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: 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

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
Magic numbers in super heavy nuclei Bogoliubov model (II)

Zhang et al. NPA753(2005)106


Contents
Contents Bogoliubov model (II)

  • 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
Deformed Halo? Deformed core? Bogoliubov model (II)

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
Hartree-Fock Bogoliubov theory Bogoliubov model (II)

  • 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
Harmonic oscillator basis and Bogoliubov model (II)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


Schroedinger woods saxon basis

V Bogoliubov model (II)WS(r)

0

r

Rmax

Schroedinger Woods-Saxon basis

Shooting Method



Dirac woods saxon basis
Dirac Woods-Saxon basis Bogoliubov model (II)


Dirac ws negative energy states
Dirac-WS: negative energy states Bogoliubov model (II)

Completeness of the basis (no contradiction with no-sea)

Underbound without inclusion of n.e. states

Results independent of basis parameters


Basis dirac ws versus schroedinger ws

Dirac WS Bogoliubov model (II)

n-max < n+max

Basis: Dirac-WS versus Schroedinger-WS

Smaller Basis!

Schroedinger WS

nFmax = nGmax + 1


Neutron density distribution 48 ca
Neutron density distribution: Bogoliubov model (II)48Ca


Spherical rela hartree calc 72 ca
Spherical Rela. Hartree calc.: Bogoliubov model (II)72Ca

SGZ, Meng & Ring,

PRC68,034323(03)

Woods-Saxon basis reproduces r space


Rmf in a woods saxon basis progress
RMF in a Woods-Saxon basis: progress Bogoliubov model (II)

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
Deformed RHB in a Woods-Saxon basis Bogoliubov model (II)

Axially deformed nuclei


Drhb matrix elements
DRHB matrix elements Bogoliubov model (II)

  • , even

    , 0

  • , even or odd

    , 0 or 1


Pairing interaction
Pairing interaction Bogoliubov model (II)

  • 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
RHB in Woods-Saxon basis for axially deformed nuclei ( Bogoliubov model (II)-force in pp channel)


How to fix the pairing strength and the pairing window
How to fix the pairing strength and the pairing window Bogoliubov model (II)

Zero pairing energy for the neutron


Convergence with e cut and compared to spherical rchb results

E Bogoliubov model (II)+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
Routines checks: comparison with available programs Bogoliubov model (II)

  • 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


Compare with spherical rchb model
Compare with spherical RCHB model Bogoliubov model (II)


Properties of 44 mg
Properties of Bogoliubov model (II)44Mg


Density distributions in 44 mg
Density distributions in Bogoliubov model (II)44Mg


Density distributions in 44 mg1
Density distributions in Bogoliubov model (II)44Mg


Density distributions in 44 mg2
Density distributions in Bogoliubov model (II)44Mg


Pairing tensor in 44 mg
Pairing tensor in Bogoliubov model (II)44Mg


Canonical single neutron states in 44 mg
Canonical single neutron states in Bogoliubov model (II)44Mg


Contents1
Contents Bogoliubov model (II)

  • 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
Analytical continuation in coupling constant Bogoliubov model (II)

Kukulin et al., 1989

Padé approximant


Analytical continuation in coupling constant1
Analytical continuation in coupling constant Bogoliubov model (II)

Zhang, Meng, SGZ, & Hillhouse,

PRC70 (2004) 034308


Analytical continuation in coupling constant2
Analytical continuation in coupling constant Bogoliubov model (II)

Zhang, Meng, SGZ, & Hillhouse,

PRC70 (2004) 034308


Real stabilization method

0 Bogoliubov model (II)

Real stabilization method

Hazi & Taylor, PRA1(1970)1109

Box boundary condition

Stable against changing of box size: resonance

Stable behavior: width


Real stabilization method1
Real stabilization method Bogoliubov model (II)

Zhang, SGZ, Meng, & Zhao, 2007

RMF (PK1)


Real stabilization method2
Real stabilization method Bogoliubov model (II)

Zhang, SGZ, Meng, & Zhao, 2007

RMF (PK1)


Comparisons
Comparisons Bogoliubov model (II)

RMF (NL3)

ACCC: analytical continuation in coupling constant

S: scattering phase shift

RSM: real stabilization method

Zhang, SGZ, Meng, & Zhao, 2007


Summary ii
Summary II Bogoliubov model (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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