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Structure of exotic nuclei from relativistic Hartree Bogoliubov model (II)

Structure of exotic nuclei from relativistic Hartree Bogoliubov model (II). Shan-Gui Zhou Email: sgzhou@itp.ac.cn ; URL: http://www.itp.ac.cn/~sgzhou Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing

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Structure of exotic nuclei from relativistic Hartree Bogoliubov model (II)

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  1. Structure of exotic nuclei from relativistic Hartree Bogoliubov model (II) Shan-Gui Zhou Email: sgzhou@itp.ac.cn; URL: http://www.itp.ac.cn/~sgzhou Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing Center of Theoretical Nuclear Physics, National Laboratory of Heavy Ion Accelerator, Lanzhou HISS-NTAA 2007 Dubna, Aug. 7-17

  2. Magic numbers in super heavy nuclei Zhang et al. NPA753(2005)106

  3. Contents • Introduction to Relativistic mean field model • Basics: formalism and advantages • Pseudospin and spin symmetries in atomic nuclei • Pairing correlations in exotic nuclei • Contribution of the continuum • BCS and Bogoliubov transformation • Spherical relativistic Hartree Bogoliubov theory • Formalism and results • Summary I • Deformed relativistic Hartree Bogoliubov theory in a Woods-Saxon basis • Why Woods-Saxon basis • Formalism, results and discussions • Single particle resonances • Analytical continuation in coupling constant approach • Real stabilization method • Summary II

  4. Deformed Halo? Deformed core? Decoupling of the core and valence nucleons？ Misu, Nazarewicz, Aberg, NPA614(97)44 11,14Be Ne isotopes … Bennaceur et al., PLB296(00)154 Hamamoto & Mottelson, PRC68(03)034312 Hamamoto & Mottelson, PRC69(04)064302 Poschl et al., PRL79(97)3841 Nunes, NPA757(05)349 Pei, Xu & Stevenson, NPA765(06)29

  5. Hartree-Fock Bogoliubov theory • Deformed non-relativistic HFB in r space • Deformed relativistic Hartree-Bogoliubov or Hartree-Fock-Bogoliubov theory in harmonic oscillator basis Terasaki, Flocard, Heenen & Bonche, NPA 621, 706 (1996) Stoitsov, Dobaczewski, Ring & Pittel, PRC61, 034311 (2000) Terán, Oberacker & Umar, PRC67, 064314 (2003) Vretenar, Lalazissis & Ring, PRL82, 4595 (1999) No deformed relativistic Hartree-Bogoliubov or Hartree-Fock-Bogoliubov theory in r space available yet

  6. Harmonic oscillator basis and r-space • Average potential in atomic nucleus • Woods-Saxon potential: no analytic solution • harmonic oscillator potential: a good approx. for stable nuclei; matrix diagonalization • Drip line nuclei: large space distribution, contribution of continuum • HO basis: localization • r-space: complicated and time-consuming (deformation and pairing) • Woods-Saxon basis: a reconciler of r-space & HO basis? • Basic idea • Numerical solutions for spherical WS potential in r space • Large-box boundary condition to discretize the continuum • WS wave functions used as a complete basis matrix diagonalization problem

  7. VWS(r) 0 r Rmax Schroedinger Woods-Saxon basis Shooting Method

  8. Spherical RMF in Schroedinger WS basis

  9. Dirac Woods-Saxon basis

  10. Dirac-WS: negative energy states Completeness of the basis (no contradiction with no-sea) Underbound without inclusion of n.e. states Results independent of basis parameters

  11. Dirac WS n-max < n+max Basis: Dirac-WS versus Schroedinger-WS Smaller Basis! Schroedinger WS nFmax = nGmax + 1

  12. Neutron density distribution: 48Ca

  13. Spherical Rela. Hartree calc.: 72Ca SGZ, Meng & Ring, PRC68,034323(03) Woods-Saxon basis reproduces r space

  14. RMF in a Woods-Saxon basis: progress SGZ, Meng & Ring,PRC68,034323(03) SGZ, Meng & Ring, AIP Conf. Proc. 865, 90 (06) SGZ, Meng & Ring, in preparation Woods-Saxon basis might be a reconciler between the HO basis and r space

  15. Deformed RHB in a Woods-Saxon basis Axially deformed nuclei

  16. DRHB matrix elements • , even , 0 • , even or odd , 0 or 1

  17. Pairing interaction • Phenomenological pairing interaction with parameters: V0, 0, and  ( = 1) Soft cutoff Bonche et al., NPA443,39 (1985) Smooth cutoff

  18. RHB in Woods-Saxon basis for axially deformed nuclei (-force in pp channel)

  19. How to fix the pairing strength and the pairing window Zero pairing energy for the neutron

  20. E+cut: 100 MeV ~16 main shells dE ~ 0.1 MeV dr ~ 0.002 fm Convergence with E+cut and compared to spherical RCHB results

  21. Routines checks: comparison with available programs • Compare with spherical RCHB model Spherical, Bogoliubov • Compare with deformed RMF in a WS basis Deformed, no pairing • Compare with deformed RMF+BCS in a WS basis Deformed, BCS for pairing

  22. Compare with spherical RCHB model

  23. Properties of 44Mg

  24. Density distributions in 44Mg

  25. Density distributions in 44Mg

  26. Density distributions in 44Mg

  27. Pairing tensor in 44Mg

  28. Canonical single neutron states in 44Mg

  29. Contents • Introduction to Relativistic mean field model • Basics: formalism and advantages • Pseudospin and spin symmetries in atomic nuclei • Pairing correlations in exotic nuclei • Contribution of the continuum • BCS and Bogoliubov transformation • Spherical relativistic Hartree Bogoliubov theory • Formalism and results • Summary I • Deformed relativistic Hartree Bogoliubov theory in a Woods-Saxon basis • Why Woods-Saxon basis • Formalism, results and discussions • Single particle resonances • Analytical continuation in coupling constant approach • Real stabilization method • Summary II

  30. Analytical continuation in coupling constant Kukulin et al., 1989 Padé approximant

  31. Analytical continuation in coupling constant Zhang, Meng, SGZ, & Hillhouse, PRC70 (2004) 034308

  32. Analytical continuation in coupling constant Zhang, Meng, SGZ, & Hillhouse, PRC70 (2004) 034308

  33. 0 Real stabilization method Hazi & Taylor, PRA1(1970)1109 Box boundary condition Stable against changing of box size: resonance Stable behavior: width

  34. Real stabilization method Zhang, SGZ, Meng, & Zhao, 2007 RMF (PK1)

  35. Real stabilization method Zhang, SGZ, Meng, & Zhao, 2007 RMF (PK1)

  36. Comparisons RMF (NL3) ACCC: analytical continuation in coupling constant S: scattering phase shift RSM: real stabilization method Zhang, SGZ, Meng, & Zhao, 2007

  37. Summary II • Deformed exotic nuclei, particularly halo • Weakly bound and large spatial extension • Continuum contributing • Deformed relativistic Hartree Bogoliubov model in a Woods-Saxon basis for exotic nuclei • W-S basis as a reconciler of the r space and the oscillator basis • Preliminary results for 44Mg • Halo in deformed nucleus tends to be spherical • Single particle resonances: bound state like methods • Analytical continuation in the coupling constant approach • Real stabilization method

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