Description of weakly bound nuclei with PTG/HFB and Gamow/HFB approaches

1. Description of weakly bound nuclei with PTG/HFB and Gamow/HFB approaches Nicolas Michel(ESNT/SPhN/CEA) Kenichi Matsuyanagi (Kyoto University) Mario Stoitsov (ORNL – University of Tennessee)

2. Plan • Scientific motivation: drip-line nuclei • Gamow states, Berggren completeness relation, Gamow Shell Model • Gamow quasi-particle states and HFB densities • Applications: Nickel chain (spherical) • Pöschl-Teller-Ginocchio (PTG) basis for loosely bound systems • Resonant structure with PTG basis • Applications: Nickel chain (spherical) Zirconium and Magnesium (deformed) • Conclusion and perspectives

3. Scientific motivation

4. Gamow states • Georg Gamow : a decay G.A. Gamow, Zs f. Phys. 51 (1928) 204; 52 (1928) 510 • Definition :

5. Complex scaling method • Radial integral calculation : complex scaling • Analytic continuation : integral independent of R and θ

6. Gamow states location Choice of contour arbitrary narrow broad

7. Completeness relations with Gamow states • Berggren completeness relation (l,j) : T. Berggren, Nucl. Phys. A 109, (1967) 205 • Continuum discretization : • N-body discretized completeness relation (all l,j) :

8. Application : He, Li and O chains • He, Li chains : valence particles above 4Hecore : H = WS (5He) + SGI 0p3/2, 0p1/2 (resonant), p3/2 and p1/2 scattering continuums SGI : Surface Gaussian Interaction : • Dependence on number of nucleons for T=0 • Spherical Gamow Hartree-Fock basis from H = WS + SGI

9. N. Michel et al., Phys. Rev C, 67 054311 (2003) Rev. Mex. Fis., 50 S2 74 (2004) Helium anomaly reproduced Bound from unbound basis

10. Halo density

11. N. Michel et al., Phys. Rev. C 70, 064313 (2004) Satisfactory results (schematic model) S-components missing E (MeV)

12. Gamow HFB space

13. Densities with Gamow HFB • HFB equations: • Complex particle and pairing densities: • HF associated bound and narrow resonant states in discrete sum

14. Quasi-particle pole states • Bound, resonant states: S matrix poles => outgoing wave function behavior

15. Quasi-particle scattering states • Scattering states: u(r):incoming and outgoing components v(r):outgoing wave function behavior

16. Gamow quasi-particle states norm • Normalization: S-matrix poles:complex scaling Scattering states: Dirac delta normalization Continuum discretization:

17. Gamow Hartree-Fock diagonalization method • Two-basis method Basisgenerated by ph part of HFB hamiltonian: B. Gall et al., Z. Phys. A348 183 (1994) • HFB matrix structure: • Diagonalization of HFB matrix in Gamow HF basis

18. Description of Nickel calculations • Considered nuclei:84Ni, 86Ni, 88Ni, 90Ni • Interaction and space: Skyrme interaction: Sly4, Ecut = 60 MeV, l: 0 →10 Rcut = 20 fm, kmax = 4 fm-1, Nscat (l,j) = 100 for GHF basis. • Interest:Resonant structure directly put in HFB basis

29. PTG basis for HFB calculations • Gamow HF(B) basis: Advantages:good asymptotics, smoothly varying continuums Inconvenients: complex arithmetic, long calculations • Weakly bound systems:realcontinuous bases sufficient • Real Gamow HF basis:problematic due to resonant structure in continuum • PTG basis:resonances replaced by bound states No resonant state in (l,j) partial wave: Hankel/Coulomb functions Smooth continuums for all partial waves Weakly bound systems asymptotics well described

30. HF/PTG potentials Accepted in Phys. Rev. C

31. HF/PTG wave functions ---- : PTG : HF r (fm) Accepted in Phys. Rev. C

32. ---- : PTG p ---- : PTG n : Box p : Box n .… : HO r (fm) r (fm) Accepted in Phys. Rev. C

33. ----- : prot. : neut. … : THO Accepted in Phys. Rev. C

34. ----- : prot. : neut. Accepted in Phys. Rev. C

35. Conclusion and perspectives • Gamow Shell Model: Hybrid method: Gamow Hartree-Fock basis necessary He, Li chains with schematic Hamiltonians Next step : realistic interactions, effective interactions with continuum • HFB expansions with Gamow and PTG bases Precise tool to study dripline heavy nuclei PTG basis near-optimal for weakly bound systems Nickel chain, 40Mg and 110Zn : spherical and deformed ground states • Conclusions and perspectives Weakly bound nuclei : fast and stable method with PTG basis Future QRPAcalculations withquasi-particle basis from HFB/PTG Gamow-HFB: problems remain for unbound nuclei