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Mean Field Methods for Nuclear Structure

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

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Mean Field Methods for Nuclear Structure

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

  2. The Random Phase Approximation in Nuclear Physics • Linear response theory: a brief reminder • Non-relativistic RPA (Skyrme) • Relativistic RPA (RMF) • Extension to QRPA • Beyond RPA . Istanbul, part 2 Nguyen Van Giai

  3. 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. . Istanbul, part 2 Nguyen Van Giai

  4. Adding a time-dependent external field: . Istanbul, part 2 Nguyen Van Giai

  5. First order response as a function of time . Istanbul, part 2 Nguyen Van Giai

  6. Two-body Green’s Function and density-density correlation function . Istanbul, part 2 Nguyen Van Giai

  7. Linear response function and Strength distribution Istanbul, part 2 Nguyen Van Giai

  8. 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 . Istanbul, part 2 Nguyen Van Giai

  9. TDHF and RPA (1) Istanbul, part 2 Nguyen Van Giai

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

  11. Residual p-h interaction Istanbul, part 2 Nguyen Van Giai

  12. 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 . Istanbul, part 2 Nguyen Van Giai

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

  14. Convection current distributions GQR in 208Pb Low-lying 2+ in 208Pb Istanbul, part 2 Nguyen Van Giai

  15. Finite temperature Applications: evolution of escape widths and Landau damping of IVGDR with temperature . Istanbul, part 2 Nguyen Van Giai

  16. RPA on a p-h basis Istanbul, part 2 Nguyen Van Giai

  17. A and B matrices Istanbul, part 2 Nguyen Van Giai

  18. 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 . Istanbul, part 2 Nguyen Van Giai

  19. 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 . Istanbul, part 2 Nguyen Van Giai

  20. 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)) . Istanbul, part 2 Nguyen Van Giai

  21. 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) . Istanbul, part 2 Nguyen Van Giai

  22. The QRPA Green’s Function Istanbul, part 2 Nguyen Van Giai

  23. External field and Strength distribution Istanbul, part 2 Nguyen Van Giai

  24. 2+ states in 120Sn Istanbul, part 2 Nguyen Van Giai

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

  26. 3- states in 120Sn, with smearing Istanbul, part 2 Nguyen Van Giai

  27. Istanbul, part 2 Nguyen Van Giai

  28. Relativistic RPA on top of Relativistic Mean Field Istanbul, part 2 Nguyen Van Giai

  29. Fermi states and Dirac states Istanbul, part 2 Nguyen Van Giai

  30. Single-particle spectrum Istanbul, part 2 Nguyen Van Giai

  31. The Hartree polarization operator Istanbul, part 2 Nguyen Van Giai

  32. Fermi and Dirac contributions Istanbul, part 2 Nguyen Van Giai

  33. The RRPA polarization operator • Generalized meson propagator for density-dependent case (Z.Y. Ma et al., 1997) . Istanbul, part 2 Nguyen Van Giai

  34. Diagrammatic representation Istanbul, part 2 Nguyen Van Giai

  35. 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) Istanbul, part 2 Nguyen Van Giai

  36. Including continuum in RRPA Istanbul, part 2 Nguyen Van Giai

  37. Effect of Dirac states on ISGMR Istanbul, part 2 Nguyen Van Giai

  38. Effect of Dirac states on ISGQR Istanbul, part 2 Nguyen Van Giai

  39. Effect of Dirac states on IVGDR Istanbul, part 2 Nguyen Van Giai

  40. 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. Istanbul, part 2 Nguyen Van Giai

  41. Lectures on:Mean Field Methods for Nuclear StructureList 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 Istanbul, part 2

  42. 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 Istanbul, part 2

  43. Collaborators • Nicu Sandulescu (Bucharest) • Marcella Grasso (Catania, Orsay) • Elias Khan (Orsay) • Gianluca Colò (Milano) • Hiro Sagawa (Aizu) • Zhongyu Ma (Beijing) Istanbul, part 2

  44. 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. Istanbul, part 2 Nguyen Van Giai

  45. 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 . Istanbul, part 2 Nguyen Van Giai

  46. Beyond RPA (3) • Particle-vibration coupling Istanbul, part 2 Nguyen Van Giai

  47. Effect of particle-vibration coupling Istanbul, part 2 Nguyen Van Giai

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