1. Collective modes of excitation in deformed neutron-rich nuclei Kenichi Yoshida @Saclay 18-20 May, 2009

2. Contents  Uniqueness in deformed neutron-rich nuclei  Deformed HFB+QRPA  Collective modes in neutron-rich Mg isotopes beyond N=20  Collectivity in nuclei at around the island of inversion  Summary and perspectives

3. Uniqueness in neutron-rich nuclei Shallow Fermi level  Spatially extended structure of the single-(quasi)particle wave functions  New shell structures Appearance of new magic numbers/disappearance of traditional magic numbers New regions of deformation Neutron skins and halos Weak binding Continuum coupling

4. M.V. Stoitsov et al., Phys. Rev. C68(2003) 054312 Neutron-rich Mg region between N=20 and 28 Systematic HFB calculation R. Rodríguez-Guzmán et al., NPA709(2002)201 New shell structures – onset of deformation D1S

5. Shallow Fermi level  Spatially extended structure of the single-(quasi)particle wave functions  New shell structures Appearance of new magic numbers/disappearance of traditional magic numbers New regions of deformation Neutron skins and halos  Pairing in the continuum M.Yamagami, PRC72(2005)064308 M.Matsuo et al., PRC71(2005)064326 Changes the spatial structure of the quasiparticle wave functions Emerges the di-neutron correlation Uniqueness in neutron-rich nuclei “Pairing anti-halo effect” K.Bennaceur et al., PLB496(2000)154

6. Collective modes unique in deformed neutron-rich nuclei Neutron excess IS and IV mixing modes Neutron-excitation dominant modes Neutron-skin excitation modes Deformation Mixing of modes with different angular momenta Quadrupole vib. Monopole vib. + Pairing vib. In deformed neutron-rich nuclei with superfluidity ??

7. Continuum Pairing Deformation Self-consistency Collective excitation modes =coherent superposition of 2qp (1p-1h) excitations Stable nuclei Drip-line nuclei Neutron excess Microscopic model required

8. J.Dobaczewski, H.Flocard and J.Treiner, NPA422(1984)103 A.Bulgac, FT-194-1980 (Institute of Atomic Physics, Bucharest) Cf. BCS Unphysical nucleon-gas problem in drip-line nuclei  Mean-field Hamiltonian  Pairing field mixed-type delta interaction SkM* interaction One can properly treat the pairing correlation in the continuum. The coordinate-space Hartree-Fock-Bogoliubov theory 0 11-point formula for derivative Simple Appropriate for describing the spatially extended structure of wavefunctions H.O. basis We solve the HFB equations directly on the 2D lattice. Theoretical framework – quasiparticles in a deformed potential

9. Quasiparticle basisHFB equations particle-hole channel: We neglect the residual spin-orbit and Coulomb interactions. particle-particle channel: Residual interactions KY, N.Van Giai, PRC78(2008)064316 Theoretical framework – quasiparticle RPA

10. Neutron-rich Mg isotopes beyond N=20 SkM*+mixed-type pairing (V 0 =-295 MeV fm 3 ) 34 Mg 36 Mg 38 Mg 40 Mg 0.350.310.290.28 0.410.390.380.36 Isoscalar transition strengths

11. Intrinsic transition densities to the excited 0 + state g.s. half density positive trans. density negative trans. density

12. Sensitive to the shell structure Microscopically calculated Experiments 34 Mg: K.Yoneda et al., PLB499(2001)233 36 Mg: A.Gade et al., PRL99(2007)072502 Low-energy spectra

13. Quadrupole excitations KY, arXiv:0902.3053 34 Mg 36 Mg 38 Mg 40 Mg 1.571.581.821.91 1.41 1.551.79 KY, M.Yamagami, K.Matsuyanagi, NPA 779(2006)99

14. Mechanism of the soft K=0 + mode 34 Mg 40 Mg 2228 KY, M.Yamagami, PRC77(2008)044312 [321]3/2 [202]3/2 [310]1/2 [303]7/2 Ground state Excited state Transition matrix element Opposite sign Enhancement Two level model (Bohr-Mottelson)

15. Neutron-pair transition strengths in 34 Mg Monopole pairingQuadrupole pairing

16. Neutron single-particle energies of 64 Cr Potential energy surfaces (SkM*) M.Stoitsov et al., Comp.Phys.Comm.167(2005)43 The HFB solver “HFBTHO” (v1.66p) Prolate orbital Oblate orbital Neutron-rich Cr and Fe isotopes at around N=40

17. N=40 KY and M.Yamagami, PRC77(2008)044312 Deformed-WS+Bogoliubov+QRPA Soft K=0 + mode in neutron-rich Cr and Fe isotopes

18. Magicity at N=20 J.A.Church et al.,PRC72(2005)054320 Low-lying 2 + state: 885keV ( 32 Mg), 659keV ( 34 Mg) Large B(E2;0 + → 2 + ): 447e 2 fm 4 ( 32 Mg), 541e 2 fm 4 ( 34 Mg) T.Motobayashi et al.,PLB346(1995)9 Breaking of the N=20 spherical magic number Shell inversion Importance of the continuum coupling and pair correlations, M.Yamagami and N.Van Giai, PRC69(2004)034301

19. The island of inversion Y.Utsuno et al., PRC64(2001)011301R N=20 Where is the border located? What is the signature? The gyromagnetic factor measurement The beta-decay study of 33 Mg V.Tripathi et al., PRL101(2008)142504 P.Himpe et al., PLB643(2006)257 “ 33 Al has a certain amount of the 2p2h intruder configuration” The electric quadrupole moment Direct information on the nuclear deformation has been measured at GANIL. T.Nagatomo et al., ENAM’08 conference E.K.Waburton et al., PRC41(1990)1147

20. Particle-vibration coupling Microscopic particle-vibration coupling model Solutions of the Skyrme-HFB+QRPA equations Change of the density due to the collective vibrations To first order in the change of the density, the difference of the potential is evaluated to be

21. Particle-vibration coupling The vacuum is defined as The density variation In a second quantized form using the RPA modes The coupling interaction can be derived from the Skyrme EDF. In the present calculation, the Landau-Migdal approximation is employed. N.Van Giai, H.Sagawa, PLB106(1981)379 The Landau-Migdal parameters are seen in

22. Description of odd A nuclei The nuclear Hamiltonian is diagonalized within the subspace The eigenstate of the odd-A systems: The electric quadrupole moment:

23. Quadrupole moment of neutron-rich Al isotopes KY, PRC79(2009)054303 SkM*+mixed-type pairing (V 0 =-295 MeV fm 3 ) spherical Experiment 31 Al at RIKEN: D. Nagae et al., PRC79(2009)027301

24. Summary  Deformed ground state in 34,36,38,40 Mg Soft K=0 + mode Soft K=0 + modeespecially in 34,40 Mg Sensitive to the neutron number (shell structure around the Fermi level) In the deformation region, where the orbitals both of up- sloping and of down-sloping exist. pairing vibration beta vibration of neutrons The coherent coupling between the pairing vibration and the beta vibration of neutrons 2D-Skyrme Hartree-Fock-Bogoliubov + quasiparticle RPA Giant monopole resonance Giant monopole resonance Two-peak structure at around 15 MeV and 25 MeV Mixed with GQR (K=0)  Core polarization in 31,33,35 Al Neutron pairing correlations across N=20 play an important role for the polarization effect.

25.  Neutron-pair transition strengths Matrix elements for the 2qp transition upper components The upper components of the HFB wavefunctions Perspectives In drip-line nuclei, it is strongly affected by the continuum. A good tool for investigating the continuum

26. The lower-lying resonance consists of two modes. Resonance associated with the K=0 component of the GQR (Non-collective) Neutron excitation mode  New kinds of resonances in deformed drip-line nuclei Isoscalar neutron

27.  Novel picture of single-(quasi)particles T.Misu et al.,NPA614(1997)44 I.Hamamoto, PRC69,041306 (2004) d5/2 s1/2 “s-wave dominance” in weak binding

28. p-h (2qp) excitations into the continuum pairing correlations in the continuum s-wave dominant levels in the continuum?? KY and K.Hagino, PRC72(2005)064311 The Gamow state in a deformed potential