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 PowerPoint PPT Presentation

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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

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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)


  • 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 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




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 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.,

Phys. Rev C, 67 054311 (2003)

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

Helium anomaly


Bound from unbound basis

Halo density

N. Michel et al., Phys. Rev. C 70, 064313 (2004)



(schematic model)



E (MeV)

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

  • 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

    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

  • 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: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

Accepted in Phys. Rev. C

HF/PTG wave functions

---- : PTG

: HF

r (fm)

Accepted in Phys. Rev. C

---- : PTG p

---- : PTG n

: Box p

: Box n

.… : HO

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 QRPAcalculations withquasi-particle basis from HFB/PTG

    Gamow-HFB: problems remain for unbound nuclei

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