# Presentation: 014174

## Gamow HFB description of loosely bound and resonant medium and heavy nuclei

_**Nicolas Michel**_** ****(Kyoto University)**
**Kenichi Matsuyanagi**** (Kyoto University)**
**Mario Stoitsov**** (ORNL – University of Tennessee)**

## Plan 

- Scientific motivation: drip-line nuclei
- Gamow states, Berggren completeness relation
- Gamow quasi-particle states and HFB densities
- Applications: Nickel chain and one proton emitters (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 

- Scientific motivation

## Gamow states

- _Georg Gamow_ :  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**
**arbitrary**

**narrow**

**broad**

## Completeness relations with Gamow states 

- _Berggren completeness relation_ (l,j) :
-     T. Berggren, Nucl. Phys. A **109**, (1967) 205
- _Continuum discretization_ :

## Gamow HFB space

- HF _bound_, resonant

- bound ( < 0)

## 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_
-     Basis generated 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

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## Spherical proton emitters with Skyrme density functional

- _Simple case:_ 185Bi. Pairing for neutrons only, protons ~ Gamow HF
-      Imaginary part of HF proton potential removed.
-      Real bound states, positive width for resonant states
-      Sly4 interaction, pairing window of 60 MeV
- _185__Bi:_  _HFB/Skyrme:_ E = 2.720 MeV, Γ = 9.228 10-11 keV  (0h9/2)
-                  _Experimental:_ Sp = 1.540 MeV, Γ ~ 10-17 keV (Phys. Rev. Lett., **76** (1996) 592)
- _Proton emitters:_ Fit of density functionals with respect to resonant s.p. levels

## 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

## HF/PTG potentials 

- Submitted to Phys. Rev. C

- HF/PTG potentials

## HF/PTG wave functions

- r (fm)

- Submitted to Phys. Rev. C

- ----     : PTG
-          :  HF

- r (fm)

- r (fm)

- Submitted to Phys. Rev. C

- Submitted to Phys. Rev. C

## Conclusion and perspectives

- _HFB expansions with Gamow and PTG bases:_
-      Precise tool to study dripline heavy nuclei
-      Continuum fully taken into account
-      PTG basis near-optimal for weakly bound systems
- _First applications_
-      Nickel chain close to neutron dripline
-      Proton emitter without proton pairing : 185Bi
-      Deformed nuclei with Mg and Zn, prolate and oblate deformation
- _Conclusions and perspectives_
-       Weakly bound nuclei : fast and stable method with PTG basis
-       Problems remain for unbound nuclei