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					      Description of drip-line
       nuclei with GSM and
     Gamow/HFB frameworks


                  Nicolas Michel (CEA/IRFU/SPhN)
                     Marek Ploszajszak (GANIL)
                    Witek Nazarewicz (UT/ORNL)
               Kenichi Matsuyanagi (Kyoto University)
            Mario Stoitsov (ORNL – University of Tennessee)



May 20th, 2009   Importance of continuum for nuclei close to   Nicolas Michel
                                 drip-line
                                     Plan
•   Scientific motivation: drip-line nuclei
•   Gamow states definition and normalization, Berggren basis
•   Spectroscopic factors, overlap functions
•   6He/5He, 7He/6He, 18O*/17O*, 6Be/5Li

•   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)
•   Conclusion and perspectives


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             Scientific motivation




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                    Gamow states
• Georg Gamow : simple model of a decay
   G.A. Gamow, Zs f. Phys. 51 (1928) 204; 52 (1928) 510


• Definition :




• Straightforward generalization to non-local potentials (HF)

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                   Complex scaling
• 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:




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          Complex energy states
                                                    Berggren completeness relation
                         Im(k)

             bound
                              narrow resonances

                                                                                Re(k)

antibound states
                                      L+ : arbitrary contour



capturing states              broad resonances
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    Completeness relations with
          Gamow states
• Berggren completeness relation (l,j) :
  T. Berggren, Nucl. Phys. A 109, (1967) 205




• Continuum discretization :



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     Application : O, He and Be isotopes

• O isotopes : valence particles above 16O core : H = T + WS (17O) + SDI
  0d5/2, 1s1/2 (bound), 0d3/2 (resonant), d3/2 scattering continuum



• He and Be isotopes : valence particles above 4He core
                          H = T + WS (5He/5Li) + SGI
   0p3/2, 0p1/2 (resonant), p3/2 and p1/2 scattering continuums
   SGI : Surface Gaussian Interaction :




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      Spectroscopic factors in GSM

• One particle emission channel: (l,j,p/n)
• Basis-independent definition:



• Experimental : cusps appear in cross-sections
• Standard : representation dependence (n,l,j,p/n)
• 5He / 6He, 5Li / 6Be : non resonant components necessary.


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      Spectroscopic factor:
                                                                                          VSGI
                                                                                         varies



 Nucleon-nucleon correlations
          measure                                                                         WS
                                                                                         varies

       Universal behavior
   close to emission threshold

                                                                                          VSGI
                                                                                         varies


                 N. Michel et al.,
Phys. Rev. C, (Rap. Comm.), 75 031301(R) (2007)


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                            Real WS fit

                Complex WS fit
                                                                         O(r)




N.Michel, W.Nazarewicz, M.Ploszajczak,
Nucl. Phys. A 794 29 (2007)
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                                                                       WS
                                                                      varies




N.Michel, W.Nazarewicz, M.Ploszajczak,
Nucl. Phys. A 794 29 (2007)
  May 20th, 2009       Importance of continuum for nuclei close to   Nicolas Michel   12
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                        VSGI varies




N.Michel, W.Nazarewicz, M.Ploszajczak,
Nucl. Phys. A 794 29 (2007)
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6Be/5Li             Mirror reactions : 6He/5He and 6Be/5Li




                                 6He/5He




          VSGI varies                                   VSGI varies
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                     Mirror reactions : 6He/5He and 6Be/5Li
6Be/5Li




                                                                            VSGI varies


                                                                            6He/5He




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                   Mirror reactions : 6He/5He and 6Be/5Li

6Be/5Li




                                                                     6He/5He




                                                                         VSGI varies




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                   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)
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     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

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              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

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                 Gamow HFB framework:
                   direct integration




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


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             HF and PTG potentials




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




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            HF/PTG wave functions




                                                                      ----        : PTG
                                                                                  : HF




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


                                 r (fm)
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              Long axis



                                                                                        PTG
           Short axis                                                                   basis

                                                                                            densities
                                                                                          ----- : prot.
                                                                                                : neut.
                                                                                        … : THO


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

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                                                                               PTG
                                                                               basis

                                                                                    Pairing
                                                                                   densities
                                                                                 ----- : prot.
                                                                                       : neut.
                                                                               … : THO

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


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                                                        Gamow HFB
                                                          Nickel
                                                         densities
                                                                  Black solid line :
                                                                     HFB box

                                                                Dashed green line :
                                                                  GHFB coord.

                                                                  Dotted blue line :
                                                                     GHF basis



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

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                                 drip-lines
                                                        Gamow HFB
                                                          Nickel
                                                          pairing
                                                         densities
                                                                  Black solid line :
                                                                     HFB box

                                                                Dashed green line :
                                                                  GHFB coord.

                                                                  Dotted blue line :
                                                                     GHF basis

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

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       Conclusion and perspectives
•   GSM : Standard shell model simplicity and power beyond Hilbert space.
    Complete description of loosely bound and resonant states..
    Spectroscopic factors: continum coupling necessary
                            Cusps: clear l-dependence
                            Important differences between proton and neutron case

•   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
    Nickel chain close to neutron dripline
    Deformed nuclei with Mg and Zn, prolate and oblate deformation

•   Perspectives
     GSM: Coherent framework unifying structure and reaction, many body open channels.
     Realistic interactions and DMRG method to be applied in calculations.
     PTG/HFB, Gamow/HFB: mean field and beyond (QRPA)


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