Description of weakly bound nuclei with ptg hfb and gamow hfb approaches
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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). Plan. Scientific motivation: drip-line nuclei

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Description of weakly bound nuclei with ptg hfb and gamow hfb approaches

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)


Plan Gamow/HFB approaches

  • 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


Scientific motivation Gamow/HFB approaches


Gamow states
Gamow states Gamow/HFB approaches

  • Georg Gamow : a decay

    G.A. Gamow, Zs f. Phys. 51 (1928) 204; 52 (1928) 510

  • Definition :


Complex scaling method
Complex scaling method Gamow/HFB approaches

  • Radial integral calculation : complex scaling

  • Analytic continuation : integral independent of R and θ


Gamow states location
Gamow states location Gamow/HFB approaches

Choice of contour

arbitrary

narrow

broad


Completeness relations with gamow states
Completeness relations with Gamow states Gamow/HFB approaches

  • 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
Application : He, Li and O chains Gamow/HFB approaches

  • 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


N. Michel et al., Gamow/HFB approaches

Phys. Rev C, 67 054311 (2003)

Rev. Mex. Fis., 50 S2 74 (2004)

Helium anomaly

reproduced

Bound from unbound basis


Halo density Gamow/HFB approaches


N. Michel et al., Phys. Rev. C Gamow/HFB approaches70, 064313 (2004)

Satisfactory

results

(schematic model)

S-components

missing

E (MeV)


Gamow hfb space
Gamow HFB space Gamow/HFB approaches


Densities with gamow hfb
Densities with Gamow HFB Gamow/HFB approaches

  • HFB equations:

  • Complex particle and pairing densities:

  • HF associated bound and narrow resonant states in discrete sum


Quasi particle pole states
Quasi-particle pole states Gamow/HFB approaches

  • Bound, resonant states:

    S matrix poles => outgoing wave function behavior


Quasi particle scattering states
Quasi-particle scattering states Gamow/HFB approaches

  • Scattering states:

    u(r):incoming and outgoing components

    v(r):outgoing wave function behavior


Gamow quasi particle states norm
Gamow Gamow/HFB approachesquasi-particle states norm

  • Normalization:

    S-matrix poles:complex scaling

    Scattering states: Dirac delta normalization

    Continuum discretization:


Gamow hartree fock diagonalization method
Gamow Hartree-Fock diagonalization method Gamow/HFB approaches

  • 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


Description of nickel calculations
Description of Nickel calculations Gamow/HFB approaches

  • 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
PTG basis for HFB calculations Gamow/HFB approaches

  • 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


Hf ptg potentials
HF/PTG potentials Gamow/HFB approaches

Accepted in Phys. Rev. C


Hf ptg wave functions
HF/PTG wave functions Gamow/HFB approaches

---- : PTG

: HF

r (fm)

Accepted in Phys. Rev. C


---- : PTG p Gamow/HFB approaches

---- : PTG n

: Box p

: Box n

.… : HO

r (fm)

r (fm)

Accepted in Phys. Rev. C


----- : prot. Gamow/HFB approaches

: neut.

… : THO

Accepted in Phys. Rev. C


----- : prot. Gamow/HFB approaches

: neut.

Accepted in Phys. Rev. C


Conclusion and perspectives
Conclusion and perspectives Gamow/HFB approaches

  • 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


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