12 June, 2006Istanbul, part I1 Mean Field Methods for Nuclear Structure Part 1: Ground State Properties: Hartree-Fock and Hartree-Fock- Bogoliubov Approaches Part 2: Nuclear Excitations: The Random Phase Approximation Nguyen Van Giai Institut de Physique Nucléaire Université Paris-Sud, Orsay

12 June, 2006Istanbul, part I2 Outline of part 1 - Introduction - Non-relativistic energy density functional - Densities and Potentials - HF and HFB in spherical symmetry - Illustrative examples - Summary Nguyen Van Giai

12 June, 2006Istanbul, part I3 Microscopic approaches to many-body, finite nuclear systems Theoretical models based on effective interactions between nucleons: - Nuclear shell model - Mean field approaches (and beyond): -Non-Relativistic (Skyrme forces, Gogny force) -Relativistic (RMF,RHF) - Molecular dynamics going away from stability regions, we need a theoretical framework which can be predictive and able to handle new situations (continuum, pairing correlations in continuum). the Hartree-Fock + Random Phase Approximation (and their extensions to include pairing effects) can be used from unstable nuclei to neutron star crust. Nguyen Van Giai

12 June, 2006Istanbul, part I4 Hartree-Fock, and HF-Bogoliubov for systems with pairing correlations Nguyen Van Giai

12 June, 2006Istanbul, part I5 Energy Density Functional in Hartree-Fock Nguyen Van Giai

12 June, 2006Istanbul, part I6 Effective Interaction: Skyrme force particle-hole channel: particle- particle channel: Skyrme interaction zero-range Nguyen Van Giai Pairing channel:

12 June, 2006Istanbul, part I7 Densities Normal density, or density matrix Abnormal density, or pairing tensor Nguyen Van Giai

12 June, 2006Istanbul, part I8 One-body densities in Hartree-Fock Nguyen Van Giai

12 June, 2006Istanbul, part I9 The Energy Density Functional Nguyen Van Giai

12 June, 2006Istanbul, part I10 The Skyrme-HF equations Variations with respect to single-particle wave functions: Nguyen Van Giai

12 June, 2006Istanbul, part I11 The Skyrme-HF effective masses Nguyen Van Giai

12 June, 2006Istanbul, part I12 The Skyrme-HF central potentials Nguyen Van Giai

12 June, 2006Istanbul, part I13 The spin-orbit and Coulomb potentials Nguyen Van Giai

12 June, 2006Istanbul, part I14 The center-of-mass correction Nguyen Van Giai

12 June, 2006Istanbul, part I15 Spherical case: radial equations in r-space Nguyen Van Giai

12 June, 2006Istanbul, part I16 Nguyen Van Giai Densities Potentials Effective masses Spin-orbit potentials From: Bender et al., Revs.Mod.Phys., 75, 121(2003)

12 June, 2006Istanbul, part I17 Nguyen Van Giai N-Z A=N+Z Binding Energy Errors From: Bender et al., Revs.Mod.Phys., 75, 121(2003)

12 June, 2006Istanbul, part I18 2-neutron separation energies Nguyen Van Giai From: Bender et al., Revs.Mod.Phys., 75, 121(2003)

12 June, 2006Istanbul, part I19 Nguyen Van Giai Single-particle energies From: Bender et al., Revs.Mod.Phys., 75, 121(2003)

12 June, 2006Istanbul, part I20 r.m.s. radii Nguyen Van Giai From: Bender et al., Revs.Mod.Phys., 75, 121(2003)

12 June, 2006Istanbul, part I21 Generalization to Hartree-Fock- Bogoliubov Nguyen Van Giai

12 June, 2006Istanbul, part I22 HFB densities in spherical case Nuclear density Abnormal (or pairing) density Kinetic energy density Spin density Nguyen Van Giai

12 June, 2006Istanbul, part I23 The Hartree-Fock-Bogoliubov Equations Nguyen Van Giai

12 June, 2006Istanbul, part I24 Hartree-Fock field and pairing field Nguyen Van Giai

12 June, 2006Istanbul, part I25 Finite-Temperature HFB E nuc = E Skyrme + E pair [ ,  ] f i =( 1+e Ei/kT ) -1  T (r) = V pair  T (r) Nguyen Van Giai where :

12 June, 2006Istanbul, part I26 Quasiparticle continuum Nguyen Van Giai

12 June, 2006Istanbul, part I27 Treatment of quasiparticle continuum (1) Nguyen Van Giai

12 June, 2006Istanbul, part I28 Treatment of quasiparticle continuum (2) Nguyen Van Giai

12 June, 2006Istanbul, part I29 Treatment of quasiparticle continuum (3) Nguyen Van Giai

12 June, 2006Istanbul, part I30 Discretization by box boundary condition Alternatively, one can enclose the system in a box of radius R. The quasiparticle spectrum is calculated with the boundary condition that the wave function vanishes at r=R. One thus obtains a discrete set of states forming a complete basis in the box. Nguyen Van Giai

12 June, 2006Istanbul, part I31 illustration: Ni isotopes Nguyen Van Giai

12 June, 2006Istanbul, part I32 E. Khan, N. Sandulescu Nguyen Van Giai

12 June, 2006Istanbul, part I33 Inner Crust Matter Crystal lattice structures ~   ~   ~ 0.5   Nguyen Van Giai

12 June, 2006Istanbul, part I34 Elementary cells Wigner-Seitz cellElementary cellLattice Nguyen Van Giai

12 June, 2006Istanbul, part I35 N.Sandulescu, Nguyen Van Giai,R.J.Liotta, Phys.Rev.C69(2004) Density in the Wigner-Seitz Cells Nguyen Van Giai

12 June, 2006Istanbul, part I36 Pairing Field in the Wigner-Seitz Cells N.Sandulescu, Phys.Rev.C70 (2004) Nguyen Van Giai

12 June, 2006Istanbul, part I37 SUMMARY A self-consistent theory of nuclear ground states. Pairing and continuum effects are treated. Applications to the description of unstable nuclei. Applications to the physics of the inner crust of neutron stars. Nguyen Van Giai

12 June, 2006Istanbul, part I38 Lectures on: Mean Field Methods for Nuclear Structure List of references for further reading 1. P. Ring, P. Schuck, “The Nuclear Many-Body Problem”, Springer-Verlag (New York, 1980) 2. Hartree-Fock calculations with Skyrme’s interaction. I: spherical nuclei, D. Vautherin, D.M. Brink, Phys. Rev. C 5, 626 (1972) 3. Hartree-Fock calculations with Skyrme’s interaction. II: axially deformed nuclei, D. Vautherin, Phys. Rev. C 7, 296 (1973) 4. A Skyrme parametrization from subnuclear to neutron star densities, E. Chabanat, P. Bonche, P. Haensel, J. Meyer, R. Schaeffer: Part I, Nucl. Phys. A 627, 710 (1997); Part II, Nucl. Phys. A 635, 231 (1998); Erratum to Part II, Nucl. Phys. A 643, 441 (1998) 5. Self-consistent mean-field models for nuclear structure, M. Bender, P.-H. Heenen, P.-G. Reinhard, Revs. Mod. Phys. 75, 121 (2003) 6. Hartree-Fock-Bogoliubov description of nuclei near the neutron drip line, J. Dobaczewski, H. Flocard, J. Treiner, Nucl.Phys. A 422, 103 (1984) 7. Mean-field description of ground state properties of drip line nuclei: pairing and continuum effects, J. Dobaczewski, W. Nazarewicz, T.R. Werner, J.-F. Berger, C.R. Chinn, J. Dechargé, Phys. Rev. C 53, 2809 (1996) 8. Pairing and continuum effects in nuclei close to the drip line, M. Grasso, N. Sandulescu, N. Van Giai, R. Liotta, Phys. Rev. C 64, (2001) 9. Nuclear response functions, G.F. Bertsch, S.F. Tsai, Phys. Rep. 12 C (1975) 10. A self-consistent description of the giant resonances including the particle continuum, K.F. Liu, N. Van Giai, Phys. Lett. B 65, 23 (1976) 11. Continuum quasiparticle random phase approximation and the time-dependent HFB approach, E. Khan, N. Sandulescu, M. Grasso, N. Van Giai, Phys. Rev. C 66, (2002) 12. Self-Consistent Description of Multipole Strength in Exotic Nuclei I: Method, J. Terasaki, J. Engel, M. Bender, J. Dobaczewski, W. Nazarewicz, M. Stoitsov, Phys. Rev. C 71, (2005) 13. Self-consistent description of multipole strength: systematic calculations, J. Terasaki, J. Engel, ArXiv nucl-th/