1. 15 June, 2006Istanbul, part 21 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

2. 15 June, 2006Istanbul, part 22 The Random Phase Approximation in Nuclear Physics 1.Linear response theory: a brief reminder 2.Non-relativistic RPA (Skyrme) 3.Relativistic RPA (RMF) 4.Extension to QRPA 5.Beyond RPA. Nguyen Van Giai

3. 15 June, 2006Istanbul, part 23 Linear Response Theory In the presence of a time-dependent external field, the response of the system reveals the characteristics of the eigenmodes. In the limit of a weak perturbing field, the linear response is simply related to the exact two-body Green’s function. The RPA provides an approximation scheme to calculate the two-body Green’s function.. Nguyen Van Giai

4. 15 June, 2006Istanbul, part 24 Adding a time-dependent external field:. Nguyen Van Giai

5. 15 June, 2006Istanbul, part 25 First order response as a function of time. Nguyen Van Giai

6. 15 June, 2006Istanbul, part 26 Two-body Green’s Function and density-density correlation function. Nguyen Van Giai

7. 15 June, 2006Istanbul, part 27 Linear response function and Strength distribution Nguyen Van Giai

8. 15 June, 2006Istanbul, part 28 Main results: The knowledge of the retarded Green’s function gives access to: Excitation energies of eigenmodes (the poles) Transition probabilities (residues of the response function) Transition densities (or form factors), transition currents, etc… of each excited state. Nguyen Van Giai

9. 15 June, 2006Istanbul, part 29 TDHF and RPA (1) Nguyen Van Giai

10. 15 June, 2006Istanbul, part 210 TDHF and RPA (2) And by comparing with p.6 Nguyen Van Giai

11. 15 June, 2006Istanbul, part 211 Residual p-h interaction Nguyen Van Giai

12. 15 June, 2006Istanbul, part 212 Analytic summation of single- particle continuum 1) u, w are regular and irregular solutions satisfying appropriate asymptotic conditions 2) This analytic summation is not possible if potential U is non-local. Nguyen Van Giai

13. 15 June, 2006Istanbul, part 213 Transition densities and divergence of transition currents Solid: GQR Dashed: low-lying 2+ Dotted: empirical Nguyen Van Giai

14. 15 June, 2006Istanbul, part 214 Convection current distributions GQR in 208PbLow-lying 2+ in 208Pb Nguyen Van Giai

15. 15 June, 2006Istanbul, part 215 Finite temperature Applications: evolution of escape widths and Landau damping of IVGDR with temperature. Nguyen Van Giai

16. 15 June, 2006Istanbul, part 216 RPA on a p-h basis Nguyen Van Giai

17. 15 June, 2006Istanbul, part 217 A and B matrices Nguyen Van Giai

18. 15 June, 2006Istanbul, part 218 Restoration of symmetries Many symmetries are broken by the HF mean- field approximation: translational invariance, isospin symmetry, particle number in the case of HFB, etc… If RPA is performed consistently, each broken symmetry gives an RPA (or QRPA) state at zero energy (the spurious state) The spurious state is thus automatically decoupled from the physical RPA excitations This is not the case in phenomenological RPA. Nguyen Van Giai

19. 15 June, 2006Istanbul, part 219 Sum rules For odd k, RPA sum rules can be calculated from HF, without performing a detailed RPA calculation. k=1: Thouless theorem k=-1: Constrained HF k=3: Scaling of HF. Nguyen Van Giai

20. 15 June, 2006Istanbul, part 220 QRPA (1) The scheme which relates RPA to linearized TDHF can be repeated to derive QRPA from linearized Time-Dependent Hartree-Fock- Bogoliubov (cf. E. Khan et al., Phys. Rev. C 66, 024309 (2002)) Fully consistent QRPA calculations, except for 2- body spin-orbit, can be performed (M. Yamagami, NVG, Phys. Rev. C 69, 034301 (2004)). Nguyen Van Giai

21. 15 June, 2006Istanbul, part 221 QRPA (2) If Vpp is zero-range, one needs a cut-off in qp space, or a renormalisation procedure a la Bulgac. Then, one cannot sum up analytically the qp continuum up to infinity If Vpp is finite range (like Gogny force) one cannot solve the Bethe-Salpeter equation in coordinate space It is possible to sum over an energy grid along the positive axis ( Khan - Sandulescu et al., 2002). Nguyen Van Giai

22. 15 June, 2006Istanbul, part 222 The QRPA Green’s Function Nguyen Van Giai

23. 15 June, 2006Istanbul, part 223 External field and Strength distribution Nguyen Van Giai

24. 15 June, 2006Istanbul, part 224 2+ states in 120Sn Nguyen Van Giai

25. 15 June, 2006Istanbul, part 225 2+ states in 120Sn, with smearing Nguyen Van Giai

26. 15 June, 2006Istanbul, part 226 3- states in 120Sn, with smearing Nguyen Van Giai

27. 15 June, 2006Istanbul, part 227 Nguyen Van Giai

28. 15 June, 2006Istanbul, part 228 Relativistic RPA on top of Relativistic Mean Field Nguyen Van Giai

29. 15 June, 2006Istanbul, part 229 Fermi states and Dirac states Nguyen Van Giai

30. 15 June, 2006Istanbul, part 230 Single-particle spectrum Nguyen Van Giai

31. 15 June, 2006Istanbul, part 231 The Hartree polarization operator Nguyen Van Giai

32. 15 June, 2006Istanbul, part 232 Fermi and Dirac contributions Nguyen Van Giai

33. 15 June, 2006Istanbul, part 233 The RRPA polarization operator Generalized meson propagator for density-dependent case (Z.Y. Ma et al., 1997). Nguyen Van Giai

34. 15 June, 2006Istanbul, part 234 Diagrammatic representation Nguyen Van Giai

35. 15 June, 2006Istanbul, part 235 RRPA and TDRMF One can derive RRPA from the linearized version of the time-dependent RMF At each time, one assumes the no-sea approximation, i.e., ones keeps only the positive energy states These states are expanded on the complete set (at positive and negative energies) of states calculated at time t=0 This is how the Dirac states appear in RRPA. How important are they? From the linearized TDRMF one obtains the matrix form of RRPA, but the p-h configuration space is much larger than in RPA!. P.Ring, Z. Ma, NVG, et al. Nucl. Phys. A 694, 249 (2001) Nguyen Van Giai

36. 15 June, 2006Istanbul, part 236 Including continuum in RRPA Nguyen Van Giai

37. 15 June, 2006Istanbul, part 237 Effect of Dirac states on ISGMR Nguyen Van Giai

38. 15 June, 2006Istanbul, part 238 Effect of Dirac states on ISGQR Nguyen Van Giai

39. 15 June, 2006Istanbul, part 239 Effect of Dirac states on IVGDR Nguyen Van Giai

40. 15 June, 2006Istanbul, part 240 Concluding Remarks More studies are needed in the following topics: 1.In non-relativistic approach: - RPA, QRPA for deformed systems. - second RPA. 2.In relativistic approach: - RPA, QRPA on top of RHF. - deformed systems. - particle-vibration coupling. Nguyen Van Giai

41. 15 June, 2006Istanbul, part 241 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, 064321 (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, 024309 (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, 034310 (2005) 13. Self-consistent description of multipole strength: systematic calculations, J. Terasaki, J. Engel, ArXiv nucl-th/0603062

42. 15 June, 2006Istanbul, part 242 Skyrme Hartree-Fock Method: Computer Programs - P.-G. Reinhard, in Computational Nuclear Physics 1 (eds. K. Langanke, J.A. Maruhn, S.E. Koonin), Springer ‘93 - Spherical, SHF+BCS (monopole pairing) - ev8: Bonche, Flocard, and Heenen, Comp. Phys. Comm. 171(’05)49 - 3D mesh, SHF+BCS (density dependent pairing) - K. Bennaceur and J. Dobaczewski, Comp. Phys. Comm. 168(’05)96 - Spherical SHFB with density dependent pairing - M.V. Stoitsov, J. Dobaczewski, W. Nazarewicz, P. Ring, Comp. Phys. Comm. 167(’05)43 Axially deformed SHFB with density dependent pairing - transformed HO basis - J. Dobaczewski and P. Olbratowski, Comp. Phys. Comm. 158(’04)158 -Axially deformed SHFB with density dependent pairing - deformed HO basis Special thanks to Kouichi Hagino

43. 15 June, 2006Istanbul, part 243 Collaborators Nicu Sandulescu (Bucharest) Marcella Grasso (Catania, Orsay) Elias Khan (Orsay) Gianluca Colò (Milano) Hiro Sagawa (Aizu) Zhongyu Ma (Beijing)

44. 15 June, 2006Istanbul, part 244 Beyond RPA (1) Large amplitude collective motion: Generator Coordinate Method RPA can describe escape widths if continuum is treated, and it contains Landau damping, but spreading effects are not in the picture Spreading effects are contained in Second RPA Some applications called Second RPA are actually Second TDA: consistent SRPA calculations of nuclei are still waited for. Nguyen Van Giai

45. 15 June, 2006Istanbul, part 245 Beyond RPA (2) There exist models to approximate SRPA: The quasiparticle-phonon model (QPM) of Soloviev et al. Recently, attempts to calculate with Skyrme forces (A. Severyukhin et al.) The ph-phonon model: see G. Colo. Importance of correcting for Pauli principle violation Not much done so far in relativistic approaches. Nguyen Van Giai

46. 15 June, 2006Istanbul, part 246 Beyond RPA (3) Particle-vibration coupling Nguyen Van Giai

47. 15 June, 2006Istanbul, part 247 Effect of particle-vibration coupling Nguyen Van Giai