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) April 7-10, 2008 ESNT/SPhN/CEA

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 Zirconium and Magnesium (deformed) Conclusion and perspectives

Scientific motivation

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

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

Gamow states location Choice of contour arbitrary narrow broad

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) :

Application : He, Li and O chains He, Li chains : valence particles above 4He core : 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

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

Halo density

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

Gamow HFB space

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

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

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

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

Gamow Hartree-Fock diagonalization method Two-basis method Basis generated 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

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

PTG basis for HFB calculations Gamow HF(B) basis: Advantages: good asymptotics, smoothly varying continuums Inconvenients: complex arithmetic, long calculations Weakly bound systems: real continuous 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

HF/PTG potentials Accepted in Phys. Rev. C

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

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

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

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

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 QRPA calculations with quasi-particle basis from HFB/PTG Gamow-HFB : problems remain for unbound nuclei