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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) 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
Magic numbers in super heavy nuclei Zhang et al. NPA753(2005)106
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
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
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
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
VWS(r) 0 r Rmax Schroedinger Woods-Saxon basis Shooting Method
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
Dirac WS n-max < n+max Basis: Dirac-WS versus Schroedinger-WS Smaller Basis! Schroedinger WS nFmax = nGmax + 1
Spherical Rela. Hartree calc.: 72Ca SGZ, Meng & Ring, PRC68,034323(03) Woods-Saxon basis reproduces r space
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
Deformed RHB in a Woods-Saxon basis Axially deformed nuclei
DRHB matrix elements • , even , 0 • , even or odd , 0 or 1
Pairing interaction • Phenomenological pairing interaction with parameters: V0, 0, and ( = 1) Soft cutoff Bonche et al., NPA443,39 (1985) Smooth cutoff
RHB in Woods-Saxon basis for axially deformed nuclei (-force in pp channel)
How to fix the pairing strength and the pairing window Zero pairing energy for the neutron
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
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
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
Analytical continuation in coupling constant Kukulin et al., 1989 Padé approximant
Analytical continuation in coupling constant Zhang, Meng, SGZ, & Hillhouse, PRC70 (2004) 034308
Analytical continuation in coupling constant Zhang, Meng, SGZ, & Hillhouse, PRC70 (2004) 034308
0 Real stabilization method Hazi & Taylor, PRA1(1970)1109 Box boundary condition Stable against changing of box size: resonance Stable behavior: width
Real stabilization method Zhang, SGZ, Meng, & Zhao, 2007 RMF (PK1)
Real stabilization method Zhang, SGZ, Meng, & Zhao, 2007 RMF (PK1)
Comparisons RMF (NL3) ACCC: analytical continuation in coupling constant S: scattering phase shift RSM: real stabilization method Zhang, SGZ, Meng, & Zhao, 2007
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