RIKEN06 Beyond Relativistic Mean Field Theory RIKEN March

RIKEN-06 Beyond Relativistic Mean Field Theory RIKEN, March 20, 2006 Peter Ring Technical University Munich 20. 03. 2006 RIKEN, March 2006: Mean field theories and beyond. 1

Content: Basics of Relativistic Mean Field Theory Relativistic RPA in deformed nuclei - D. Pena Arteaga A new method to carry out variation after projection - E. Lopes, R. Rossignoli, J. Sheikh, Applications of Generator Coordinate Method: - T. Niksic, D. Vretenar Coupling of low-lying collective states: - E. Litvinova Conclusions and outlook 20. 03. 2006 RIKEN, March 2006: Mean field theories and beyond. 2

Covariant density functional theory Walecka model Nucleons are coupled by exchange of mesons through an effective Lagrangian (EFT) (J , T)=(0+, 0) Sigma-meson: attractive scalar field 20. 03. 2006 (J , T)=(1 -, 0) (J , T)=(1 -, 1) Omega-meson: short-range repulsive RIKEN, March 2006: Mean field theories and beyond. Rho-meson: isovector field 3

Point-Coupling Models σ J=0, T=0 ω J=1, T=0 δ J=0, T=1 ρ J=1, T=1 Point-coupling model Manakos and Mannel, Z. Phys. 330, 223 (1988) Bürvenich, Madland, Maruhn, Reinhard, PRC 65, 044308 (2002 20. 03. 2006 RIKEN, March 2006: Mean field theories and beyond. 4

Density Effective density dependence: dependence non-linear potential: Boguta and Bodmer, NPA. 431, 3408 (1977) NL 1, NL 3. . density dependent coupling constants: 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 g g(r(r)) 20. 03. 2006 DD-ME 1, DD-ME 2 RIKEN, March 2006: Mean field theories and beyond. 5

Density Three relativistic models: dependence Meson exchange with non-linear meson couplings: Boguta and Bodmer, NPA. 431, 3408 (1977) NL 1, NL 3, TM 1, . . Meson exchange with density dependent coupling constants: R. Brockmann and H. Toki, PRL 68, 3408 (1992) DD-ME 1, DD-ME 2 Point-coupling models with density dependent coupling constants: Manakos and Mannel, Z. Phys. 330, 223 (1988) PC-1, …. 20. 03. 2006 RIKEN, March 2006: Mean field theories and beyond. 6

static RMF eq. time reversal) The static limit (with for the nucleons we find the static Dirac equation for the mesons we find the Helmholtz equations 20. 03. 2006 No-sea approxim. ! RIKEN, March 2006: Mean field theories and beyond. 7

TDRMF: Eq. Time dependent mean field theory: No-sea approxim. ! and similar equations for the ρ- and A-field 20. 03. 2006 RIKEN, March 2006: Mean field theories and beyond. 8

208 Pb breathing mode: Monopole motion K∞=211 K∞=271 K∞=355 20. 03. 2006 RIKEN, March 2006: Mean field theories and beyond. 9

RRPA Relativistic RPA for excited states drph, drah Small amplitude limit: ground-state density drhp, drha RRPA matrices: the same effective interaction determines the Dirac-Hartree single-particle spectrum and the residual interaction 20. 03. 2006 RIKEN, March 2006: Mean field theories and beyond. Interaction: 10

ph ah-configurations Eph ≈ 10 Me. V Fermi sea Selfconsistent Symmetries Eah ≈ -1200 Me. V ah Dirac sea <ph|Vω|ah> ≈ 0 for vector mesons <ph|Vσ|ah> ≠ 0 for scalar mesons 1. size of matrix. element 2. the same quantum numbers possible 3. large degeneracy P. Ring et al. , NPA 694 (2001) 249 20. 03. 2006 RIKEN, March 2006: Mean field theories and beyond. 11

Pb: GMR/GDR G. A. Lalazissis et al, PRC 71, 024312 (2005) 20. 03. 2006 RIKEN, March 2006: Mean field theories and beyond. DD-ME 2 12

IV-GDR in Sn-isotopes DD-ME 2 Sn: GDR DD-ME 2 G. A. Lalazissis et al, PRC 71, 024312 (2005) 20. 03. 2006 RIKEN, March 2006: Mean field theories and beyond. 13

20 Ne IV-GDRinindeformed Ne-20 (dots) IV-GDR NL 3 20. 03. 2006 RIKEN, March 2006: Mean field theories and beyond. 14

16 O IV-GDR O-16 (test) IV-GDR in in 20. 03. 2006 RIKEN, March 2006: Mean field theories and beyond. 15

BM 1 strength in Ne-20 Scissor mode in 20 Ne NL 3 20. 03. 2006 RIKEN, March 2006: Mean field theories and beyond. 16

20 Ne Ne-20 Scissor mode inin at Eδρ 1+=5. 66 Me. V n p Transition density δρ(r, z) NL 3 20. 03. 2006 RIKEN, March 2006: Mean field theories and beyond. 17

20 Ne Spurious modes in Ne-20 1 _ 1+ Eph=hω 1+ NL 3 20. 03. 2006 RIKEN, March 2006: Mean field theories and beyond. 18

pygmy-resonance in Ne-26 26 Ne Pygmy-Resonance in deformed GANIL n i im l re p 20. 03. 2006 y r a THEORY RIKEN, March 2006: Mean field theories and beyond. 19

Beyond Mean Field * 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 J. Sheikh and P. Ring, Nucl. Phys. A 665 (2000) 71 E. Lopes, Ph. D 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 20. 03. 2006 RIKEN, March 2006: Mean field theories and beyond. 20

decomposition Halo wave function in of the|Li> canonical basis: |CN|2 2 20. 03. 2006 4 6 RIKEN, March 2006: Mean field theories and beyond. 8 10 N 21

projected DFT Projected density functionals: projected denstiy functional: density dependent interactions !!! analytic expressions projected HFB-equations (variation after projection): J. Sheikh and P. Ring NPA 665 (2000) 71 20. 03. 2006 RIKEN, March 2006: Mean field theories and beyond. 22

Halo-formation in Ne-isotopes projected Ne-chain pairing energies binding energies rms-radii L. Lopes, Ph. D Thesis, TUM, 2002 20. 03. 2006 RIKEN, March 2006: Mean field theories and beyond. 23

Generator Coordinate Method (GCM) GCM-method 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: 20. 03. 2006 RIKEN, March 2006: Mean field theories and beyond. 24

potential surface: Hg-194 Energy surfaces in 194 Hg: 20. 03. 2006 RIKEN, March 2006: Mean field theories and beyond. 25

details of matrix norm-matrix in GCM Norm in GCM 20. 03. 2006 RIKEN, March 2006: Mean field theories and beyond. 26

unprojected GCM (N-dependence) unprojected GCM: N-dependence 20. 03. 2006 RIKEN, March 2006: Mean field theories and beyond. 27

unprojected GCM: q-steps unprojected GCM (Δq-dependence) 20. 03. 2006 RIKEN, March 2006: Mean field theories and beyond. 28

GCM-wavefunctions: Hg-194 GCM wave functions 20. 03. 2006 RIKEN, March 2006: Mean field theories and beyond. 29

J-projected energy surfaces: Hg-194 20. 03. 2006 RIKEN, March 2006: Mean field theories and beyond. 30

32 Mg: potential surface: in Mg-32 Energy surfaces 20. 03. 2006 RIKEN, March 2006: Mean field theories and beyond. 31

single-particle spectra: single particle spectra. Mg-32 20. 03. 2006 RIKEN, March 2006: Mean field theories and beyond. 32

J-projectedsurfaces: potential. Mg-32 surfaces 20. 03. 2006 RIKEN, March 2006: Mean field theories and beyond. 33

Mg-32 Gaussian. GOA: operlap approximation 20. 03. 2006 RIKEN, March 2006: Mean field theories and beyond. 34

J-projected Normnorm Mg-32 J-projected 20. 03. 2006 RIKEN, March 2006: Mean field theories and beyond. 35

rotational energy correction: Mg-32 rotational energy corrections 20. 03. 2006 RIKEN, March 2006: Mean field theories and beyond. 36

J-projected GCM spectra; Mg-32 J-projected GCM-spectrum 20. 03. 2006 RIKEN, March 2006: Mean field theories and beyond. 37

JGCM-wavefunctions: Mg-32 J-projected GCM wave functions: 20. 03. 2006 RIKEN, March 2006: Mean field theories and beyond. 38

Particle-Vibrational Coupling: energy dependent self-energy: RPA-modes mean field = pole part μ + + μ single particle strength: 20. 03. 2006 RIKEN, March 2006: Mean field theories and beyond. 39

209 Bi Distribution of single-particle strength in fragmentation in 209 -Bi 20. 03. 2006 RIKEN, March 2006: Mean field theories and beyond. 40

20. 03. 2006 RIKEN, March 2006: Mean field theories and beyond. 41

Width of Giant Resonances The full response contains energy dependent parts coming from vibrational couplings. Self energy g – phonon amplitudes (QRPA) ph interaction amplitude 20. 03. 2006 RIKEN, March 2006: Mean field theories and beyond. 42

Decay-width of the Giant Resonances E 1 photoabsorption cross section 20. 03. 2006 RIKEN, March 2006: Mean field theories and beyond. 43

GMRin in. RRPA+RRPA-PC GMR and RRPA-PC 20. 03. 2006 RIKEN, March 2006: Mean field theories and beyond. 44

GDR in Pb, RRPA-PC The Giant Dipole Resonance in 208 Pb 20. 03. 2006 RIKEN, March 2006: Mean field theories and beyond. 45

GDR in Dipole Sn, RRPA+RRPA-PC The Giant Resonance in 132 Sn 20. 03. 2006 RIKEN, March 2006: Mean field theories and beyond. 46

Energy and Width GDR: comparison withofexperiment 20. 03. 2006 RIKEN, March 2006: Mean field theories and beyond. 47

Conclusions: Deformed RPA- and QRPA calculations are now possible. They are very important for nuclei far from stability Conclusions 1 ------- 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 2 p-1 h or 2 p-2 h 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 20. 03. 2006 RIKEN, March 2006: Mean field theories and beyond. 48

-----Open Problems: 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) 20. 03. 2006 RIKEN, March 2006: Mean field theories and beyond. 49