1. 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)
April 7-10, 2008
ESNT/SPhN/CEA

2. 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
Zirconium and Magnesium (deformed)
Conclusion and perspectives

3. Scientific motivation

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

5. Complex scaling method
Radial integral calculation : complex scaling
Analytic continuation : integral independent of R and θ

6. Gamow states location
Choice of contour
arbitrary
narrow
broad

7. 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) :

8. Application : He, Li and O chains
He, Li chains : valence particles above 4He core : 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

9. Bound from unbound basis
N. Michel et al.,
Phys. Rev C, (2003)
Rev. Mex. Fis., 50 S2 74 (2004)
Helium anomaly
reproduced
Bound from unbound basis

10. Halo density

11. E (MeV) Satisfactory results (schematic model) S-components missing
N. Michel et al., Phys. Rev. C 70, (2004)
Satisfactory
results
(schematic model)
E (MeV)
S-components
missing

12. Gamow HFB space

13. Densities with Gamow HFB
HFB equations:
Complex particle and pairing densities:
HF associated bound and narrow resonant states in discrete sum

14. Quasi-particle pole states
Bound, resonant states:
S matrix poles => outgoing wave function behavior

15. Quasi-particle scattering states
u(r): incoming and outgoing components
v(r): outgoing wave function behavior

16. Gamow quasi-particle states norm
Normalization:
S-matrix poles: complex scaling
Scattering states: Dirac delta normalization
Continuum discretization:

17. Gamow Hartree-Fock diagonalization method
Two-basis method
Basis generated by ph part of HFB hamiltonian:
B. Gall et al., Z. Phys. A (1994)
HFB matrix structure:
Diagonalization of HFB matrix in Gamow HF basis

18. 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

29. PTG basis for HFB calculations
Gamow HF(B) basis:
Advantages: good asymptotics, smoothly varying continuums
Inconvenients: complex arithmetic, long calculations
Weakly bound systems: real continuous 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

30. HF/PTG potentials
Accepted in Phys. Rev. C

31. HF/PTG wave functions
: PTG
: HF
r (fm)
Accepted in Phys. Rev. C

32. .… : HO ---- : PTG p ---- : PTG n : Box p : Box n r (fm) r (fm)
Accepted in Phys. Rev. C

33. : prot.
: neut.
… : THO
Accepted in Phys. Rev. C

34. : prot.
: neut.
Accepted in Phys. Rev. C

35. 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 QRPA calculations with quasi-particle basis from HFB/PTG
Gamow-HFB : problems remain for unbound nuclei