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

1. 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)

2. 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

3. Scientific motivation

4. 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)

5. Complex scaling • Normalization: complex scaling • Analytic continuation: integral independent of R and θ • Normalization of bound and resonant states: • Scattering states: normalization impossiblewith complex scaling Normalization with Dirac delta:

6. Complex energy states Berggren completeness relation Im(k) bound narrowresonances Re(k) antibound states L+ : arbitrary contour broadresonances capturing states

7. Completeness relations with Gamow states • Berggren completeness relation (l,j) : T. Berggren, Nucl. Phys. A 109, (1967) 205 • Continuum discretization :

8. Application : O, He and Be isotopes • O isotopes: valence particles above 16Ocore : H = T + WS (17O) + SDI • 0d5/2, 1s1/2 (bound), 0d3/2(resonant), d3/2 scattering continuum • He and Be isotopes : valence particles above 4Hecore • H = T + WS (5He/5Li) + SGI • 0p3/2, 0p1/2 (resonant), p3/2 and p1/2 scattering continuums • SGI : Surface Gaussian Interaction : 8

9. 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. 9

10. Universal behavior close to emission threshold Spectroscopic factor: VSGI varies Nucleon-nucleon correlations measure WS varies VSGI varies N. Michel et al., Phys. Rev. C, (Rap. Comm.), 75 031301(R) (2007) 10

11. Real WS fit Complex WS fit O(r) N.Michel, W.Nazarewicz, M.Ploszajczak, Nucl. Phys. A 794 29 (2007) 11

12. WS varies N.Michel, W.Nazarewicz, M.Ploszajczak, Nucl. Phys. A 794 29 (2007) 12

13. VSGIvaries N.Michel, W.Nazarewicz, M.Ploszajczak, Nucl. Phys. A 794 29 (2007) 13

14. Mirror reactions : 6He/5He and 6Be/5Li 6Be/5Li 6He/5He VSGIvaries VSGIvaries 14

15. Mirror reactions : 6He/5He and 6Be/5Li 6Be/5Li VSGIvaries 6He/5He 15

16. Mirror reactions : 6He/5He and 6Be/5Li 6Be/5Li 6He/5He VSGIvaries 16

17. HFB framework • HFB ground state : product of independent quasi-particles • HFB equations: • Standard methods of resolution HO basis diagonalization: wellbound states only THO basis diagonalization: basis dependencefromscalingfunction ? Direct integration: veryprecise, but long (box boundary conditions)

18. 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) • Berggrenquasi-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

19. Gamow HFB framework: HF basis diagonalization • 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 PTG or Gamow HF basis

20. Gamow HFB framework:direct integration N. Michel, K.Matsuyanagi, M. Stoitsov Phys. Rev. C, 78 044319 (2008)

21. HF and PTG potentials M. Stoitsov, N. Michel, K.Matsuyanagi, Phys. Rev. C, 77, 054301 (2008)

22. HF/PTG wave functions ---- : PTG : HF M. Stoitsov, N. Michel, K.Matsuyanagi, Phys. Rev. C, 77, 054301 (2008) r (fm)

23. Long axis PTG basis Short axis densities ----- : prot. : neut. … : THO M. Stoitsov, N. Michel, K.Matsuyanagi, Phys. Rev. C, 77, 054301 (2008)

24. PTG basis Pairing densities ----- : prot. : neut. … : THO M. Stoitsov, N. Michel, K.Matsuyanagi, Phys. Rev. C, 77, 054301 (2008)

25. Gamow HFB Nickel densities Black solid line : HFB box Dashed green line : GHFB coord. Dottedblue line : GHF basis N. Michel, K.Matsuyanagi, M. Stoitsov Phys. Rev. C, 78 044319 (2008)

26. Gamow HFB Nickel pairing densities Black solid line : HFB box Dashed green line : GHFB coord. Dottedblue line : GHF basis N. Michel, K.Matsuyanagi, M. Stoitsov Phys. Rev. C, 78 044319 (2008)

27. Conclusion and perspectives • GSM : Standard shell model simplicity and power beyond Hilbert space. Complete description of looselybound 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 Continuumfully 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)