The Relativistic Hartree-Fock and RPA with exchange

1
The Relativistic Hartree-Fock and RPA with
exchange

• Nguyen Van Giai
• IPN Orsay, Université Paris-Sud

2
Outline

• 1 . What is Relativistic Hartree-Fock (RHF)
• 2. The density-dependent RHF model
• 3. Nuclear ground states
• 4. Effect of pi-nucleon coupling on s.p. states
• 5. Effect of rho-pion coupling
• 6. Charge-exchange spin excitations

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The self-consistent mean field approach
Non-Relativistic Relativistic

• - build an effective Lagrangian with nucleon and
  meson fields.
• - find energy extrema by applying variational
  principle to all fields.
• - no-sea approximation to avoid cumbersome
  renormalisation scheme.

• - build V(eff), or directly an energy density
  functional.
• - find local energy minima by Hartree-Fock, or
  Kohn-Sham.

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RMF (Hartree) and RHF (Hartree-Fock)

• Relativistic Mean Field
• - include Hartree contributions
• - neglect Fock contributions (no effect of pion,
  etc)
• - no-sea approximation
• Relativistic Hartree-Fock
• - include Hartree and Fock contributions (pion,
  tensor part of rho can contribute)
• - no-sea approximation

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Lagrangian
A.Bouyssy, J.F. Mathiot, NVG, S. Marcos,
Phys.Rev.C 36, 380 (1987)

• Lagrangian density
• where
• Variational Principle

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Density-dependence of Coupling constants
Fig. 4 Isovector channels
Fig. 3 Isoscalar channels

• Sigma, omega, rho couplings weaker in HF than in
  Hartree
• Pi coupling- exists only in HF
• - becomes weak in medium

7
Coupling strengths with all mesons
W. Long, H. Sagawa, NVG, J. Meng, Phys. Rev. C
76, 034314 (2007)
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Finite Nuclei

• RHF mean field and single-particle states are
  calculated self-consistently by solving
  iteratively the Hartree-Fock equations in
  coordinate space.
• Pairing correlations are included by the BCS
  method with a zero-range, density-dependent
  pairing interaction.

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Binding energies of Sn isotopes
10
Density distributions of magic nuclei
N. Van Giai
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The importance of pion tensor coupling on
single-particle spectra

• It is experimentally found that the shell
  structure (magic numbers N or Z 8, 14, 16, 20,
  etc) changes in  exotic  nuclei
• Shell model analysis suggests that these changes
  are related to the tensor part of the N-N
  interaction
• The RHF approach provides the framework for
  studying the effect of the tensor force induced
  by the pion coupling.

12
Non-relativistic reduction leads to central and
tensor components of one-pion exchange potential
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Theory and Experiment

• Importance of tensor forces on s.p. spectra has
  been stressed in literature
• - shell model context
• T. Otsuka, T. Suzuki, R. Fujimoto, H. Grawe,
  Phys.Rev.Lett.95, 232502 (2005)
• - Skyrme-HF context
• G. Colò, H. Sagawa, S. Fracasso, P.F. Bortignon,
  Phys. Lett. B 646, 227 (2007)
• D.M. Brink and F. Stancu, Phys. Rev. C 75, 064311
  (2007)
• - Experiments on N82 isotones and Z50 isotopes
• J.P. Schiffer et al., Phys. Rev. Lett. 92, 162501
  (2004)

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E(n1i13/2)-E(n1h9/2) in N82 isotones
Expshell closure at Z64 Theory shell closure
At Z58
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E(p1h11/2)-E(p1g7/2) in Z50 isotopes
Particle-vibration coupling brings up mean field
results by 500 keV around N82
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Central and Tensor pion contributions to
E(1i13/2)-E(1h9/2)
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Tensor pion importance of valence orbitals
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Adding a rho-tensor coupling
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Disappearance of artificial gap at N,Z58 (I)
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Disappearance of artificial gap at N,Z58 (II)
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Charge-exchange excitations
What is charge exchange excitations?
example 208Pb (3He, t) 208Bi
H. Akimune et al. PRC 1995.
IAS ?S0 ?L0 GTR ?S1 ?L0 SDR ?S1 ?L1

• Charge exchange excitations can
• provide direct information on the spin and
  spin-isospin properties of the effective nuclear
  interaction.
• determine the neutron rms radius by SDR or IAS
  and GTR.
• calculate the GT ß-decay rates to decide the path
  of the r-process.

A. Krasznahorkay et al. PRL 1999 K. Yako et al.
PRC 2006 D. Vretenar et al. PRL 2003.
J. Engel et al. PRC 1999.
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Theory background

• About 30 years ago, Skyrme-RPA model was extended
  to the charge exchange channel.
• Recently, a fully self-consistent CE Skyrme-QRPA
  model has been developed.
• CE Relativistic (Q)RPA model has been developed
  based on relativistic mean field (RMF) theory.
• Our goal establish a fully self-consistent CE
  RRPA model based on relativistic Hartree-Fock
  (RHF) approach.

N. Auerbach, N. Van Giai and A. Yeverechyahu,
1980.
S. Fracasso and G. Colo, PRC 2005.
C. De Conti et al., PLB 1998 D. Vretenar et al.,
PRL 2003 N. Parr et al., PRC 2004.
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The Landau-Migdal g parameter

• g ' 1/3 corresponds to complete cancellation of
  the contact term due to the pion pseudovector
  coupling this is what is done in RHF-RPA
• In RMFRPA model, g'0.6 is treated as an
  adjusted parameter to reproduce the GTR
  excitation energy of 208Pb.

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RMF-RPA and RHF-RPA

• Self-consistent RPA the particle-hole residual
  interaction is derived from the same Lagrangian
  as ground state.

RHF RPA
one meson exchange
RMF RPA
p
p
p
h
h
h
-
p
h
p
h
p
h
exchange terms
?
?
?
?
?
no in-medium effect
re-fitting
? g1/3
?
?
C. De Conti et al. PLB 1998. D. Vretenar et al.
PRL 2003. N. Parr et al. PRC 2004.
For the first time, a fully self-consistent RRPA
model is established.
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IAS and GTR

• RHFRPA results for excitation energies of IAS
  and GTR. All units are in MeV.

exp aD.J. Horen et al. 1980 H. Akimune et al.
1995. bD.E. Bainum et al. 1980 T. Wakasa et al.
1997. cB.D. Anderson et al. 1985.
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Spin Dipole Resonance in 90Zr

• RHFRPA calculations reproduce the strength
  distributions well up to around 30 MeV in both
  channels.
• 2p-2h configurations are missing.

exp K. Yako et al. PRC 2006.
Tab Charge exchange SD sum rule value and
neutron skin thickness of 90Zr in RHFRPA model.
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About sum rules

• Neutron skin thickness can be determined by
  model-independent SD sum rule.

BUT In relativistic theory with no-sea
approximation, the left hand side must be
calculated including also the negative energy
excitations!
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Conclusion

• Pion and rho-tensor induced interactions treated
  in Relativistic Hartree-Fock
• Importance of tensor components on single
  particle spectra
• Tensor matrix elements important if large j
  orbitals are involved
• Same effective Lagrangian can be used for
  describing charge-exchange modes (spin dipole
  resonances) in RPA
• Next task RPA with rho-tensor coupling

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Collaborators

• Wenhui LONG (Peking University and University of
  Aizu)
• Haozhao LIANG (Peking University and Université
  Paris-Sud)
• Jie MENG (Peking University)
• Hiro SAGAWA (University of Aizu)
• THANK YOU !

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• Neutron skin thickness can be determined by
  model-independent SD sum rule.