1. Giant and Pygmy Resonance in Relativistic Approach The Sixth China-Japan Joint Nuclear Physics May 16-20, 2006 Shanghai Zhongyu Ma China Institute of Atomic Energy, Beijing Collaborators: Ligang Cao, Baoqiu Chen, Jun Liang

2. Introduction  Nucleus moving away from the valley of  -stability diffuse neutron: neutron skin, hallo structure, new magic numbers, new modes of excitations, etc.  Significant interest on low-energy excited states GDR (Coulomb excitations) restoring force proportional to the symmetry energy Pygmy resonance Loosely bound neutron coherently oscillate against the p-n core neutron density distribution, neutron radius, skin et al. density dependence of symmetry energy Astrophysical implications

3. Fully Consistent RRPA  RRPA -- Consistent in sense: ph residual interaction determined from the same Lagrangian for g.s. RRPA polarization operator i= , ,   i =1,  ,    3 for , , , respectively consistent to RMF no sea approx. Include both ph pairs and  h pairs Z. Y. Ma, et al., Nucl. Phys. A703(2002)222 RRPA TDRMF at small amplitude limit TDRMF at each time no sea approximation  is calculated in a stable complete set basis P.Ring et al., Nucl. Phys. A694(2001)249

4. Fig: ISGMR NL1,NL3,TM1,NLSH Z.Y. Ma, et al., Nucl. Phys. A686(2001)173 M.E. of vector fields coupling  h and ph----Largely reduced due to Dirac str. Cancellation of the  &  fields --- not take place, Large M.E. coupling  h and ph exist

5. Treatment of the continuum  Resonant states in the continuum Metastable states in the centrifugal & Coulomb Barrier  Discretization of the continuum Expansion on Harmonic Oscillator basis Box approximation: set a wall at a large distance  Exact treatment of the continuum Set up a proper boundary condition Single particle resonance with energy and width Green’s function method  Scattering phase shift Centrifugal & Coulomb Nuclear Pot. Total

6. Scattering phase shift  Boundary conditions: Normalized by phase shift:  =  /2 resonant state  For proton, Dirac Coulomb functions have to be solved for  ~1 Z large large diff. from norn Coulomb wf Cao & Ma PRC66(02)024311 W. Grainer “Rel. Quantum Mach.”

7. Example of resonance states More resonant states for p than those for n due to the Coulomb barrier

8. Resonant continuum in pairing correlations  Pairing correlations play a crucial role in MF models for open shell HF+BCS and RMF+BCS simple successful in nuclei when  F not close to the continuum HFB and RHB important in nuclei near the drip-line HF eq. + gap eq. are solved simultaneously states in continuum are discretized in both methods  Resonant states : HFB eq. are solved with exact boundary conditions Grasso, Sandulescu, Nguyen, PRC64(2001)064321  Discretization of the continuum overestimates pairing corr.  Effect of the continuum on pairing --- mainly by a few resonant states in the continuum RMF+BCS with resonant states including widths

9. BCS with the continuum Gap equation : Nucleon densities: Continuum level density

10. Pairing correlation energy  BCS are good in the vicinity of the stable line.  Width effects are large for nuclei far from the stability line. Cao, Ma, Eur. Phys. J. A 22 (2004)189

11. Quasi-particle RRPA  Response function  Unperturbed polarization operator BCS occupation prob. Outside the pairing active space Positive unoccu. states occu. states Negative states

12. Ni-isotopes Extended RMF+BCS s.p. resonant states 2d 5/2,2d 3/2,1g 7/2,2f 7/2,1h 11/2, G=20.5/A MeV Cao, Ma, Modern Phys. Lett. A19(2004)2845

13. IVGDR Ni-isotopes 58 Ni – 64 Ni are stable vibration of p-n Ni-isotopes A=70~96 The response functions of IVGDR in QRRPA Loosely bound neutron coherently oscillate against the p-n core E H ~ 16 MeV low-lying dipole <10 MeV

14. IVGDR in Ni-isotopes Cao, Ma, Modern Phys. Lett. A19(2004)2845 GDR restoring force proportional to the symmetry energy Linear dep. on the neutron skin

15. Experiments on GDR  Gibelin and Beaumel (Orsay), exp. at RIKEN inelastic scattering of 26 Ne + 208 Pb 60 MeV/u 26 Ne secondary beam Dominated by Coulomb excitations selective for E1 transitions. Thesis of J. Gibelin IPNO-T-05-11 Future work: 28 Ne + 208 Pb  Theoretical investigation – practical significance Cao, Ma, PRC71(05)034305

16. Properties of 26,28 Ne Extended RMF+BCS with NL3 GQR check the validity of spherical assumption

17. 26 Ne, 28 Ne IVGDR Cao, Ma, PRC71(05)034305

18. Sum rule Low-lying GDR in 26 Ne exhaust about 4.9% of TRK sum rule 28 Ne 5.8%

19. Comparisons of Low-lying dipole state in 26 Ne Authors Methods Shape Result Elias(Orsay) SHF+BCS+ spherical 11.7 not coll. QRPA(RF) Cao, Ma(CIAE) RMF+BCS(R.) spherical 8.4(5%) coll. PRC71(2005)034305 +QRRPA(RF) Peru(CEA) def. HFB(Gogny) Spherical 10.7 coll. +QRPA(Matrix) Ring(TUM) def. RHB +QRRPA Deformed 7.9 9.3 less coll. (Matrix) Exp.(Gibelin,Beaumel) measure ? ~9(5%) IPNO-T-05-11 Preliminary

20. Symmetry Energy and GDR Restoring force of GDR Symmetry energy in NM All parameters give very good description of g.s. properties, NM saturations Centroid energy of GDR E cen =m 1 /m 0 Linear dep on the symmetry energy at saturation energy May give constraint: 33 MeV< a sym (  0 )<37 MeV

21. Density dep. of symmetry energy Non linear  -  coupling Todd, Pickarewicz, PRC67(03) Modify the poorly known density dep. of symmetry energy Without changing the agreement with existing NM, g.s. properties Softening of the symmetry energy NL3 B/A=16.24MeV a sym =37.3 MeV  0 =.148fm -3 (k F =1.3fm -1 ) K=272 MeV a sym =25.67 MeV at  =.1fm -3 (k F =1.15fm -1 )

22. Ground state properties in 132 Sn vv B/A(MeV)r p (fm)r n (fm)r n -r p 00 a sym (MeV) a sym (  =0.1) 08.3204.6434.9890.3460.14837.3425.68 0.0158.3404.6474.9490.3020.14834.0525.68 0.0258.3464.6534.9260.2730.14832.4125.68 Exp.8.355 B/A, r p slightly changed a sym softened r n -r p becomes small

23. Pygmy Resonance & Symmetry Energy E peak (Pygmy)=8.0 MeV above one n separation energy E peak (GDR)= 13.8, 14.0, 14.2 MeV Adrich et al. PRL95(05)132501 GDR : peak energy is shifted dep. on the symmetry energy at  0 Pygmy resonance is kept unchanged at 8.0 MeV It may set up a constraint on the density dep. of symmetry energy GSI

24. Summary  Theoretical investigations on Pygmy resonance in quasi-particle RPA non-relativistic QRPA relativistic approaches QRRPA  Pairing correlation is important, coupling to the continuum Extended RMF+BCS the s.p. resonance in the continuum including widths  GDR -- restoring force is proportional to the symmetry energy systematic study 33 MeV < a sym (  0 ) < 37 MeV  New excitation modes in exotic nuclei Pygmy modes are related to neutron skin and density dependence of symmetry energy

25. Thanks