1. Nicolas Michel Importance of continuum for nuclei close to drip-line May 20th, 2009 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. May 20th, 2009 Importance of continuum for nuclei close to drip-lines Importance of continuum for nuclei close to drip-lines Nicolas Michel 2 Plan Scientific motivation: drip-line nuclei Scientific motivation: drip-line nuclei Gamow states definition and normalization, Berggren basis Gamow states definition and normalization, Berggren basis Spectroscopic factors, overlap functions Spectroscopic factors, overlap functions 6 He/ 5 He, 7 He/ 6 He, 18 O*/ 17 O*, 6 Be/ 5 Li 6 He/ 5 He, 7 He/ 6 He, 18 O*/ 17 O*, 6 Be/ 5 Li Gamow HFB framework: HF basis diagonalization, direct integration Gamow HFB framework: HF basis diagonalization, direct integration Applications: Nickel chain Applications: Nickel chain Pöschl-Teller-Ginocchio (PTG) basis for loosely bound systems Pöschl-Teller-Ginocchio (PTG) basis for loosely bound systems Applications: Nickel chain (spherical) Applications: Nickel chain (spherical) Zirconium and Magnesium (deformed) Zirconium and Magnesium (deformed) Conclusion and perspectives Conclusion and perspectives

3. May 20th, 2009 Importance of continuum for nuclei close to drip-lines Importance of continuum for nuclei close to drip-lines Nicolas Michel 3 Scientific motivation

4. May 20th, 2009 Importance of continuum for nuclei close to drip-lines Importance of continuum for nuclei close to drip-lines Nicolas Michel 4 Gamow states Georg Gamow : simple model of  decay Georg Gamow : simple model of  decay G.A. Gamow, Zs f. Phys. 51 (1928) 204; 52 (1928) 510 G.A. Gamow, Zs f. Phys. 51 (1928) 204; 52 (1928) 510 Definition : Definition : Straightforward generalization to non-local potentials (HF) Straightforward generalization to non-local potentials (HF)

5. May 20th, 2009 Importance of continuum for nuclei close to drip-lines Importance of continuum for nuclei close to drip-lines Nicolas Michel Normalization: complex scaling Normalization: complex scaling Analytic continuation: integral independent of R and θ Analytic continuation: integral independent of R and θ Normalization of bound and resonant states: Normalization of bound and resonant states: Scattering states: normalization impossible with complex scaling Scattering states: normalization impossible with complex scaling Normalization with Dirac delta: Normalization with Dirac delta: 5 Complex scaling

6. May 20th, 2009 Importance of continuum for nuclei close to drip-lines Importance of continuum for nuclei close to drip-lines Nicolas Michel 6 Complex energy states bound broad resonances narrow resonances L + : arbitrary contour antibound states capturing states Im(k) Re(k) Berggren completeness relation

7. May 20th, 2009 Importance of continuum for nuclei close to drip-lines Importance of continuum for nuclei close to drip-lines Nicolas Michel 7 Completeness relations with Gamow states Berggren completeness relation (l,j) : Berggren completeness relation (l,j) : T. Berggren, Nucl. Phys. A 109, (1967) 205 T. Berggren, Nucl. Phys. A 109, (1967) 205 Continuum discretization : Continuum discretization :

8. May 20th, 2009 Importance of continuum for nuclei close to drip-lines Importance of continuum for nuclei close to drip-lines Nicolas Michel O isotopes : valence particles above 16 O core : H = T + WS ( 17 O) + SDI O isotopes : valence particles above 16 O core : H = T + WS ( 17 O) + SDI 0d 5/2, 1s 1/2 (bound), 0d 3/2 (resonant), d 3/2 scattering continuum 0d 5/2, 1s 1/2 (bound), 0d 3/2 (resonant), d 3/2 scattering continuum He and Be isotopes : valence particles above 4 He core He and Be isotopes : valence particles above 4 He core H = T + WS ( 5 He/ 5 Li) + SGI H = T + WS ( 5 He/ 5 Li) + SGI 0p 3/2, 0p 1/2 (resonant), p 3/2 and p 1/2 scattering continuums 0p 3/2, 0p 1/2 (resonant), p 3/2 and p 1/2 scattering continuums SGI : Surface Gaussian Interaction : SGI : Surface Gaussian Interaction : 8 Application : O, He and Be isotopes

9. May 20th, 2009 Importance of continuum for nuclei close to drip-lines Importance of continuum for nuclei close to drip-lines Nicolas Michel 9 Spectroscopic factors in GSM One particle emission channel: (l,j,  ) One particle emission channel: (l,j,  ) Basis-independent definition: Basis-independent definition: Experimental : cusps appear in cross-sections Experimental : cusps appear in cross-sections Standard : representation dependence (n,l,j,  ) Standard : representation dependence (n,l,j,  ) 5 He / 6 He, 5 Li / 6 Be : non resonant components necessary. 5 He / 6 He, 5 Li / 6 Be : non resonant components necessary.

10. May 20th, 2009 Importance of continuum for nuclei close to drip-lines Importance of continuum for nuclei close to drip-lines Nicolas Michel 10 Spectroscopic factor: Nucleon-nucleon correlations measure Universal behavior close to emission threshold N. Michel et al., Phys. Rev. C, (Rap. Comm.), 75 031301(R) (2007) V SGI varies varies WSvaries

11. May 20th, 2009 Importance of continuum for nuclei close to drip-lines Importance of continuum for nuclei close to drip-lines Nicolas Michel 11 Real WS fit Complex WS fit O(r) N.Michel, W.Nazarewicz, M.Ploszajczak, Nucl. Phys. A 794 29 (2007) Nucl. Phys. A 794 29 (2007)

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

13. May 20th, 2009 Importance of continuum for nuclei close to drip-lines Importance of continuum for nuclei close to drip-lines Nicolas Michel 13 N.Michel, W.Nazarewicz, M.Ploszajczak, Nucl. Phys. A 794 29 (2007) Nucl. Phys. A 794 29 (2007) V SGI varies

14. May 20th, 2009 Importance of continuum for nuclei close to drip-lines Importance of continuum for nuclei close to drip-lines Nicolas Michel 14 V SGI varies Mirror reactions : 6 He/ 5 He and 6 Be/ 5 Li 6 He/ 5 He 6 Be/ 5 Li

15. May 20th, 2009 Importance of continuum for nuclei close to drip-lines Importance of continuum for nuclei close to drip-lines Nicolas Michel 15 Mirror reactions : 6 He/ 5 He and 6 Be/ 5 Li V SGI varies 6 Be/ 5 Li 6 He/ 5 He

16. May 20th, 2009 Importance of continuum for nuclei close to drip-lines Importance of continuum for nuclei close to drip-lines Nicolas Michel 16 V SGI varies Mirror reactions : 6 He/ 5 He and 6 Be/ 5 Li 6 Be/ 5 Li 6 He/ 5 He

17. May 20th, 2009 Importance of continuum for nuclei close to drip-lines Importance of continuum for nuclei close to drip-lines Nicolas Michel HFB framework HFB ground state : product of independent quasi-particles HFB ground state : product of independent quasi-particles HFB equations: HFB equations: Standard methods of resolution Standard methods of resolution HO basis diagonalization: well bound states only HO basis diagonalization: well bound states only THO basis diagonalization: basis dependence from scaling function ? THO basis diagonalization: basis dependence from scaling function ? Direct integration: very precise, but long (box boundary conditions) Direct integration: very precise, but long (box boundary conditions) 17

18. May 20th, 2009 Importance of continuum for nuclei close to drip-lines Importance of continuum for nuclei close to drip-lines Nicolas Michel 18 PTG/HFB and Gamow/HFB models Continuous basis methods Continuous basis methods Complex-energy formalism Complex-energy formalism Bound, resonant and scattering states (Berggren basis) Bound, resonant and scattering states (Berggren basis) Berggren basis of HF particle states (two-basis method) Berggren basis of HF particle states (two-basis method) Berggren quasi-particles states calculated in coordinate space Berggren quasi-particles states calculated in coordinate space Real-energy formalism Real-energy formalism Basis generated by a Pöschl-Teller-Ginocchio (PTG) potential Basis generated by a Pöschl-Teller-Ginocchio (PTG) potential Examples: Ni chain, 110 Zr and 40 Mg, HFB density functional: Sly4 + surface pairing Examples: Ni chain, 110 Zr and 40 Mg, HFB density functional: Sly4 + surface pairing orbital momentum: l=0 to l=10, E cut = 60 MeV orbital momentum: l=0 to l=10, E cut = 60 MeV

19. May 20th, 2009 Importance of continuum for nuclei close to drip-lines Importance of continuum for nuclei close to drip-lines Nicolas Michel 19 Gamow HFB framework: HF basis diagonalization Two-basis method Two-basis method Basis generated by ph part of HFB hamiltonian: Basis generated by ph part of HFB hamiltonian: B. Gall et al., Z. Phys. A348 183 (1994) B. Gall et al., Z. Phys. A348 183 (1994) HFB matrix structure: HFB matrix structure: Diagonalization of HFB matrix in PTG or Gamow HF basis Diagonalization of HFB matrix in PTG or Gamow HF basis

20. May 20th, 2009 Importance of continuum for nuclei close to drip-lines Importance of continuum for nuclei close to drip-lines Nicolas Michel 20 Gamow HFB framework: direct integration N. Michel, K.Matsuyanagi, M. Stoitsov Phys. Rev. C, 78 044319 (2008)

21. May 20th, 2009 Importance of continuum for nuclei close to drip-lines Importance of continuum for nuclei close to drip-lines Nicolas Michel 21 HF and PTG potentials M. Stoitsov, N. Michel, K.Matsuyanagi, Phys. Rev. C, 77, 054301 (2008)

22. May 20th, 2009 Importance of continuum for nuclei close to drip-lines Importance of continuum for nuclei close to drip-lines Nicolas Michel 22 HF/PTG wave functions r (fm) ---- : PTG : HF : HF M. Stoitsov, N. Michel, K.Matsuyanagi, Phys. Rev. C, 77, 054301 (2008)

23. May 20th, 2009 Importance of continuum for nuclei close to drip-lines Importance of continuum for nuclei close to drip-lines Nicolas Michel 23 densities ----- : prot. : neut. : neut. … : THO M. Stoitsov, N. Michel, K.Matsuyanagi, Phys. Rev. C, 77, 054301 (2008) PTGbasis Long axis Short axis

24. May 20th, 2009 Importance of continuum for nuclei close to drip-lines Importance of continuum for nuclei close to drip-lines Nicolas Michel 24 M. Stoitsov, N. Michel, K.Matsuyanagi, Phys. Rev. C, 77, 054301 (2008) Pairingdensities ----- : prot. : neut. : neut. … : THO PTGbasis

25. May 20th, 2009 Importance of continuum for nuclei close to drip-lines Importance of continuum for nuclei close to drip-lines Nicolas Michel 25 Black solid line : HFB box Dashed green line : GHFB coord. GHFB coord. Dotted blue line : Dotted blue line : GHF basis GHF basis Gamow HFB Nickeldensities N. Michel, K.Matsuyanagi, M. Stoitsov Phys. Rev. C, 78 044319 (2008)

26. May 20th, 2009 Importance of continuum for nuclei close to drip-lines Importance of continuum for nuclei close to drip-lines Nicolas Michel 26 Black solid line : HFB box Dashed green line : GHFB coord. GHFB coord. Dotted blue line : Dotted blue line : GHF basis GHF basis Gamow HFB Nickelpairingdensities N. Michel, K.Matsuyanagi, M. Stoitsov Phys. Rev. C, 78 044319 (2008)

27. May 20th, 2009 Importance of continuum for nuclei close to drip-lines Importance of continuum for nuclei close to drip-lines Nicolas Michel 27 Conclusion and perspectives GSM : Standard shell model simplicity and power beyond Hilbert space. GSM : Standard shell model simplicity and power beyond Hilbert space. Complete description of loosely bound and resonant states.. Complete description of loosely bound and resonant states.. Spectroscopic factors: continum coupling necessary Spectroscopic factors: continum coupling necessary Cusps: clear l-dependence Cusps: clear l-dependence Important differences between proton and neutron case Important differences between proton and neutron case HFB expansions with Gamow and PTG bases: HFB expansions with Gamow and PTG bases: Precise tool to study dripline medium and heavy nuclei Precise tool to study dripline medium and heavy nuclei Continuum fully taken into account Continuum fully taken into account PTG basis very good for weakly bound systems PTG basis very good for weakly bound systems Nickel chain close to neutron dripline Nickel chain close to neutron dripline Deformed nuclei with Mg and Zn, prolate and oblate deformation Deformed nuclei with Mg and Zn, prolate and oblate deformation Perspectives Perspectives GSM: Coherent framework unifying structure and reaction, many body open channels. GSM: Coherent framework unifying structure and reaction, many body open channels. Realistic interactions and DMRG method to be applied in calculations. Realistic interactions and DMRG method to be applied in calculations. PTG/HFB, Gamow/HFB: mean field and beyond (QRPA) PTG/HFB, Gamow/HFB: mean field and beyond (QRPA)