Mean Field Theory

Contents
Preface
Introduction
1. Self-consistent mean field approximations
	1.1 Variational principle
	1.2 Effective interparticle interaction
		1.2.1 Nuclear matter
		1.2.2 Finite nuclei
		1.2.3 Landau’s effective interaction
	1.3 Energy density functional
		1.3.1 Potential energy due to Skyrme forces
		1.3.2 Spin-orbit potential energy
		1.3.3 Coulomb energy
		1.3.4 Energy density
	1.4 Hartree-Fock equations for Skyrme forces
		1.4.1 Nuclear matter properties in terms of Skyrme force parameters
		1.4.2 Skyrme parameters in terms of nuclear matter properties
		1.4.3 Determining the Skyrme force parameters
2. Many-body problem in phase space
	2.1 Weyl’s and Wigner’s representations
		2.1.1 Weyl representation
		2.1.2 Wigner representation
		2.1.3 Composition formula
	2.2 Wigner distribution function
	2.3 Application of the Wigner transformation
		2.3.1 Kinetic equations
		2.3.2 Pauli blocking
		2.3.3 Heavy-ion collisions
	2.4 Energy weighted smearing
	2.5 Temperature smearing
	2.6 Semi-classical approximation
	2.7 Density of internal kinetic energy
	2.8 Grand canonical potential density
3. Fluid dynamics approach
	3.1 Wigner transform to TDHF equations of motion
	3.2 Local approximation
	3.3 Quasi-equilibrium approximation
	3.4 Dynamic distortion of the Fermi surface
	3.5 Adiabatic approximation and rotational motion
	3.6 Isovector mode
	3.7 Collisionless Landau-Vlasov kinetic equation
	3.8 Response function for nuclear Fermi-liquid
	3.9 Pairing correlation and nuclear fluid dynamics
4. Static properties of nuclei
	4.1 Thomas-Fermi approximation for finite nuclei
		4.1.1 Particular cases
		4.1.2 Exact solution. Numerical procedure
		4.1.3 Finite nuclei
	4.2 Spherical nuclei. Shell effects
	4.3 Equation of state of hot nuclear matter
	4.4 Surfaces of equilibrium in hot nuclear matter
	4.5 Finite nuclei
	4.6 Nuclear parameters for hot nuclei
	4.7 Nuclear level density in hot nuclei
		4.7.1 The single-particle level density g()
			4.7.1.1 Phase-shift approach
			4.7.1.2 Green function approach
			4.7.1.3 Semiclassical methods for g()
			4.7.1.4 Energy dependent potential well
			4.7.1.5 Numerical illustration
			4.7.1.6 Effective mass
		4.7.2 Level density parameter a
		4.7.3 Preequilibrium decay within the exciton model
5. Direct variational method
	5.1 Energy functional for asymmetric nuclei
		5.1.1 Profile function
	5.2 Neutron excess and symmetry energy
	5.3 Line of beta-stability. Coulomb and symmetry energies
	5.4 Deformed nuclei
		5.4.1 Shape parametrization
		5.4.2 Deformation and Coulomb energies
		5.4.3 Diffuse layer effects
		5.4.4 Neck formation. Total kinetic energy of fission fragments
	5.5 Gibbs-Tolman approach
		5.5.1 Dividing surface and equimolar radius
		5.5.2 Surface energy within Gibbs-Tolman approach
	5.6 Equation of state and radii of finite nuclei
		5.6.1 Equation of state of finite nuclei
		5.6.2 Pressure in finite nuclei
		5.6.3 Nuclear radii
		5.6.4 Skin and halo effects
6. Small amplitude dynamics: Quantum approach
	6.1 Hartree-Fock based random phase approximation (HF-RPA)
		6.1.1 Random phase approximation
		6.1.2 Strength function and transition density
		6.1.3 Spurious state mixing
	6.2 Application of HF+RPA to Giant Multipole Resonances
		6.2.1 Folding-model distorted wave Born approximation (FM-DWBA)
		6.2.2 DWBA calculations of excitation cross-section
		6.2.3 Effects of violations of self-consistency in RPA calculations
			6.2.3.1 Self-consistent continuum RPA results
			6.2.3.2 Consequences of violations of self-consistency
		6.2.4 Sensitivity of excitation energies of giant resonances to properties of nuclear matter
7. Small amplitude dynamics in phase space
	7.1 Fluid dynamic approximation. Compression modes
		7.1.1 Giant monopole resonance and the nuclear incompressibility coefficient
		7.1.2 Isoscalar giant dipole resonance
		7.1.3 Macroscopic and quantum transition densities
			7.1.3.1 Quantum derivations
			7.1.3.2 Macroscopic transition density
			7.1.3.3 Semi-classical Fermi-liquid approach
	7.2 Isoscalar GMR at L ≥ 2
		7.2.1 Incompressible Fermi liquid. Nuclear shape vibrations
	7.3 Vortex motion
	7.4 Surface excitations in a semi-infinite Fermi liquid
	7.5 The effect of finite surface layer
	7.6 Isovector giant resonances
8. Relaxation processes
	8.1 Kinetic equation
	8.2 Relaxation and viscosity of nuclear matter
		8.2.1 Regime of frequent collisions (first-sound regime)
		8.2.2 Regime of rare collisions (zero-sound regime)
	8.3 Isovector giant resonances in presence of relaxation processes
		8.3.1 Strength function
		8.3.2 Energy-weighted sums and centroid energies
	8.4 Finite size effects
		8.4.1 Spherical harmonic oscillator (HO) potential
		8.4.2 Woods-Saxon (WS) potential
	8.5 Nuclear transport properties
		8.5.1 Response function
		8.5.2 Transport coefficients
	8.6 Fluctuations
	8.7 Particle emission
9. Instabilities and large amplitude motion
	9.1 Bulk and surface instabilities of a Fermi-liquid drop
		9.1.1 Bulk instability
		9.1.2 Surface instability
	9.2 Non-Markovian large amplitude dynamics
	9.3 Nuclear fission
		9.3.1 Motion near the barrier
		9.3.2 Descent from the barrier
	9.4 Quantum description
		9.4.1 Density matrix in the moving frame
		9.4.2 Occupation probabilities for the terms
		9.4.3 Change of energy
		9.4.4 Kinetic coefficients and spectral smearing
		9.4.5 Dissipation energy
		9.4.6 Equation of motion
10. Dynamics of hot nuclei
	10.1 Sound propagation in hot nuclear matter
	10.2 Landau’s damping
		10.2.1 Dispersion relation
		10.2.2 Viscosity
	10.3 Collisional damping of isovector mode
		10.3.1 Response function
		10.3.2 Width of IVGDR
	10.4 Giant multipole resonances in hot nuclei
	10.5 Boiling of nuclear matter
		10.5.1 Critical bubbles in overheated nuclear matter
		10.5.2 Surface of equilibrium
		10.5.3 Heterophase fluctuations and boiling
		10.5.4 Bubble collapse
Appendix A Fermi Integrals
Appendix B Nuclear mean field Vq(r) with Skyrme interaction
Appendix C Skyrme energy functional
Appendix D Generalized Fermi integral and coefficients κi(δ)
Appendix E Expansion coefficients ci,j
Appendix F Amplitude of Fermi-surface distortion
Appendix G Internal response function
Appendix H Linear response function and kinetic coefficients
Appendix I Liquid drop viscosity
Appendix J Fit by an oscillator response function
Appendix K Fluctuation-dissipation theorem
Appendix L Collisional integral for isovector mode
Bibliography