I. BASIC ELEMENTS AND MODELS
1. Elementary concepts of nuclear physics ....................... 3
1.1. The force between two nucleons .......................... 3
1.1.1. Possible forms of the interaction ................ 4
1.1.2. The radial dependence of the interaction ......... 5
1.1.3. The role of sub-nuclear degrees of freedom ....... б
1.2. The model of the Fermi gas .............................. 7
1.2.1. Many-body properties in the ground state ......... 9
1.2.2. Two-body correlations in a homogeneous system ... 11
1.3. Basic properties of finite nuclei ...................... 14
1.3.1. The interaction of nucleons with nuclei ......... 14
1.3.2. The optical model ............................... 19
1.3.3. The liquid drop model ........................... 23
2. Nuclear matter as a Fermi liquid ............................ 30
2.1. A first, qualitative survey ............................ 30
2.1.1. The inadequacy of Hartree-Fock with bare
interactions .................................... 30
2.1.2. Short-range correlations ........................ 32
2.1.3. Properties of nuclear matter in chiral
dynamics ........................................ 34
2.1.4. How dense is nuclear matter? .................... 35
2.2. The independent pair approximation ..................... 36
2.2.1. The equation for the one-body wave functions .... 36
2.2.2. The total energy in terms of two-body wave
functions ....................................... 37
2.2.3. The Bethe-Goldstone equation .................... 38
2.2.4. The G-matrix .................................... 42
2.3. Brueckner-Hartree-Fock approximation (BHF) ............. 44
2.3.1. BHF at finite temperature ....................... 48
2.4. A variational approach based on generalized Jastrow
functions .............................................. 48
2.4.1. Extension to finite temperature ................. 50
2.5. Effective interactions of Skyrme type .................. 52
2.5.1. Expansion to small relative momenta ............. 52
2.6. The nuclear equation of state (EOS) .................... 55
2.6.1. An energy functional with three-body forces ..... 55
2.6.2. The EOS with Skyrme interactions ................ 56
2.6.3. Applications in astrophysics .................... 59
2.7. Transport phenomena in the Fermi liquid ................ 61
2.7.1. Semi-classical transport equations .............. 62
3. Independent particles and quasiparticles in finite nuclei ... 67
3.1. Hartree-Fock with effective forces ..................... 67
3.1.1. H-F with the Skyrme interaction ................. 67
3.1.2. Constrained Hartree-Fock ........................ 68
3.1.3. Other effective interactions .................... 69
3.2. Phenomenological single particle potentials ............ 70
3.2.1. The spherical case .............................. 70
3.2.2. The deformed single particle model .............. 73
3.3. Excitations of the many-body system .................... 78
3.3.1. The concept of particle-hole excitations ........ 78
3.3.2. Pair correlations ............................... 79
4. From the shell model to the compound nucleus ................ 85
4.1. Shell model with residual interactions ................. 85
4.1.1. Nearest level spacing ........................... 86
4.2. Random Matrix Model .................................... 87
4.2.1. Gaussian ensembles of real symmetric matrices ... 87
4.2.2. Eigenvalues, level spacings and eigenvectors .... 89
4.2.3. Comments on the RMM ............................. 91
4.3. The spreading of states into more complicated
configurations ......................................... 93
4.3.1. A schematic model ............................... 94
4.3.2. Strength functions .............................. 95
4.3.3. Time-dependent description ...................... 98
4.3.4. Spectral functions for single particle motion ... 99
5. Shell effects and Strutinsky renormalization ............... 104
5.1. Physical background ................................... 106
5.1.1. The independent particle picture, once more .... 107
5.1.2. The Strutinsky energy theorem .................. 108
5.2. The Strutinsky procedure .............................. 109
5.2.1. Formal aspects of smoothing .................... 109
5.2.2. Shell correction to level density and ground
state energy ................................... 110
5.2.3. Further averaging procedures ................... 113
5.3. The static energy of finite nuclei .................... 115
5.4. An excursion into periodic-orbit theory (POT) ......... 119
5.5. The total energy at finite temperature ................ 122
5.5.1. The smooth part of the energy at small
excitations .................................... 123
5.5.2. Contributions from the oscillating level
density ........................................ 124
6. Average collective motion of small amplitude ............... 128
6.1. Equation of motion from energy conservation ........... 129
6.1.1. Induced forces for harmonic motion ............. 129
6.1.2. Equation of motion ............................. 131
6.1.3. One-particle one-hole excitations .............. 133
6.2. The collective response function ...................... 134
6.2.1. Collective response and sum rules for stable
systems ........................................ 137
6.2.2. Generalization to several dimensions ........... 139
6.2.3. Mean field approximation for an effective
two-body interaction ........................... 141
6.2.4. Isovector modes ................................ 143
6.3. Rotations as degenerate vibrations .................... 143
6.4. Microscopic origin of macroscopic damping ............. 145
6.4.1. Irreversibility through energy smearing ........ 146
6.4.2. Relaxation in a Random Matrix Model ............ 150
6.4.3. The effects of "collisions" on nucleonic
motion ......................................... 150
6.5. Damped collective motion at thermal excitations ....... 154
6.5.1. The equation of motion at finite thermal
excitations .................................... 154
6.5.2. The strict Markov limit ........................ 157
6.5.3. The collective response for quasi-static
processes ...................................... 160
6.5.4. An analytically solvable model ................. 164
6.6. Temperature dependence of nuclear transport ........... 166
6.6.1. The collective strength distribution at
finite T ....................................... 166
6.6.2. Diabatic models ................................ 171
6.6.3. T-dependence of transport coefficients ......... 177
6.7. Rotations at finite thermal excitations ............... 185
7. Transport theory of nuclear collective motion .............. 190
7.1. The locally harmonic approximation .................... 191
7.2. Equilibrium fluctuations of the local oscillator ...... 194
7.3. Fluctuations of the local propagators ................. 196
7.3.1. Quantal diffusion coefficients from the FDT .... 200
7.4. Fokker-Planck equations for the damped harmonic
oscillator ............................................ 203
7.4.1. Stationary solutions for oscillators ........... 203
7.4.2. Dynamics of fluctuations for stable modes ...... 206
7.4.3. The time-dependent solutions for unstable
modes and their physical interpretation ........ 207
7.5. Quantum features of collective transport from the
microscopic point of view ............................. 209
7.5.1. Quantized Hamiltonians for collective motion ... 210
7.5.2. A non-perturbative Nakajima-Zwanzig approach ... 217
II.COMPLEX NUCLEAR SYSTEMS
8. The statistical model for the decay of excited nuclei ...... 225
8.1. Decay of the compound nucleus by particle emission .... 225
8.1.1. Transition rates ............................... 225
8.1.2. Evaporation rates for light particles .......... 228
8.2. Fission ............................................... 229
8.2.1. The Bohr-Wheeler formula ....................... 229
8.2.2. Stability conditions in the macroscopic
limit .......................................... 232
9. Pre-equilibrium reactions .................................. 235
9.1. An illustrative, realistic prototype .................. 236
9.2. A sketch of existing theories ......................... 242
9.2.1. Comments ....................................... 244
10.Level densities and nuclear thermometry .................... 246
10.1.Darwin-Fowler approach for theoretical models ......... 246
10.1.1.Level densities and Strutinsky
renormalization ................................ 247
10.1.2.Dependence on angular momentum ................. 251
10.1.3.Microscopic models with residual
interactions ................................... 252
10.2.Empirical level densities ............................. 254
10.3.Nuclear thermometry ................................... 257
11.Large-scale collective motion at finite thermal
excitations ................................................ 262
11.1.Global transport equations ............................ 262
11.1.1.Fokker-Planck equations ........................ 262
11.1.2.Over-damped motion ............................. 265
11.1.3.Langevin equations ............................. 267
11.1.4.Probability distribution for collective
variables ...................................... 269
11.2.Transport coefficients for large-scale motion ......... 270
11.2.1.The LHA at level crossings and avoided crossings .... 275
11.2.2.Thermal aspects of global motion ............... 278
12.Dynamics of fission at finite temperature .................. 280
12.1.Transitions between potential wells ................... 280
12.1.1.Transition rate for over-damped motion ......... 281
12.2.The rate formulas of Kramers and Langer ............... 283
12.3.Escape time for strongly damped motion ................ 288
12.4.A critical discussion of timescales ................... 291
12.4.1.Transient- and saddle-scission times ........... 293
12.4.2.Implications from the concept of the MFPT ...... 296
12.5.Inclusion of quantum effects .......................... 299
12.5.1.Quantum decay rates within the LHA ............. 300
12.5.2.Rate formulas for motion treated self-
consistently ................................... 302
12.5.3.Quantum effects in collective transport,
a true challenge ............................... 307
13.Heavy-ion collisions at low energies ....................... 308
13.1.Transport models for heavy-ion collisions ............. 309
13.1.1.Commonly used inputs for transport equations ... 313
13.2.Differential cross sections ........................... 317
13.3.Fusion reactions ...................................... 319
13.3.1.Micro- and macroscopic formation
probabilities .................................. 321
13.4.Critical remarks on theoretical approaches and their
assumptions ........................................... 326
14.Giant dipole excitations ................................... 330
14.1.Absorption and radiation of the classical dipole ...... 330
14.2.Nuclear dipole modes .................................. 332
14.2.1.Extension to quantum mechanics ................. 333
14.2.2.Damping of giant dipole modes .................. 333
III.MESOSCOPIC SYSTEMS
15.Metals and quantum wires ................................... 341
15.1.Electronic transport in metals ........................ 341
15.1.1.The Drude model and basic definitions .......... 341
15.1.2.The transport equation and electronic
conductance .................................... 342
15.2.Quantum wires ......................................... 344
15.2.1.Mesoscopic systems in semiconductor
heterostructures ............................... 344
15.2.2.Two-dimensional electron gas ................... 345
15.2.3.Quantization of conductivity for ballistic
transport ...................................... 346
15.2.4.Physical interpretation and discussion ......... 348
16.Metal clusters ............................................. 350
16.1.Structure of metal clusters ........................... 350
16.2.Optical properties .................................... 351
16.2.1.Cross sections for scattering of light ......... 353
16.2.2.Optical properties for the jellium model ....... 354
16.2.3.The infinitely deep square well ................ 355
17.Energy transfer to a system of independent fermions ........ 361
17.1.Forced energy transfer within the wall picture ........ 361
17.1.1.Energy transfer at finite frequency ............ 363
17.1.2.Fermions inside billiards ...................... 365
17.2.Wall friction by Strutinsky smoothing ................. 366
IV.THEORETICAL TOOLS
18.Elements of reaction theory ................................ 373
18.1.Potential scattering .................................. 373
18.1.1.The T-matrix ................................... 373
18.1.2.Phase shifts for central potentials ............ 379
18.1.3.Inelastic processes ............................ 380
18.2.Generalization to nuclear reactions ................... 382
18.2.1.Reaction channels .............................. 382
18.2.2.Cross section .................................. 383
18.2.3.The T-matrix for nuclear reactions ............. 384
18.2.4.Isolated resonances ............................ 385
18.2.5.Overlapping resonances ......................... 389
18.2.6.T-matrix with angular momentum coupling ........ 389
18.3.Energy averaged amplitudes ............................ 390
18.3.1.The optical model .............................. 390
18.3.2.Intermediate structure through doorway
resonances ..................................... 392
18.4.Statistical theory .................................... 395
18.4.1.Porter-Thomas distribution for widths .......... 396
18.4.2.Smooth and fluctuating parts of the cross
section ........................................ 396
18.4.3.Hauser-Feshbach theory ......................... 401
18.4.4.Critique of the statistical model .............. 403
19.Density operators and Wigner functions ..................... 406
19.1.The many-body system .................................. 406
19.1.1.Hilbert states of the many-body system ......... 406
19.1.2.Density operators and matrices ................. 406
19.1.3.Reduction to one- and two-body densities ....... 408
19.2.Many-body functions from one-body functions ........... 410
19.2.1. One- and two-body densities ................... 411
19.3.The Wigner transformation ............................. 412
19.3.1.The Wigner transform in three dimensions ....... 412
19.3.2.Many-body systems of indistinguishable
particles ...................................... 414
19.3.3.Propagation of wave packets .................... 415
19.3.4.Correspondence rules ........................... 416
19.3.5.The equilibrium distribution of the
oscillator ..................................... 417
20.The Hartree—Fock approximation ............................. 420
20.1.Hartree-Fock with density operators ................... 420
20.1.1.The Hartree-Fock equations ..................... 422
20.1.2.The ground state energy in HF-approximation .... 423
20.2.Hartree-Fock at finite temperature .................... 424
20.2.1 TDHF at finite T ............................... 425
21.Transport equations for the one-body density ............... 426
21.1.The Wigner transform of the von Neumann equation ...... 426
21.2.Collision terms in semi-classical approximations ...... 428
21.2.1.The collision term in the Born approximation ... 430
21.2.2.The BUU and the Landau-Vlasov equation ......... 431
21.3.Relaxation to equilibrium ............................. 432
21.3.1.Relaxation time approximation to the
collision term ................................. 434
21.3.2.A few remarks on the concept of self-
energies ....................................... 435
22.Nuclear thermostatics ...................................... 437
22.1.Elements of statistical mechanics ..................... 437
22.1.1.Thermostatics for deformed nuclei .............. 438
22.1.2.Generalized ensembles .......................... 443
22.1.3.Extremal properties ............................ 448
22.2.Level densities and energy distributions .............. 450
22.2.1.Composite systems .............................. 452
22.2.2.A Gaussian approximation ....................... 454
22.2.3.Darwin-Fowler method for the level density ..... 457
22.3.Uncertainty of temperature for isolated systems ....... 461
22.3.1.The physical background ........................ 461
22.3.2.The thermal uncertainty relation ............... 463
22.4.The lack of extensivity and negative specific heats ... 464
22.5.Thermostatics of independent particles ................ 466
22.5.1.Sommerfeld expansion for smooth level
densities ...................................... 469
22.5.2.Thermostatics for oscillating level
densities ...................................... 470
22.5.3.Influence of angular momentum .................. 472
23.Linear response theory ..................................... 475
23.1.The model of the damped oscillator .................... 475
23.2.A brief reminder of perturbation theory ............... 478
23.2.1.Transition rate in lowest order ................ 480
23.3.General properties of response functions .............. 482
23.3.1.Basic definitions .............................. 482
23.3.2.Basic properties ............................... 484
23.3.3.Dissipation of energy .......................... 486
23.3.4.Spectral representations ....................... 487
23.4.Correlation functions and the fluctuation
dissipation theorem ................................... 489
23.4.1.Basic definitions .............................. 489
23.4.2.The fluctuation dissipation theorem ............ 491
23.4.3.Strength functions for periodic
perturbations .................................. 492
23.4.4.Linear response for a Random Matrix Model ...... 493
23.5.Linear response at complex frequencies ................ 496
23.5.1.Relation to thermal Green functions ............ 497
23.5.2.Response functions for unstable modes .......... 498
23.5.3.Equilibrium fluctuations of the oscillator ..... 499
23.6.Susceptibilities and the static response .............. 501
23.6.1.Static perturbations of the local
equilibrium .................................... 501
23.6.2.Isothermal and adiabatic susceptibilities ...... 504
23.6.3.Relations to the static response ............... 505
23.7.Linear irreversible processes ......................... 507
23.7.1.Relaxation functions ........................... 507
23.7.2.Variation of entropy in time ................... 510
23.7.3.Time variation of the density operator ......... 512
23.7.4.Onsager relations for macroscopic motion ....... 515
23.8.Kubo formula for transport coefficients ............... 518
24.Functional integrals ....................................... 522
24.1. Path integrals in quantum mechanics .................. 522
24.1.1. Time propagation in quantum mechanics ......... 522
24.1.2.Semi-classical approximation to the
propagator ..................................... 525
24.1.3.The path integral as a functional .............. 531
24.2.Path integrals for statistical mechanics .............. 532
24.2.1.The classical limit of statistical mechanics ... 535
24.2.2.Quantum corrections ............................ 536
24.3.Green functions and level densities ................... 539
24.3.1.Periodic orbit theory .......................... 540
24.3.2.The level density for regular and chaotic
motion ......................................... 542
24.4.Functional integrals for many-body systems ............ 543
24.4.1.The Hubbard-Stratonovich transformation ........ 543
24.4.2.The high temperature limit and quantum
corrections .................................... 546
24.4.3.The perturbed static path approximation
(PSPA) ......................................... 548
25.Properties of Langevin and Fokker—Planck equations ......... 554
25.1.The Brownian particle, a heuristic approach ........... 554
25.1.1.Langevin equation .............................. 554
25.1.2.Fokker-Planck equations ........................ 557
25.1.3.Cumulant expansion and Gaussian
distributions .................................. 559
25.2.General properties of stochastic processes ............ 560
25.2.1.Basic concepts ................................. 561
25.2.2.Markov processes and the Chapman-Kolmogorov
equation ....................................... 562
25.2.3.Fokker-Planck equations from the Kramers-
Moyal expansion ................................ 564
25.2.4.The master equation ............................ 568
25.3.Non-linear equations in one dimension ................. 570
25.3.1.Transport equations for multiplicative noise ... 570
25.3.2.Properties of the general Fokker-Planck
equation ....................................... 572
25.4.The mean first passage time ........................... 573
25.4.1.Differential equation for the MFPT ............. 574
25.5.The multidimensional Kramers equation ................. 575
25.5.1.Gaussian solutions in curvilinear
coordinates .................................... 576
25.5.2.Time dependence of first and second moments
for the harmonic oscillator .................... 578
25.6.Microscopic approach to transport problems ............ 580
25.6.1.The Nakajima-Zwanzig projection technique ...... 580
25.6.2.Perturbative approach for factorized
coupling ....................................... 582
V. AUXILIARY INFORMATION
26.Formal means ............................................... 589
26.1.Gaussian integrals .................................... 589
26.2.Stationary phase and steepest decent .................. 590
26.3.The δ-function ........................................ 591
26.4.Fourier and Laplace transformations ................... 591
26.5.Derivative of exponential operators ................... 592
26.6.The Mori product ...................................... 592
26.7.Spin and isospin ...................................... 593
26.8.Second quantization for fermions ...................... 594
27.Natural units in nuclear physics ........................... 596
References .................................................... 597
Index ......................................................... 615
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