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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)

  • 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
gamow states
Gamow states
  • Georg Gamow : a decay

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

  • Definition :
complex scaling method
Complex scaling method
  • Radial integral calculation : complex scaling
  • Analytic continuation : integral independent of R and θ
gamow states location
Gamow states location

Choice of contour




completeness relations with gamow states
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
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


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



(schematic model)



E (MeV)

densities with gamow hfb
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
Quasi-particle pole states
  • Bound, resonant states:

S matrix poles => outgoing wave function behavior

quasi particle scattering states
Quasi-particle scattering states
  • Scattering states:

u(r):incoming and outgoing components

v(r):outgoing wave function behavior

gamow quasi particle states norm
Gamow quasi-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
  • 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
  • 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 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

Accepted in Phys. Rev. C

hf ptg wave functions
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
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