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E-raamat: Physics of Warm Nuclei: With Analogies to Mesoscopic Systems [Oxford Scholarship Online e-raamatud]

(Department of Physics, Technical University of Munich)
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Physics of Warm Nuclei: With Analogies to Mesoscopic SystemsSuurem pilt
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The text describes in detail modern developments in theories of small many body systems like nuclear complexes in accelerator experiments or metal clusters.

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.
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
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 6-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
Professor Dr. Helmut Hofmann (retired) Department of Physics Technical University of Munich

Diploma (Universität Heidelberg, 1968) Dr. rer.nat. (Universität Heidelberg, 1971) Dr. rer.nat. habil. (Technische Universität München, 1979) Privatdozent (Technische Universität München, 1993) apl Professor (Technische Universität München, 2001)
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