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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

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

- Definition :

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

Choice of contour

arbitrary

narrow

broad

- Berggren completeness relation (l,j) :
T. Berggren, Nucl. Phys. A 109, (1967) 205

- Continuum discretization :
- N-body discretized completeness relation (all l,j) :

- 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

reproduced

Bound from unbound basis

Halo density

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

Satisfactory

results

(schematic model)

S-components

missing

E (MeV)

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

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

- Scattering states:
u(r):incoming and outgoing components

v(r):outgoing wave function behavior

- Normalization:
S-matrix poles:complex scaling

Scattering states: Dirac delta normalization

Continuum discretization:

- 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

- 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

- 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

Accepted in Phys. Rev. C

---- : 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

- 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