# Presentation: 472871

## Description of medium and heavy drip-line nuclei within the PTG/HFB and Gamow/HFB frameworks 

**Description of medium and heavy drip-line nuclei within the PTG/HFB and Gamow/HFB frameworks **

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

## Plan 

- Plan

- Scientific motivation: drip-line nuclei
- Gamow states definition and normalization, Berggren basis
- Gamow HFB framework: HF basis diagonalization, direct integration
- Applications: Nickel chain
- Pöschl-Teller-Ginocchio (PTG) basis for loosely bound systems
- Applications: Nickel chain (spherical)
-                             Zirconium and Magnesium (deformed)
- Optimized HFB matrix diagonalization : Takagi factorization
- Conclusion and perspectives

## Scientific motivation 

- Scientific motivation

## Gamow states

- _Georg Gamow_ : simple model of  decay
-      G.A. Gamow, Zs f. Phys. **51** (1928) 204; **52** (1928) 510
- _Definition_ :
- Straightforward generalization to non-local potentials (HF)

## 
Normalization: complex scaling

Analytic continuation: integral independent of R and θ

Normalization of bound and resonant states: 

Scattering states: normalization impossible with complex scaling
      Normalization with Dirac delta: 

- _Normalization_: complex scaling
- _Analytic continuation_: integral _independent_ of R and θ
- _Normalization of bound and resonant states: _
- _Scattering states:_ normalization impossible with complex scaling
-       Normalization with Dirac delta:

- Complex scaling

## Complex energy states

- bound

- broad resonances

- narrow resonances

- L+ : arbitrary contour

- antibound states

- capturing states

**Im(k)**

**Re(k)**

**Berggren completeness relation**

## 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_ :

## HFB framework

- _HFB ground state _: product of independent quasi-particles
- _HFB equations: _
- _Standard methods of resolution_
-       HO basis diagonalization: well bound states only
-       THO basis diagonalization: basis dependence from scaling function ?
-       Direct integration: very precise, but long (box boundary conditions)

## PTG/HFB and Gamow/HFB models

- _Continuous basis methods_
-        _Complex-energy formalism_
-        Bound, resonant and scattering states (Berggren basis)
-        Berggren basis of HF particle states  (two-basis method)
-        Berggren quasi-particles states calculated in coordinate space
-        _Real-energy formalism_
-        Basis generated by a Pöschl-Teller-Ginocchio (PTG) potential
- _Examples:_ Ni chain, 110Zr and 40Mg, HFB density functional: Sly4 + surface pairing
-                         orbital momentum: l=0 to l=10, Ecut = 60 MeV

## Gamow HFB framework: HF basis diagonalization

- Gamow HFB framework: HF basis diagonalization

- _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 PTG or Gamow HF basis_

## Gamow HFB framework:direct integration

- Gamow HFB framework:direct integration

- N. Michel, K.Matsuyanagi, M. Stoitsov
- Phys. Rev. C, **78**  044319 (2008)

## HF and PTG potentials

- M. Stoitsov, N. Michel, K.Matsuyanagi,
- Phys. Rev. C, **77**, 054301 (2008)

## HF/PTG wave functions

- r (fm)

- ----     : PTG
-          :  HF

- M. Stoitsov, N. Michel, K.Matsuyanagi,
- Phys. Rev. C, **77**, 054301 (2008)

## PTG 

- _densities_
- -----  : prot.
-          : neut.
- ... :  THO

- M. Stoitsov, N. Michel, K.Matsuyanagi,
- Phys. Rev. C, **77**, 054301 (2008)

- PTG
- basis

- Long axis

- Short axis

## PTG 

- M. Stoitsov, N. Michel, K.Matsuyanagi,
- Phys. Rev. C, **77**, 054301 (2008)

- _Pairing_
- _densities_
- -----  : prot.
-          : neut.
- ... :  THO

- PTG
- basis

## Gamow HFB

- Black solid line :
- HFB box
- Dashed green line :
-  GHFB coord.
-  Dotted blue line :
-  GHF basis

- _Gamow HFB_
- Nickel
- densities

- N. Michel, K.Matsuyanagi, M. Stoitsov
- Phys. Rev. C, **78**  044319 (2008)

## Gamow HFB

- Black solid line :
- HFB box
- Dashed green line :
-  GHFB coord.
-  Dotted blue line :
-  GHF basis

- _Gamow HFB_
- Nickel
- pairing
- densities

- N. Michel, K.Matsuyanagi, M. Stoitsov
- Phys. Rev. C, **78**  044319 (2008)

## HFB equations

- _Supermatrix standard representation:_
-       Wasteful : 2Nx2N matrix diagonalization for N vectors
- _Equivalent formulation for real case:_
- _Takagi factorization:_
-       _Special case of SVD:
-       _    : positive eigenvalues only : quasi-particle energies
-       _                                 up to complex phases only
-       _Phase provided by phase-dependent equation

## Numerical results

- _Comparison of different methods (Householder + QL):_
- _Conclusion:_
-       Takagi factorization much faster than full diagonalization (gain ~ 3.5)
-       Speed comparable to NxN complex hermitian diagonalization
-       Divide-and-conquer or twisted factorization to be considered in the future

## Conclusion and perspectives

- _HFB expansions with Gamow and PTG bases:_
-      Precise tool to study dripline medium and heavy nuclei
-      Continuum fully taken into account
-      PTG basis very good for weakly bound systems
- _First applications_
-      Nickel chain close to neutron dripline
-      Deformed nuclei with Mg and Zn, prolate and oblate deformation
- _Perspectives_
-       PTG/HFB, Gamow/HFB: mean field and beyond (QRPA)
-       Unbound many-body states at HFB and QRPA level
-       Takagi factorization to calculate HFB wave functions