1. RIKEN, March 2006: Mean field theories and beyond.20.03.2006 1 Peter Ring RIKEN, March 20, 2006 Technical University Munich RIKEN-06 Beyond Relativistic Mean Field Theory

2. RIKEN, March 2006: Mean field theories and beyond.20.03.2006 2 Content: Basics of Relativistic Mean Field Theory Conclusions and outlook Applications of Generator Coordinate Method: - T. Niksic, D. Vretenar A new method to carry out variation after projection - E. Lopes, R. Rossignoli, J. Sheikh, Coupling of low-lying collective states: - E. Litvinova Relativistic RPA in deformed nuclei - D. Pena Arteaga

3. RIKEN, March 2006: Mean field theories and beyond.20.03.2006 3 Walecka model Nucleons are coupled by exchange of mesons through an effective Lagrangian (EFT) (J ,T)=(0 +,0) (J ,T)=(1 -,0) (J ,T)=(1 -,1) Sigma-meson: attractive scalar field Omega-meson: short-range repulsive Rho-meson: isovector field Covariant density functional theory

4. RIKEN, March 2006: Mean field theories and beyond.20.03.2006 4 Point-coupling model Point-Coupling Models σ ωδρ J=0, T=0 J=1, T=0 J=0, T=1 J=1, T=1 Manakos and Mannel, Z.Phys. 330, 223 (1988) Bürvenich, Madland, Maruhn, Reinhard, PRC 65, 044308 (2002

5. RIKEN, March 2006: Mean field theories and beyond.20.03.2006 5 Density dependence Effective density dependence: non-linear potential: density dependent coupling constants: g  g (  r)) NL1,NL3.. DD-ME1,DD-ME2 Boguta and Bodmer, NPA. 431, 3408 (1977) R.Brockmann and H.Toki, PRL 68, 3408 (1992) S.Typel and H.H.Wolter, NPA 656, 331 (1999) T. Niksic, D. Vretenar, P. Finelli, and P. Ring, PRC 56 (2002) 024306

6. RIKEN, March 2006: Mean field theories and beyond.20.03.2006 6 Density dependence Three relativistic models: Meson exchange with non-linear meson couplings: Meson exchange with density dependent coupling constants: NL1,NL3,TM1,.. DD-ME1,DD-ME2 Boguta and Bodmer, NPA. 431, 3408 (1977) R.Brockmann and H.Toki, PRL 68, 3408 (1992) Point-coupling models with density dependent coupling constants: Manakos and Mannel, Z.Phys. 330, 223 (1988) PC-1,….

7. RIKEN, March 2006: Mean field theories and beyond.20.03.2006 7 for the nucleons we find the static Dirac equation for the mesons we find the Helmholtz equations No-sea approxim. ! static RMF eq. The static limit (with time reversal)

8. RIKEN, March 2006: Mean field theories and beyond.20.03.2006 8 TDRMF: Eq. and similar equations for the ρ- and A-field Time dependent mean field theory: No-sea approxim. !

9. RIKEN, March 2006: Mean field theories and beyond.20.03.2006 9 K ∞ =271 K ∞ =355 Monopole motion K ∞ =211 breathing mode: 208 Pb

10. RIKEN, March 2006: Mean field theories and beyond.20.03.2006 10 RRPA Relativistic RPA for excited states RRPA matrices: the same effective interaction determines the Dirac-Hartree single-particle spectrum and the residual interaction  ph,   h  hp,  h  Small amplitude limit: ground-state density Interaction:

11. RIKEN, March 2006: Mean field theories and beyond.20.03.2006 11 ah ≈ 0 for vector mesons ≠ 0 for scalar mesons 1.size of matrix.element 2.the same quantum numbers possible 3.large degeneracy Dirac sea P. Ring et al., NPA 694 (2001) 249 Selfconsistent Symmetries ah-configurations ph E ph ≈ 10 MeV Fermi sea E ah ≈ -1200 MeV

12. RIKEN, March 2006: Mean field theories and beyond.20.03.2006 12 Pb: GMR/GDR DD-ME2 G.A. Lalazissis et al, PRC 71, 024312 (2005)

13. RIKEN, March 2006: Mean field theories and beyond.20.03.2006 13 Sn: GDR DD-ME2 DD-ME2 IV-GDR in Sn-isotopes G.A. Lalazissis et al, PRC 71, 024312 (2005)

14. RIKEN, March 2006: Mean field theories and beyond.20.03.2006 14 IV-GDR in Ne-20 (dots) IV-GDR in deformed 20 Ne NL3

15. RIKEN, March 2006: Mean field theories and beyond.20.03.2006 15 IV-GDR in O-16 (test) IV-GDR in 16 O

16. RIKEN, March 2006: Mean field theories and beyond.20.03.2006 16 BM1 strength in Ne-20 NL3 Scissor mode in 20 Ne

17. RIKEN, March 2006: Mean field theories and beyond.20.03.2006 17 Scissor in Ne-20 δρ Scissor mode in 20 Ne at E 1+ =5.66 MeV n p NL3 Transition density δρ(r,z)

18. RIKEN, March 2006: Mean field theories and beyond.20.03.2006 18 Spurious modes in Ne-20 E ph =hω Spurious modes in 20 Ne NL3 1_1_ 1+1+ 1+1+

19. RIKEN, March 2006: Mean field theories and beyond.20.03.2006 19 pygmy-resonance in Ne-26 Pygmy-Resonance in deformed 26 Ne preliminary GANIL THEORY

20. RIKEN, March 2006: Mean field theories and beyond.20.03.2006 20 Beyond Mean Field J. Sheikh and P. Ring, Nucl. Phys. A665 (2000) 71 E. Lopes, PhD thesis, TU München T. Niksic, D. Vretenar, and P. Ring, Phys. Rev. C (2006) in print E. Litvinova and P. Ring, Phys. Rev. C (2006) in print * Conservation of symmetries: Projection before Variation * Motion with large amplitude: Generator Coordinates * Coupling to Collective Vibrations - shifts of single particle energies - decay width of giant resonances

21. RIKEN, March 2006: Mean field theories and beyond.20.03.2006 21 |C N | 2 N 246810 decomposition of |Li> Halo wave function in the canonical basis:

22. RIKEN, March 2006: Mean field theories and beyond.20.03.2006 22 density dependent interactions !!! projected denstiy functional: analytic expressions projected HFB-equations (variation after projection): J.Sheikh and P. Ring NPA 665 (2000) 71 projected DFT Projected density functionals:

23. RIKEN, March 2006: Mean field theories and beyond.20.03.2006 23 pairing energies binding energies rms-radii L. Lopes, PhD Thesis, TUM, 2002 projected Ne-chain Halo-formation in Ne-isotopes

24. RIKEN, March 2006: Mean field theories and beyond.20.03.2006 24 Constraint Hartree Fock produces wave functions depending on a generator coordinate q GCM wave function is a superposition of Slaterdeterminants Hill-Wheeler equation: with projection: GCM-method Generator Coordinate Method (GCM)

25. RIKEN, March 2006: Mean field theories and beyond.20.03.2006 25 potential surface: Hg-194 Energy surfaces in 194 Hg:

26. RIKEN, March 2006: Mean field theories and beyond.20.03.2006 26 details of norm-matrix in GCM Norm matrix in GCM

27. RIKEN, March 2006: Mean field theories and beyond.20.03.2006 27 unprojected GCM: N-dependence unprojected GCM (N-dependence)

28. RIKEN, March 2006: Mean field theories and beyond.20.03.2006 28 unprojected GCM: q-steps unprojected GCM (Δq-dependence)

29. RIKEN, March 2006: Mean field theories and beyond.20.03.2006 29 GCM-wavefunctions: Hg-194 GCM wave functions

30. RIKEN, March 2006: Mean field theories and beyond.20.03.2006 30 J-projected energy surfaces: Hg-194

31. RIKEN, March 2006: Mean field theories and beyond.20.03.2006 31 potential surface: Mg-32 Energy surfaces in 32 Mg:

32. RIKEN, March 2006: Mean field theories and beyond.20.03.2006 32 single-particle spectra: Mg-32 single particle spectra

33. RIKEN, March 2006: Mean field theories and beyond.20.03.2006 33 J-projected surfaces: Mg-32 J-projected potential surfaces

34. RIKEN, March 2006: Mean field theories and beyond.20.03.2006 34 GOA: Mg-32 Gaussian operlap approximation

35. RIKEN, March 2006: Mean field theories and beyond.20.03.2006 35 J-projected Norm Mg-32 J-projected norm

36. RIKEN, March 2006: Mean field theories and beyond.20.03.2006 36 rotational energy correction: Mg-32 rotational energy corrections

37. RIKEN, March 2006: Mean field theories and beyond.20.03.2006 37 J-projected GCM spectra; Mg-32 J-projected GCM-spectrum

38. RIKEN, March 2006: Mean field theories and beyond.20.03.2006 38 JGCM-wavefunctions: Mg-32 J-projected GCM wave functions:

39. RIKEN, March 2006: Mean field theories and beyond.20.03.2006 39 Particle-Vibrational Coupling: energy dependent self-energy: single particle strength: + + RPA-modes μ μ mean fieldpole part =

40. RIKEN, March 2006: Mean field theories and beyond.20.03.2006 40 fragmentation in 209-Bi Distribution of single-particle strength in 209 Bi

41. RIKEN, March 2006: Mean field theories and beyond.20.03.2006 41

42. RIKEN, March 2006: Mean field theories and beyond.20.03.2006 42 Width of Giant Resonances The full response contains energy dependent parts coming from vibrational couplings. Self energy ph interaction amplitude g – phonon amplitudes (QRPA)

43. RIKEN, March 2006: Mean field theories and beyond.20.03.2006 43 Decay-width of the Giant Resonances E1 photoabsorption cross section

44. RIKEN, March 2006: Mean field theories and beyond.20.03.2006 44 GMR in RRPA+RRPA-PC GMR in RRPA and RRPA-PC

45. RIKEN, March 2006: Mean field theories and beyond.20.03.2006 45 GDR in Pb, RRPA-RRPA-PC The Giant Dipole Resonance in 208 Pb

46. RIKEN, March 2006: Mean field theories and beyond.20.03.2006 46 GDR in Sn, RRPA+RRPA-PC The Giant Dipole Resonance in 132 Sn

47. RIKEN, March 2006: Mean field theories and beyond.20.03.2006 47 Energy and Width of GDR GDR: comparison with experiment

48. RIKEN, March 2006: Mean field theories and beyond.20.03.2006 48 Conclusions 1 ------- Conclusions: Deformed RPA- and QRPA calculations are now possible. They are very important for nuclei far from stability In transitional nuclei we can do Variation after Projection a new method allows this also in relativistic calculations. Configuration mixing is possible by GCM calculations: - they require angular momentum projection - the numerical effort is large - restriction to axial symmetry at the moment essential The admixture of 2p-1h or 2p-2h states is possible by particle vibrational coupling (PVC). - this leads to fragmentation of the single particle states - enhanced level density at the Fermi surfac - enhanced width of giant resonances

49. RIKEN, March 2006: Mean field theories and beyond.20.03.2006 49 Simpler parametrizations: - point coupling - simpler pairing Improved energy functional: - Fock terms and tensor forces - why is the first order pion-exchange quenched? - is vacuum polarization important in finite nuclei Conceptual problems: - what is form of the functional, if we go beyond mean field? - density dependence and projection (Egido-poles) - density dependence and symmetries (RPA) -----Open Problems ------ Open Problems: