interaction.- 3.3 Three-particle Systems and Beyond.- 3.3.1 Three-particle Wave Functions.- 3.3.2 Extension to n-particle Wave Functions.- 3.3.3 Some Applications: Three-particle Systems.- 3.4 Non-identical Particle Systems: Isospin.- 3.4.1 Isospin: Introduction and Concepts.- 3.4.2 Isospin Formalism.- 3.4.3 Two-Body Matrix Elements with Isospin.- 4. Electromagnetic Properties in the Shell Model.- 4.1 General.- 4.2 Electric and Magnetic Multipole Operators.- 4.3 Single-particle Estimates and Examples.- 4.4 Electromagnetic Transitions in Two-particle Systems.- 4.5 Quadrupole Moments.- 4.5.1 Single-particle Quadrupole Moment.- 4.5.2 Two-particle Quadrupole Moment.- 4.6 Magnetic Dipole Moment.- 4.6.1 Single-particle Moment: Schmidt Values.- 4.6.2 Two-particle Dipole Moment.- 4.7 Additivity Rules for Static Moments.- 5. Second Quantization.- 5.1 Creation and Annihilation Operators.- 5.2 Operators in Second Quantization.- 5.3 Angular Momentum Coupling in Second Quantization.- 5.4 Hole Operators in Second Quantization.- 5.5 Normal Ordering, Contraction, Wick's Theorem.- 5.6 Application to the Hartree-Fock Formalism.- 6. Elementary Modes of Excitation: Particle-Hole Excitations at Closed Shells.- 6.1 General.- 6.2 The TDA Approximation.- 6.3 The RPA Approximation.- 6.4 Application of the Study of 1p-1h Excitations: 16O.- 7. Pairing Correlations: Particle-Particle Excitations in Open-Shell Nuclei.- 7.1 Introduction.- 7.2 Pairing in a Degenerate Single j-Shell.- 7.3 Pairing in Non-Degenerate Levels: Two-Particle Systems.- 7.4 n Particles in Non-Degenerate Shells: BCS-Theory.- 7.5 Applications of BCS.- 7.5.1 Odd-Even Mass Differences, E1qp.- 7.5.2 Energy Spectra.- 7.5.3 Electromagnetic Transitions.- 7.5.4 Spectroscopic Factors.- 7.6 Broken-Pair Model.- 7.6.1 Low-Seniority Approximation to the Shell Model.- 7.6.2 Broken-Pair or Generalized-Seniority Scheme for Semi-Magic Nuclei.- 7.6.3 Generalization to Both Valence Protons and Neutrons.- 7.7 Interacting Boson-Model Approximation to the Nuclear Shell Model.- 8. Self-Consistent Shell-Model Calculations.- 8.1 Introduction.- 8.2 Construction of a Nucleon-Nucleon Force: Skyrme Forces.- 8.2.1 Hartree-Fock Bogoliubov (HFB) Formalism for Nucleon-Nucleon Interactions Including Three-Body Forces.- 8.2.2 Application of HFB to Spherical Nuclei.- 8.2.3 The Extended Skyrme Force.- 8.2.4 Parameterization of Extended Skyrme Forces: Nuclear Ground-State Properties.- 8.3 Excited-State Properties of SkE Forces.- 8.3.1 Particle-Particle Excitations: Determination of x3.- 8.3.2 The Skyrme Interaction as a Particle-Hole Interaction.- 8.3.3 Rearrangement Effects for Density-Dependent Interactions and Applications for SkE Forces.- 9. Some Computer Programs.- 9.1 Clebsch-Gordan Coefficients.- 9.2 Wigner 6j-Symbol.- 9.3 Wigner 9j-Symbol.- 9.4 Calculation of Table of Slater Integrals.- 9.5 Calculation of ?-Matrix Element.- 9.6 Matrix Diagonalization.- 9.7 Radial Integrals Using Harmonic Oscillator Wave Functions.- 9.8 BCS Equations with Constant Pairing Strength.- A. The Angular Momentum Operator in Spherical Coordinates.- B. Explicit Calculation of the Transformation Coefficients for Three-Angular Momentum Systems.- C. Tensor Reduction Formulae for Tensor Products.- D. The Surface-Delta Interaction (SDI).- G. The Magnetic Multipole Operator.- H. A Two-Group (Degenerate) RPA Model.- I. The Condon-Shortley and Biedenharn-Rose Phase Conventions: Application to Electromagnetic Operators and BCS Theory.- 1.1 Electromagnetic Operators: Long-Wavelength Form and Matrix Elements.- 1.2 Properties of the Electromagnetic Multipole Operators Under Parity Operation,Time Reflection and Hermitian Conjugation.- 1.3 Phase Conventions in the BCS Formalism.- Problems.- References.