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[资源] From Nucleons to Nucleus - Concepts of Microscopic Nuclear Theory

Contents
Part I Particles and Holes
1 Angular Momentum Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1 Clebsch–Gordan Coefficients and 3j Symbols. . . . . . . . . . . . . . . . 3
1.2 More on Clebsch–Gordan Coefficients; 3j Symbols . . . . . . . . . . . 7
1.2.1 Clebsch–Gordan Coefficients . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.2 More Symmetry: 3j Symbols. . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.3 Relations for 3j Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3 The 6j Symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3.1 Symmetry Properties of the 6j Symbol . . . . . . . . . . . . . . . 13
1.3.2 Relations for 6j Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3.3 Explicit Expressions for 6j Symbols. . . . . . . . . . . . . . . . . . 14
1.4 The 9j Symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.4.1 Symmetry Properties of the 9j Symbol . . . . . . . . . . . . . . . 17
1.4.2 Relations for 9j Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2 Tensor Operators and the Wigner–Eckart Theorem . . . . . . . . 23
2.1 Spherical Tensor Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.1.1 Rotations of the Coordinate Axes . . . . . . . . . . . . . . . . . . . 23
2.1.2 Wigner D Functions and Spherical Tensors . . . . . . . . . . . 25
2.1.3 Contravariant and Covariant Components
of Spherical Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2 The Wigner–Eckart Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2.1 Immediate Consequences of the Wigner–Eckart Theorem 29
2.2.2 Pauli Spin Matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.3 Matrix Elements of Coupled Tensor Operators . . . . . . . . . . . . . . 32
2.3.1 Theorem I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.3.2 Theorem II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
XII Contents
3 The Nuclear Mean Field and Many-Nucleon Configurations 39
3.1 The Nuclear Mean Field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.1.1 The Mean-Field Approximation . . . . . . . . . . . . . . . . . . . . . 40
3.1.2 Phenomenological Potentials . . . . . . . . . . . . . . . . . . . . . . . . 44
3.1.3 The Spin–Orbit Interaction . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.2 Woods–Saxon Wave Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2.1 Harmonic Oscillator Wave Functions . . . . . . . . . . . . . . . . . 48
3.2.2 Diagonalization of the Woods–Saxon Hamiltonian . . . . . 50
3.3 Many-Nucleon Configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4 Occupation Number Representation . . . . . . . . . . . . . . . . . . . . . . . 63
4.1 Occupation Number Representation of Many-Nucleon States . . 63
4.1.1 Fock Space: Particle Creation and Annihilation . . . . . . . 64
4.1.2 Further Properties of Creation
and Annihilation Operators. . . . . . . . . . . . . . . . . . . . . . . . . 66
4.2 Operators and Their Matrix Elements. . . . . . . . . . . . . . . . . . . . . . 67
4.2.1 Occupation Number Representation
of One-Body Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.2.2 Matrix Elements of One-Body Operators . . . . . . . . . . . . . 68
4.2.3 Occupation Number Representation
of Two-Body Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.3 Evaluation of Many-Nucleon Matrix Elements. . . . . . . . . . . . . . . 70
4.3.1 Normal Ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.3.2 Contractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.3.3 Wick’s Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.4 Particle–Hole Representation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.4.1 Properties of Particle and Hole Operators . . . . . . . . . . . . 75
4.4.2 Particle–Hole Representation of Operators
and Excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.5 Hartree–Fock Equation from Wick’s Theorem . . . . . . . . . . . . . . . 78
4.5.1 Derivation of the Hartree–Fock Equation . . . . . . . . . . . . . 78
4.5.2 Residual Interaction; Ground-State Energy . . . . . . . . . . . 80
4.6 Hartree–Fock Eigenvalue Problem . . . . . . . . . . . . . . . . . . . . . . . . . 81
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5 The Mean-Field Shell Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.1 Valence Space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.2 One-Particle and One-Hole Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.2.1 Examples of One-Particle Nuclei . . . . . . . . . . . . . . . . . . . . 89
5.2.2 Examples of One-Hole Nuclei . . . . . . . . . . . . . . . . . . . . . . . 91
5.3 Two-Particle and Two-Hole Nuclei. . . . . . . . . . . . . . . . . . . . . . . . . 93
5.3.1 Examples of Two-Particle Nuclei . . . . . . . . . . . . . . . . . . . . 93
5.3.2 Examples of Two-Hole Nuclei . . . . . . . . . . . . . . . . . . . . . . . 97
5.4 Particle–Hole Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
Contents XIII
5.5 Isospin Representation of Few-Nucleon Systems . . . . . . . . . . . . . 105
5.5.1 General Isospin Formalism. . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.5.2 Tensor Operators in Isospin Representation . . . . . . . . . . . 107
5.5.3 Isospin Representation of Two-Particle
and Two-Hole Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.5.4 Isospin Representation of Particle–Hole Nuclei . . . . . . . . 112
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
6 Electromagnetic Multipole Moments
and Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.1 General Properties of Electromagnetic Observables . . . . . . . . . . 117
6.1.1 Transition Probability and Half-Life . . . . . . . . . . . . . . . . . 118
6.1.2 Selection Rules for Electromagnetic Transitions . . . . . . . 121
6.1.3 Single-Particle Matrix Elements
of the Multipole Operators . . . . . . . . . . . . . . . . . . . . . . . . . 122
6.1.4 Properties of the Radial Integrals. . . . . . . . . . . . . . . . . . . . 124
6.1.5 Tables of Numerical Values of Single-Particle Matrix
Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.1.6 Electromagnetic Multipole Moments . . . . . . . . . . . . . . . . . 128
6.1.7 Weisskopf Units and Transition Rates . . . . . . . . . . . . . . . . 130
6.2 Electromagnetic Transitions in One-Particle
and One-Hole Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
6.2.1 Reduced Transition Probabilities . . . . . . . . . . . . . . . . . . . . 132
6.2.2 Example: Transitions in One-Hole Nuclei
15 N and 15 O . 134
6.2.3 Magnetic Dipole Moments: Schmidt Lines . . . . . . . . . . . . 136
6.3 Electromagnetic Transitions in Two-Particle
and Two-Hole Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.3.1 Example: Transitions in Two-Particle Nuclei
18 O
and
18 Ne . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
6.4 Electromagnetic Transitions in Particle–Hole Nuclei. . . . . . . . . . 140
6.4.1 Transitions Involving Charge-Conserving Particle–
Hole Excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
6.4.2 Example: Doubly Magic Nucleus
16 O . . . . . . . . . . . . . . . . 143
6.4.3 Transitions Between Charge-Changing Particle–Hole
Excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
6.4.4 Example: Odd–Odd Nucleus
16 N . . . . . . . . . . . . . . . . . . . . 147
6.5 Isoscalar and Isovector Transitions . . . . . . . . . . . . . . . . . . . . . . . . . 148
6.5.1 Isospin Decomposition of the Electromagnetic
Decay Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
6.5.2 Example: 3 − States in
16 O . . . . . . . . . . . . . . . . . . . . . . . . . 149
6.5.3 Isospin Selection Rules in Two-Particle
and Two-Hole Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
XIV Contents
7 Beta Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
7.1 General Properties of Nuclear Beta Decay . . . . . . . . . . . . . . . . . . 157
7.2 Allowed Beta Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
7.2.1 Half-Lives, Reduced Transition Probabilities
and ft Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
7.2.2 Fermi and Gamow–Teller Matrix Elements . . . . . . . . . . . 165
7.2.3 Phase-Space Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
7.2.4 Combined β + and Electron Capture Decays . . . . . . . . . . 168
7.2.5 Decay Q Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
7.2.6 Partial and Total Decay Half-Lives; Decay Branchings . 169
7.2.7 Classification of Beta Decays. . . . . . . . . . . . . . . . . . . . . . . . 170
7.3 Beta-Decay Transitions in One-Particle and One-Hole Nuclei. . 171
7.3.1 Matrix Elements and Reduced Transition Probabilities . 171
7.3.2 Application to Beta Decay of
15 O; Other Examples . . . . 172
7.4 Beta-Decay Transitions in Particle–Hole Nuclei. . . . . . . . . . . . . . 174
7.4.1 Beta Decay to and from the Even–Even Ground State . 174
7.4.2 Application to Beta Decay of
56 Ni . . . . . . . . . . . . . . . . . . . 175
7.4.3 Beta-Decay Transitions Between Two Particle–Hole
States. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
7.4.4 Application to Beta Decay of
16 N . . . . . . . . . . . . . . . . . . . 178
7.5 Beta-Decay Transitions in Two-Particle and Two-Hole Nuclei . 180
7.5.1 Transition Amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
7.5.2 Application to Beta Decay of
6 He . . . . . . . . . . . . . . . . . . . 183
7.5.3 Application to the Beta-Decay Chain
18 Ne → 18 F → 18 O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
7.5.4 Further Examples: Beta Decay in A = 42
and A = 54 Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
7.6 Forbidden Unique Beta Decay. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
7.6.1 General Aspects of First-Forbidden Beta Decay . . . . . . . 188
7.6.2 First-Forbidden Unique Beta Decay. . . . . . . . . . . . . . . . . . 190
7.6.3 Application to First-Forbidden Unique Beta Decay
of
16 N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
7.6.4 Higher-Forbidden Unique Beta Decay . . . . . . . . . . . . . . . . 193
7.6.5 Application to Third-Forbidden Unique Beta Decay
of
40 K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
7.6.6 Forbidden Unique Beta Decay in Few-Particle
and Few-Hole Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
7.6.7 Forbidden Non-Unique Beta Decays . . . . . . . . . . . . . . . . . 201
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
8 Nuclear Two-Body Interaction and Configuration Mixing. . 205
8.1 General Properties of the Nuclear Two-Body Interaction . . . . . 205
8.1.1 Coupled Two-Body Interaction Matrix Elements . . . . . . 206
8.1.2 Relations for Coupled Two-Body Matrix Elements. . . . . 209
8.1.3 Different Types of Two-Body Interaction . . . . . . . . . . . . . 210
Contents XV
8.1.4 Central Forces with Spin and Isospin Dependendence . . 212
8.2 Separable Interactions; the Surface Delta Interaction . . . . . . . . . 213
8.2.1 Multipole Decomposition of a General Separable
Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
8.2.2 Two-Body Matrix Elements of the Surface Delta
Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
8.3 Configuration Mixing in Two-Particle Nuclei . . . . . . . . . . . . . . . . 219
8.3.1 Matrix Representation of an Eigenvalue Problem . . . . . . 219
8.3.2 Solving the Eigenenergies of a Two-by-Two Problem . . . 221
8.3.3 Matrix Elements of the Hamiltonian
in the Two-Nucleon Basis . . . . . . . . . . . . . . . . . . . . . . . . . . 223
8.3.4 Solving the Eigenvalue Problem
for a Two-Particle Nucleus . . . . . . . . . . . . . . . . . . . . . . . . . 224
8.3.5 Application to A = 6 Nuclei . . . . . . . . . . . . . . . . . . . . . . . . 225
8.3.6 Application to A = 18 Nuclei . . . . . . . . . . . . . . . . . . . . . . . 228
8.4 Configuration Mixing in Two-Hole Nuclei. . . . . . . . . . . . . . . . . . . 231
8.4.1 Diagonalization of the Residual Interaction
in a Two-Hole Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
8.4.2 Application to A = 14 Nuclei . . . . . . . . . . . . . . . . . . . . . . . 233
8.4.3 Application to A = 38 Nuclei . . . . . . . . . . . . . . . . . . . . . . . 234
8.5 Electromagnetic and Beta-Decay Transitions in Two-Particle. . 236
8.5.1 Transition Amplitudes With Configuration Mixing. . . . . 236
8.5.2 Application to Beta Decay of
6 He . . . . . . . . . . . . . . . . . . . 237
8.5.3 Application to E2 Decays in
18 O and 18 Ne. . . . . . . . . . . . 239
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
9 Particle–Hole Excitations and the Tamm–Dancoff
Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
9.1 The Tamm–Dancoff Approximation . . . . . . . . . . . . . . . . . . . . . . . . 243
9.1.1 Justification of the TDA: Brillouin’s Theorem . . . . . . . . . 243
9.1.2 Derivation of Explicit Expressions for the TDA Matrix . 246
9.1.3 Tabulated Values of Particle–Hole Matrix Elements . . . . 248
9.1.4 TDA as an Eigenvalue Problem; Properties
of the Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
9.2 TDA for General Separable Forces . . . . . . . . . . . . . . . . . . . . . . . . . 253
9.2.1 Schematic Model; Dispersion Equation . . . . . . . . . . . . . . . 253
9.2.2 The Schematic Model for T = 0 and T = 1 . . . . . . . . . . . 256
9.2.3 The Schematic Model with the Surface
Delta Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
9.2.4 Application to 1 − Excitations in
4 He . . . . . . . . . . . . . . . . 258
9.3 Excitation Spectra of Doubly Magic Nuclei . . . . . . . . . . . . . . . . . 260
9.3.1 Block Decomposition of the TDA Matrix . . . . . . . . . . . . . 260
9.3.2 Application to 1 − States in
4 He . . . . . . . . . . . . . . . . . . . . . 260
9.3.3 Application to Excited States in
16 O . . . . . . . . . . . . . . . . . 262
9.3.4 Further Examples:
40 Ca and 48 Ca . . . . . . . . . . . . . . . . . . . 263
XVI Contents
9.4 Electromagnetic Transitions in Doubly Magic Nuclei . . . . . . . . . 265
9.4.1 Transitions to the Particle–Hole Ground State . . . . . . . . 266
9.4.2 Non-Energy-Weighted Sum Rule . . . . . . . . . . . . . . . . . . . . 267
9.4.3 Application to Octupole Transitions in
16 O . . . . . . . . . . . 268
9.4.4 Collective Transitions in the TDA . . . . . . . . . . . . . . . . . . . 271
9.4.5 Application to Octupole Transitions in
40 Ca . . . . . . . . . . 272
9.4.6 E1 Transitions: Isospin Breaking
in the Nuclear Mean Field . . . . . . . . . . . . . . . . . . . . . . . . . . 274
9.4.7 Transitions Between Two TDA Excitations . . . . . . . . . . . 277
9.4.8 Application to the 5 −
1
→ 3 −
1
Transition in
40 Ca. . . . . . . . 277
9.5 Electric Transitions on the Schematic Model . . . . . . . . . . . . . . . . 278
9.5.1 Transition Amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
9.5.2 Application to Electric Dipole Transitions in
4 He . . . . . . 280
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
10 Charge-Changing Particle–Hole Excitations
and the pnTDA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
10.1 The Proton–Neutron Tamm–Dancoff Approximation . . . . . . . . . 287
10.1.1 Structure of the pnTDA Matrix . . . . . . . . . . . . . . . . . . . . . 287
10.1.2 Application to
4
1 H 3
and
4
3 Li 1
. . . . . . . . . . . . . . . . . . . . . . . . 289
10.1.3 Further Examples: States of
16
7 N 9
and
40
19 K 21
. . . . . . . . . . 290
10.2 Electromagnetic Transitions in the pnTDA . . . . . . . . . . . . . . . . . 294
10.2.1 Transition Amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
10.2.2 Application to the E2 Transition 0 −
1
→ 2 −
gs
in
16
7 N 9
. . . . 294
10.3 Beta-Decay Transitions in the pnTDA . . . . . . . . . . . . . . . . . . . . . 296
10.3.1 Transitions to the Particle–Hole Vacuum . . . . . . . . . . . . . 296
10.3.2 First-Forbidden Unique Beta Decay of
16
7 N 9 . . . . . . . . . . . 297
10.3.3 Transitions between Particle–Hole States . . . . . . . . . . . . . 298
10.3.4 Allowed Beta Decay of
16
7 N 9
to Excited States in
16
8 O 8
. 299
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
11 The Random-Phase Approximation . . . . . . . . . . . . . . . . . . . . . . . . 305
11.1 The Equations-of-Motion Method . . . . . . . . . . . . . . . . . . . . . . . . . 305
11.1.1 Derivation of the Equations of Motion . . . . . . . . . . . . . . . 306
11.1.2 Derivation of the Hartree–Fock Equations by the EOM. 310
11.2 Sophisticated Particle–Hole Theories: The RPA . . . . . . . . . . . . . 312
11.2.1 Derivation of the RPA Equations by the EOM . . . . . . . . 312
11.2.2 Explicit Form of the Correlation Matrix . . . . . . . . . . . . . . 315
11.2.3 Numerical Tables of Correlation Matrix Elements . . . . . 317
11.3 Properties of the RPA Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 318
11.3.1 RPA Energies and Amplitudes . . . . . . . . . . . . . . . . . . . . . . 318
11.3.2 The RPA Ground State . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
11.3.3 RPA One-Particle Densities. . . . . . . . . . . . . . . . . . . . . . . . . 324
11.4 RPA Solutions of the Schematic Separable Model . . . . . . . . . . . . 327
11.4.1 The RPA Dispersion Equation . . . . . . . . . . . . . . . . . . . . . . 327
Contents XVII
11.4.2 Application to 1 − Excitations in
4 He . . . . . . . . . . . . . . . . 329
11.4.3 The Degenerate Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331
11.5 RPA Description of Doubly Magic Nuclei . . . . . . . . . . . . . . . . . . . 332
11.5.1 Examples of the RPA Matrices . . . . . . . . . . . . . . . . . . . . . . 332
11.5.2 Diagonalization of the RPA Supermatrix
by Similarity Transformations. . . . . . . . . . . . . . . . . . . . . . . 335
11.5.3 Application to 1 − Excitations in
4 He Carried Through . 337
11.5.4 The 1 − Excitations of
4 He Revisited . . . . . . . . . . . . . . . . . 340
11.5.5 Further Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342
11.6 Electromagnetic Transitions in the RPA Framework . . . . . . . . . 344
11.6.1 Transitions to the RPA Ground State . . . . . . . . . . . . . . . . 344
11.6.2 Extreme Collective Model . . . . . . . . . . . . . . . . . . . . . . . . . . 346
11.6.3 Octupole Decay in
16 O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
11.6.4 The Energy-Weighted Sum Rule. . . . . . . . . . . . . . . . . . . . . 349
11.6.5 Sum Rule for the Octupole Transitions in
16 O. . . . . . . . . 352
11.6.6 Electric Transitions to the RPA Ground State
on the Schematic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
11.6.7 Electric Dipole Transitions in
4 He on the Schematic
Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354
11.6.8 The Degenerate Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355
11.6.9 Electromagnetic Transitions Between Two RPA
Excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356
11.6.10The E2 Transition 5 −
1
→ 3 −
1
in
40 Ca . . . . . . . . . . . . . . . . . 359
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
Part II Quasiparticles
12 Nucleon Pairing and Seniority . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369
12.1 Evidence of Nucleon Pairing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370
12.2 The Pure Pairing Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372
12.3 Two-Particle Spectrum of the Pure Pairing Force . . . . . . . . . . . . 374
12.4 Seniority Model of the Pure Pairing Force . . . . . . . . . . . . . . . . . . 376
12.4.1 Derivation of the Seniority-Zero Spectrum . . . . . . . . . . . . 376
12.4.2 Spectra of Seniority-One and Seniority-Two States . . . . 377
12.4.3 States of Higher Seniority . . . . . . . . . . . . . . . . . . . . . . . . . . 379
12.4.4 Application of the Seniority Model to 0f 7/2 -Shell Nuclei 380
12.5 The Two-Level Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381
12.5.1 The Pair Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382
12.5.2 Matrix Elements of the Pairing Hamiltonian . . . . . . . . . . 383
12.5.3 Application to a Two-Particle System . . . . . . . . . . . . . . . . 386
12.6 Two Particles in a Valence Space of Many j Shells . . . . . . . . . . . 387
12.6.1 Dispersion Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387
12.6.2 The Three-Level Case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389
XVIII Contents
13 BCS Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391
13.1 BCS Quasiparticles and Their Vacuum . . . . . . . . . . . . . . . . . . . . . 391
13.1.1 The BCS Ground State . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392
13.1.2 BCS Quasiparticles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393
13.2 Occupation Number Representation
for BCS Quasiparticles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394
13.2.1 Contraction Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395
13.2.2 Quasiparticle Representation of the Nuclear
Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395
13.3 Derivation of the BCS Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 398
13.3.1 BCS as a Constrained Variational Problem . . . . . . . . . . . 398
13.3.2 The Gap Equation and the Quasiparticle Mean Field . . 400
13.4 Properties of the BCS Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 403
13.4.1 Physical Meaning of the Basic Parameters . . . . . . . . . . . . 403
13.4.2 Particle Number and Its Fluctuations . . . . . . . . . . . . . . . . 404
13.4.3 Odd–Even Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406
13.5 Solution of the BCS Equations for Simple Models . . . . . . . . . . . 406
13.5.1 Single j Shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407
13.5.2 The Lipkin Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408
13.5.3 Example: The Lipkin Model for Two j =
7
2
Shells . . . . . 411
13.5.4 The Two-Level Model for Two j =
7
2
Shells . . . . . . . . . . . 412
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414
14 Quasiparticle Mean Field: BCS and Beyond . . . . . . . . . . . . . . . 417
14.1 Numerical Solution of the BCS Equations . . . . . . . . . . . . . . . . . . 417
14.1.1 Iterative Numerical Procedure . . . . . . . . . . . . . . . . . . . . . . 418
14.1.2 Application to Nuclei in the d-s and f-p-0g 9/2 Shells . . . 420
14.2 Lipkin–Nogami BCS Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430
14.2.1 The Lipkin–Nogami Model Hamiltonian . . . . . . . . . . . . . . 430
14.2.2 Derivation of the Lipkin–Nogami BCS Equations . . . . . . 433
14.3 Lipkin–Nogami BCS Theory in Simple Models . . . . . . . . . . . . . . 436
14.3.1 Single j Shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436
14.3.2 The Lipkin Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438
14.3.3 Example: The j =
7
2
Case . . . . . . . . . . . . . . . . . . . . . . . . . . 441
14.4 The Two-Level Model for j = j ? =
7
2
. . . . . . . . . . . . . . . . . . . . . . . 443
14.5 Application of Lipkin–Nogami Theory
to Realistic Calculations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447
15 Transitions in the Quasiparticle Picture . . . . . . . . . . . . . . . . . . . 449
15.1 Quasiparticle Representation of a One-Body Transition
Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449
15.2 Transition Densities for Few-Quasiparticle Systems . . . . . . . . . . 450
15.2.1 Transitions Between One-Quasiparticle States . . . . . . . . . 450
Contents XIX
15.2.2 Transitions Between a Two-Quasiparticle State
and the BCS Vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451
15.2.3 Transitions Between Two-Quasiparticle States . . . . . . . . 451
15.3 Transitions in Odd-A Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452
15.3.1 Transition Amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453
15.3.2 Beta and Gamma Decays in the A = 25 Chain
of Isobars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455
15.3.3 Beta Decays in the A = 63 Chain of Isobars . . . . . . . . . . 457
15.4 Transitions Between a Two-Quasiparticle State
and the BCS Vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459
15.4.1 Formalism for Transition Amplitudes . . . . . . . . . . . . . . . . 459
15.4.2 Beta and Gamma Decays in the A = 30 Chain
of Isobars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463
15.5 Transitions Between Two-Quasiparticle States. . . . . . . . . . . . . . . 467
15.5.1 Electromagnetic Transitions . . . . . . . . . . . . . . . . . . . . . . . . 467
15.5.2 Beta-Decay Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469
15.5.3 Beta Decay of
30 P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472
15.5.4 Magnetic Dipole Decay in
30 P . . . . . . . . . . . . . . . . . . . . . . 473
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474
16 Mixing of Two-Quasiparticle Configurations . . . . . . . . . . . . . . . 479
16.1 Quasiparticle Representation of the Residual Interaction. . . . . . 479
16.2 Derivation of the Quasiparticle-TDA Equation . . . . . . . . . . . . . . 484
16.3 General Properties of QTDA Solutions . . . . . . . . . . . . . . . . . . . . . 490
16.3.1 Orthogonality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490
16.3.2 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491
16.4 Excitation Spectra of Open-Shell Even–Even Nuclei. . . . . . . . . . 491
16.4.1 Explicit Form of the QTDA Matrix . . . . . . . . . . . . . . . . . . 492
16.4.2 Excitation Energies of 2 + States in
24 Mg . . . . . . . . . . . . . 493
16.4.3 Pairing Strength Parameters from Empirical Pairing
Gaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498
16.4.4 Excitation Spectrum of
24
12 Mg 12 . . . . . . . . . . . . . . . . . . . . . . 501
16.4.5 Excitation Spectra of the Mirror Nuclei
30
14 Si 16
and
30
16 S 14 502
16.4.6 Excitation Spectrum of
66 Zn . . . . . . . . . . . . . . . . . . . . . . . . 503
16.5 Electromagnetic Transitions to the Ground State . . . . . . . . . . . . 506
16.5.1 Decay Amplitude. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506
16.5.2 E2 Decay of the Lowest 2 + State in
24 Mg . . . . . . . . . . . . 507
16.5.3 Collective States and Electric Transitions. . . . . . . . . . . . . 509
16.6 QTDA Sum Rule for Electromagnetic Transitions . . . . . . . . . . . 513
16.6.1 Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513
16.6.2 Examples of the NEWSR in the 0d-1s and 0f-1p-0g 9/2
Shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514
16.7 Transitions Between QTDA Excited States . . . . . . . . . . . . . . . . . 515
16.7.1 Transition Amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515
16.7.2 Example: The 0 +
1
→ 2 +
1
Transition in
24 Mg . . . . . . . . . . . 516
XX Contents
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519
17 Two-Quasiparticle Mixing in Odd–Odd Nuclei . . . . . . . . . . . . . 523
17.1 The Proton–Neutron QTDA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523
17.1.1 Equation of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524
17.1.2 Properties of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525
17.2 Excitation Spectra of Open-Shell Odd–Odd Nuclei . . . . . . . . . . . 525
17.2.1 1 + States in the Mirror Nuclei
24 Na and 24 Al . . . . . . . . . 526
17.2.2 Energy Spectra in the d-s and f-p-0g 9/2 Shells. . . . . . . . . 527
17.2.3 Average Particle Number in the pnQTDA . . . . . . . . . . . . 530
17.3 Electromagnetic Transitions in the pnQTDA . . . . . . . . . . . . . . . . 533
17.3.1 Decay of the 2 +
1
State in
24 Na . . . . . . . . . . . . . . . . . . . . . . 534
17.4 Beta-Decay Transitions in the pnQTDA . . . . . . . . . . . . . . . . . . . . 537
17.4.1 Transitions to and from an Even–Even Ground State . . 537
17.4.2 Gamow–Teller Beta Decay of
30 S . . . . . . . . . . . . . . . . . . . . 538
17.4.3 The Ikeda Sum Rule and the pnQTDA. . . . . . . . . . . . . . . 541
17.4.4 Examples of the Ikeda Sum Rule . . . . . . . . . . . . . . . . . . . . 545
17.4.5 Gamow–Teller Giant Resonance . . . . . . . . . . . . . . . . . . . . . 548
17.4.6 Beta-Decay Transitions Between a QTDA
and a pnQTDA State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550
17.4.7 Gamow–Teller Beta Decay of
30 P. . . . . . . . . . . . . . . . . . . . 550
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553
18 Two-Quasiparticle Mixing by the QRPA . . . . . . . . . . . . . . . . . . . 557
18.1 The QRPA Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557
18.1.1 Derivation of the QRPA Equations by the EOM. . . . . . . 558
18.1.2 Explicit Form of the Correlation Matrix . . . . . . . . . . . . . . 559
18.2 General Properties of QRPA Solutions . . . . . . . . . . . . . . . . . . . . . 562
18.2.1 QRPA Energies and Wave Functions . . . . . . . . . . . . . . . . . 563
18.2.2 The QRPA Ground State and Transition Densities . . . . 567
18.3 QRPA Description of Open-Shell Even–Even Nuclei. . . . . . . . . . 569
18.3.1 Structure of the Correlation Matrix . . . . . . . . . . . . . . . . . . 569
18.3.2 Excitation Energies of 2 + States in
24 Mg . . . . . . . . . . . . . 570
18.3.3 Further Examples in the 0d-1s Shell . . . . . . . . . . . . . . . . . 572
18.3.4 Spurious Contributions to 1 − States . . . . . . . . . . . . . . . . . 573
18.4 Electromagnetic Transitions
in the QRPA Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575
18.4.1 Transitions to the QRPA Ground State . . . . . . . . . . . . . . 575
18.4.2 E2 Decays in the 0d-1s and 0f-1p-0g 9/2 Shells . . . . . . . . . 577
18.4.3 Energy-Weighted Sum Rule of the QRPA. . . . . . . . . . . . . 579
18.4.4 Electric Quadrupole Sum Rule in
24 Mg . . . . . . . . . . . . . . 580
18.4.5 Electromagnetic Transitions Between Two QRPA
Excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582
18.4.6 Electric Quadrupole Transition 4 +
1
→ 2 +
1
in
24 Mg. . . . . . 584
18.4.7 Collective Vibrations and Rotations . . . . . . . . . . . . . . . . . 587
Contents XXI
18.5 Collective Vibrational Two-Phonon States . . . . . . . . . . . . . . . . . . 587
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 590
19 Proton–Neutron QRPA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595
19.1 The pnQRPA Equation and its Basic Properties. . . . . . . . . . . . . 595
19.1.1 The pnQRPA Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595
19.1.2 Basic Properties of the Solutions of the pnQRPA
Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596
19.2 Description of Open-Shell Odd–Odd Nuclei
by the pnQRPA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599
19.2.1 Low-Lying 1 + States in
24 Na and 24 Al . . . . . . . . . . . . . . . 599
19.2.2 Other Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602
19.3 Average Particle Number in the pnQRPA . . . . . . . . . . . . . . . . . . 604
19.4 Electromagnetic Transitions in the pnQRPA . . . . . . . . . . . . . . . . 605
19.4.1 Transition Amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605
19.4.2 Decay of the 2 +
1
State in
24 Na . . . . . . . . . . . . . . . . . . . . . . 606
19.5 Beta-Decay Transitions in the pnQRPA Framework . . . . . . . . . . 608
19.5.1 Transitions Involving the Even–Even Ground State . . . . 608
19.5.2 Gamow–Teller Decay of the 1 +
1
Isomer in
24 Al . . . . . . . . 610
19.6 The Ikeda Sum Rule for the pnQRPA . . . . . . . . . . . . . . . . . . . . . . 612
19.6.1 Derivation of the Sum Rule . . . . . . . . . . . . . . . . . . . . . . . . . 612
19.6.2 Examples of the Sum Rule . . . . . . . . . . . . . . . . . . . . . . . . . 613
19.7 Beta-Decay Transitions Between a QRPA
and a pnQRPA State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616
19.7.1 Derivation of the Transition Amplitude . . . . . . . . . . . . . . 616
19.7.2 The Gamow–Teller Decay
24 Al(1 +
1 ) →
24 Mg(2 +
1 ) . . . . . . 618
19.7.3 First-Forbidden Unique Beta Decay in the 0f-1p-0g 9/2
Shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 620
19.7.4 Empirical Particle–Hole and Particle–Particle Forces . . . 624
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 629
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633

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