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

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

Shan-Gui Zhou

Email: [email protected]; 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

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-WSSmaller 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

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 resultsRoutines 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

0Real stabilization method

Hazi & Taylor, PRA1(1970)1109

Box boundary condition

Stable against changing of box size: resonance

Stable behavior: width

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

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