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The Physics of Warm Nuclei: With Analogies to Mesoscopic Systems
(Oxford Studies in Nuclear Physics)
By
Helmut Hofmann
Publisher
Oxford University Press, USA
Number Of Pages: 624
Publication Date: 2008-06-15
ISBN-10 / ASIN: 0198504012
ISBN-13 / EAN: 9780198504016
Product Description
This book offers a comprehensive survey of basic elements of nuclear dynamics at low energies and discusses similarities to mesoscopic systems. It addresses systems with finite excitations of their internal degrees of freedom, so that their collective motion exhibits features typical for transport processes in small and isolated systems. The importance of quantum aspects is examined with respect to both the microscopic damping mechanism and the nature of the transport equations. The latter must account for the fact that the collective motion is self-sustained. This implies highly nonlinear couplings between internal and collective degrees of freedom --- different to assumptions made in treatments known in the literature. A critical discussion of the use of thermal concepts is presented. The book can be considered self-contained. It presents existing models, theories and theoretical tools, both from nuclear physics and other fields, which are relevant to an understanding of the observed physical phenomena.
CONTENTS
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 6
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
xii Contents
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
Contents xiii
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
xiv Contents
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
Contents xv
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
xvi Contents
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
Contents xvii
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
xviii Contents
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 ä-function 591
26.4 Fourier and Laplace transformations 591
Contents xix
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
LinK
http://ifile.it/uy13n5l/9780198504016.rar
:gift::gift:
The Physics of Warm Nuclei: With Analogies to Mesoscopic Systems
(Oxford Studies in Nuclear Physics)
By
Helmut Hofmann
Publisher
Oxford University Press, USA
Number Of Pages: 624
Publication Date: 2008-06-15
ISBN-10 / ASIN: 0198504012
ISBN-13 / EAN: 9780198504016
Product Description
This book offers a comprehensive survey of basic elements of nuclear dynamics at low energies and discusses similarities to mesoscopic systems. It addresses systems with finite excitations of their internal degrees of freedom, so that their collective motion exhibits features typical for transport processes in small and isolated systems. The importance of quantum aspects is examined with respect to both the microscopic damping mechanism and the nature of the transport equations. The latter must account for the fact that the collective motion is self-sustained. This implies highly nonlinear couplings between internal and collective degrees of freedom --- different to assumptions made in treatments known in the literature. A critical discussion of the use of thermal concepts is presented. The book can be considered self-contained. It presents existing models, theories and theoretical tools, both from nuclear physics and other fields, which are relevant to an understanding of the observed physical phenomena.
CONTENTS
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 6
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
xii Contents
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
Contents xiii
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
xiv Contents
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
Contents xv
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
xvi Contents
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
Contents xvii
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
xviii Contents
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 ä-function 591
26.4 Fourier and Laplace transformations 591
Contents xix
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
LinK
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