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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)

slide2
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
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

arbitrary

narrow

broad

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
slide9

N. Michel et al.,

Phys. Rev C, 67 054311 (2003)

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

Helium anomaly

reproduced

Bound from unbound basis

slide11

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

Satisfactory

results

(schematic model)

S-components

missing

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

slide32

---- : PTG p

---- : PTG n

: Box p

: Box n

.… : HO

r (fm)

r (fm)

Accepted in Phys. Rev. C

slide33

----- : prot.

: neut.

… : THO

Accepted in Phys. Rev. C

slide34

----- : 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

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