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Shell Model in Complex Energy Plane

Shell Model in Complex Energy Plane. N. Sandulescu. Institute of Atomic Physics, Bucharest. Espace de Structure Nucleaire Theorique, Saclay. Resonances and virtual states: Berggren representation Shell model with resonances and virtual states Application: the structure of 11 Li.

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Shell Model in Complex Energy Plane

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  1. Shell Model in Complex Energy Plane N. Sandulescu Institute of Atomic Physics, Bucharest Espace de Structure Nucleaire Theorique, Saclay • Resonances and virtual states: Berggren representation • Shell model with resonances and virtual states • Application: the structure of 11Li *Collaborators: R.Id. Betan (Rosario) , R.J.Liotta (Stockholm), T. Vertse (Debrecen) **Similar work: N. Michel, W.Nazarewicz, M. Ploszajczak, K. Bennaceur,…

  2. Resonances and virtual states 10Li virtual state resonances 9Li

  3. Single-particle resonant states 79Ni 78Ni

  4. Resonant states Decaying state (Gamow,1928) divergence ! « Capturing » state:

  5. Resonant states General defintion ( Siegert, 1939) « Resonances »: out-going solutions Time-reversed solutions :

  6. Poles of S-matrix Im k • k-plane : Re k • energy plane: « anti-resonance » « anti-bound » Re E « resonance » « crazy »

  7. Gamow states : normalisation • Bi-orthogonal set : • Regularisation: Zeldovich (’60) ; Gyarmati & Vertse (1971) complex quantity ! • Matrix elements: • Note : Gamow functions rigged Hilbert space

  8. Berggren representation • Real-energy axis: • Complex-energy plane: Re k L (T. Berggren, Nucl. Phys. A108,265,1968)

  9. Two-particle resonances ; (R.Betan, R.J.Liotta, N.S., T. Vertse, Phys.Rev. Lett. 89, 042501, 2002)

  10. Single-particle states

  11. Two-particle states

  12. Two-particle resonant states 80Ni 78Ni

  13. Two-particle resonant states ( R.Betan, R.J.Liotta, N.S., T. Vertse, Phys. Rev. Lett. 89, 042501, 2002 )

  14. Two-particle resonant states ( R.Betan, R.J.Liotta, N.S., T. Vertse, Phys. Rev. Lett. 89, 042501,2002 )

  15. Resonances and anti-bound states 10Li anti-bound resonances

  16. Anti-bound states Im k Re k • definition: • wave function: (A.B.Migdal et al, Sov.J.Nucl.Phys. 14, 488, 1872 )

  17. Energy contours in Berggren representation Anti-bound state L Resonant states L

  18. Resonances and anti-bound states 10Li anti-bound resonances Note: does a unique mean field exist ? NO !

  19. Effective mean fields for 10Li H. Esbensen, G.F. Bertsch, K. Hencken, Phys.Rev.C56(1997)3054 Particle-vibration couplings: N. Vinh Mau, Nucl. Phys. A592(1995)33 F. Barranco et al, Eur. Phys. J. A11(2001)385 J.C.Pacheco, N. Vinh Mau, Pys.Rev.C65(2002)044004

  20. Ground state of 11Li: pole structure (R.Betan, R.J.Liotta, N.S., T. Vertse, Phys. Lett. B584, 48, 2004 )

  21. Two-particle resonant states in 11Li (R.Betan, R.J.Liotta, N.S., T. Vertse, Phys. Lett. B584, 48, 2004 )

  22. Conclusions Main advantages of shell model in complex energy plane: • based on relevant continuum configurations • direct access to multi-particle resonant states Open problems : • multi-particle resonant states: decays channels ? • efficient truncation schemes for large systems ? - Density Matrix Renormalisation Group ( N. Michel, W. Nazarewicz, M. Ploszajczak, J. Rotureau, nucl-th/0401036) - Lee-Suzuki similarity transformation - Multi-reference perturbation method ( G.Hagen, M.Hjorth-Jensen, J. Vagen, nucl-th/0410114 )

  23. Decay channels R r ……………. …………….

  24. Localisation of scattering states ( R.Betan, R.J.Liotta, N.S., T. Vertse, Phys. Lett. B584, 48, 2004 )

  25. Anti-bound states: trajectories

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