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Contents
6.2 T = 0, 1 Nuclear Matrix Elements in the Berggren Basis. . . . . . . . . . . 244
6.2.1 Dependence of Nuclear Matrix Elements on Angular
Momentum and Continuum Coupling . .. . . . . . . . . . . . . . . . . . . . 246
6.2.2 Influence of Continuum Coupling on Effective
Nuclear Matrix Elements . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 250
6.3 Effective Nucleon–Nucleon Interactions for Gamow Shell
Model Calculations in Light Nuclei . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 253
6.3.1 Study of the Helium, Lithium, and Beryllium
Isotope Chains. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 254
6.4 Three-Body Model in Berggren Basis . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 267
6.4.1 Berggren Basis Expansion in the Three-Body Model .. . . . . 267
6.4.2 Convergence Properties of Eigenvalues Calculated
in Jacobi and Cluster Orbital Shell Model Coordinates .. . . 271
6.4.3 Correlation Densities in a Three-Body Gamow Model . . . . 273
6.5 Inclusion of Continuum Couplings in the Pairing Model .. . . . . . . . . . . 276
6.5.1 Numerical Solution of the Generalized Richardson
Equations .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 284
6.6 Comparison of Gamow Shell Model and Hamiltonian
Complex Scaling Formalisms .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 291
6.6.1 Complex-Scaled Hamiltonians and Their Eigenstates . . . . . 292
6.6.2 Basis Dependence in Truncated Model Spaces .. . . . . . . . . . . . 296
6.6.3 Numerical Comparison of Eigenenergies of Gamow
Shell Model and Hamiltonian Complex Scaling
Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 299
Solutions to Exercises .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 301
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 307
7 Wave Functions in the Vicinity of Particle-Emission Threshold .. . . . . . . 313
7.1 Configuration Mixing in the Vicinity of the Dissociation
Threshold.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 313
7.2 Halo Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 317
7.2.1 Halo States in a Single-Particle Potential .. . . . . . . . . . . . . . . . . . 319
7.2.2 Effects of Nucleon–Nucleon Correlations on Halo
States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 328
7.3 Asymptotic Normalization Coefficient in Mirror Nuclei . . . . . . . . . . . . 331
7.4 Near-Threshold Behavior of Wave Functions . . . .. . . . . . . . . . . . . . . . . . . . 337
7.4.1 Two-Particle Reaction Cross Sections and Wigner
Cusps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 338
7.4.2 Spectroscopic Factors and Radial Overlap Integrals . . . . . . . 339
7.5 Mirror Nuclei and Isospin Symmetry Breaking ... . . . . . . . . . . . . . . . . . . . 350
Solutions to Exercises .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 358
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 360
Contents
6.2 T = 0, 1 Nuclear Matrix Elements in the Berggren Basis. . . . . . . . . . . 244
6.2.1 Dependence of Nuclear Matrix Elements on Angular
Momentum and Continuum Coupling . .. . . . . . . . . . . . . . . . . . . . 246
6.2.2 Influence of Continuum Coupling on Effective
Nuclear Matrix Elements . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 250
6.3 Effective Nucleon–Nucleon Interactions for Gamow Shell
Model Calculations in Light Nuclei . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 253
6.3.1 Study of the Helium, Lithium, and Beryllium
Isotope Chains. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 254
6.4 Three-Body Model in Berggren Basis . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 267
6.4.1 Berggren Basis Expansion in the Three-Body Model .. . . . . 267
6.4.2 Convergence Properties of Eigenvalues Calculated
in Jacobi and Cluster Orbital Shell Model Coordinates .. . . 271
6.4.3 Correlation Densities in a Three-Body Gamow Model . . . . 273
6.5 Inclusion of Continuum Couplings in the Pairing Model .. . . . . . . . . . . 276
6.5.1 Numerical Solution of the Generalized Richardson
Equations .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 284
6.6 Comparison of Gamow Shell Model and Hamiltonian
Complex Scaling Formalisms .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 291
6.6.1 Complex-Scaled Hamiltonians and Their Eigenstates . . . . . 292
6.6.2 Basis Dependence in Truncated Model Spaces .. . . . . . . . . . . . 296
6.6.3 Numerical Comparison of Eigenenergies of Gamow
Shell Model and Hamiltonian Complex Scaling
Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 299
Solutions to Exercises .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 301
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 307
7 Wave Functions in the Vicinity of Particle-Emission Threshold .. . . . . . . 313
7.1 Configuration Mixing in the Vicinity of the Dissociation
Threshold.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 313
7.2 Halo Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 317
7.2.1 Halo States in a Single-Particle Potential .. . . . . . . . . . . . . . . . . . 319
7.2.2 Effects of Nucleon–Nucleon Correlations on Halo
States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 328
7.3 Asymptotic Normalization Coefficient in Mirror Nuclei . . . . . . . . . . . . 331
7.4 Near-Threshold Behavior of Wave Functions . . . .. . . . . . . . . . . . . . . . . . . . 337
7.4.1 Two-Particle Reaction Cross Sections and Wigner
Cusps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 338
7.4.2 Spectroscopic Factors and Radial Overlap Integrals . . . . . . . 339
7.5 Mirror Nuclei and Isospin Symmetry Breaking ... . . . . . . . . . . . . . . . . . . . 350
Solutions to Exercises .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 358
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 360