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
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
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
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
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
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) W. Grainer “Rel. Quantum Mach.”
Example of resonance states More resonant states for p than those for n due to the Coulomb barrier
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) 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
BCS with the continuum Gap equation : Nucleon densities: Continuum level density
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
Quasi-particle RRPA Response function Unperturbed polarization operator BCS occupation prob. Outside the pairing active space Positive unoccu. states occu. states Negative states
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
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
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
Experiments on GDR Gibelin and Beaumel (Orsay), exp. at RIKEN inelastic scattering of 26 Ne Pb 60 MeV/u 26 Ne secondary beam Dominated by Coulomb excitations selective for E1 transitions. Thesis of J. Gibelin IPNO-T Future work: 28 Ne Pb Theoretical investigation – practical significance Cao, Ma, PRC71(05)034305
Properties of 26,28 Ne Extended RMF+BCS with NL3 GQR check the validity of spherical assumption
26 Ne, 28 Ne IVGDR Cao, Ma, PRC71(05)034305
Sum rule Low-lying GDR in 26 Ne exhaust about 4.9% of TRK sum rule 28 Ne 5.8%
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) QRRPA(RF) Peru(CEA) def. HFB(Gogny) Spherical 10.7 coll. +QRPA(Matrix) Ring(TUM) def. RHB +QRRPA Deformed less coll. (Matrix) Exp.(Gibelin,Beaumel) measure ? ~9(5%) IPNO-T Preliminary
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
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 )
Ground state properties in 132 Sn vv B/A(MeV)r p (fm)r n (fm)r n -r p 00 a sym (MeV) a sym ( =0.1) Exp B/A, r p slightly changed a sym softened r n -r p becomes small
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) 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
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
Thanks