beyond the mean field with a multiparticle-multihole wave function and the gogny force
DESCRIPTION
Beyond the mean field with a multiparticle-multihole wave function and the Gogny force. N.Pillet J.-F.Berger M.Girod CEA Bruyères-le-Châtel. E.Caurier Ires Strasbourg. Nuclear Correlations. Pairing correlations (BCS-HFB). (non conservation of particle number ). - PowerPoint PPT PresentationTRANSCRIPT
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Beyond the mean field with a
multiparticle-multihole wave function
and the Gogny force
N.Pillet
J.-F.Berger
M.Girod
CEA Bruyères-le-Châtel
E.Caurier
Ires Strasbourg
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E
E
Nuclear Correlations
Pairing correlations (BCS-HFB)
Correlations associated to collective oscillations
Small amplitude (RPA)
Large amplitude (GCM)
(non conservation of particle number )
(Pauli principle not respected )
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Aim of our work
An unified treatment of the correlations beyond the mean field
•conserving the particle number
•enforcing the Pauli principle
•using the Gogny interaction
Description of the pairing-type correlations in all pairing regimes
Will the D1S force be adapted to describe correlations beyond the mean field in this approach ?
Description of particle-vibration coupling
Description of collective and non collective states
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Trial wave function
Superposition of Slater determinants corresponding to multiparticle-multihole excitations upon a given ground state of HF type
Similar to the m-scheme
Simultaneous Excitations of protons and neutrons
{d+n} are axially deformed harmonic oscillator
statesDescription of the nucleus in a deformed basis
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Some Properties of the mpmh wave function
• Importance of the different ph excitation orders ?
• Treatment of the proton-neutron residual part of the interaction
• The projected BCS wave function on particle number is a subset of the mpmh wave function
specific ph excitations (pair excitations)
specific mixing coefficients (particle coefficients x hole coefficients)
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Richardson exact solution of the Pairing hamiltonian
Picket fence model
(for one type of particle)
g
The exact solution corresponds to the multiparticle-multihole wave function including all the configurations built as pair excitations
Test of the importance of the different terms in the mpmh wave function expansion : presently pairing-type correlations (2p2h, 4p4h ...)
εi
εi+1
d
R.W. Richardson, Phys.Rev. 141 (1966) 949
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N.Pillet, N.Sandulescu, Nguyen Van Giai and J.-F.Berger , Phys.Rev. C71 , 044306 (2005)
Ground state Correlation energy
gc=0.24
ΔEcorr(BCS)~ 20%
Ecorr=E(g≠0)-E(g=0)
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Ground state
Occupation probabilities
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R.W. Richardson, Phys.Rev. 141 (1966) 949
Picket fence model
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Variational Principle
Determination of • the mixing coefficients
• the optimized single particle states used in building the
Slater determinants.Definition
Total energy
One-body density
Minimization of the energy functional
Correlation energy
Hamiltonian ijkl
kljiij
ji aaaaklVij4
1aajKiH
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Determination of the mixing coefficients
Use of the Shell Model technology !
Using Wick’s theorem, one can extract the usual mean field part and the residual part
VHHH
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h1 h2p1 p2
p1 p2 h2h1
h1 p3p1
p2 p1 h3h2
h1
h1
h2
p1
p2 p1
p2
h2
h1
h4
h3p2
p1 p3
p4
h2
h1
|n-m|=2
|n-m|=1
|n-m|=0
npnh< Φτ |:V:|Φτ>mpmh
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Determination of optimized single particle states
Use of the mean field technology !
•Iterative resolution → selfconsistent procedure
•No inert core
•Shift of single particle states with respect to those of the HF solution
In the general case, h and ρ are no longer diagonal
simultaneously
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Preliminary results with the D1S Gogny force in the case of pairing-type correlations
Ground state, β=0
(without self-consistency)
-Ecor (BCS) =0.124 MeV
-TrΔΚ ~ 2.1 MeV
Nsh = 9 Nsh = 9
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T(0,0)= 89.87%
T(0,1)= 7.50%
T(0,2)= 0.24%
T(2,0)= 0.03%
T(1,1)= 0.17%
T(1,0)= 2.19%
T(3,0) + T(0,3) + T(2,1) + T(1,2) = 0.0003%
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Ground state, β=0
(without self-consistency)
-Ecor (BCS) =0.588 MeV
-TrΔΚ ~ 6.7 MeV
Nsh = 9Nsh = 9
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T(0,0)= 82.65%
T(0,1)= 10.02%
T(0,2)= 0.56%
T(0,2)= 0.23%
T(1,1)= 0.54%
T(1,0)= 5.98%
T(3,0) + T(0,3) + T(2,1) + T(1,2) = 0.03% ~ 15 keV
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Occupation probabilities (without self-consistency)
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Occupation probabilities (without self-consistency)
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Outlook
•the effect of the selfconsistency
•more general correlations than the pairing-type ones
•connection with RPA
•excited states
•axially deformed nuclei
•e-e, e-o, o-o nuclei
•charge radii, bulk properties
.........
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Two particles-two levels model
εa= 0
εα= ε
BCS
mpmh
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Ground state, β=0
(without self-consistency)
-Ecor (BCS) =0.588 MeV
-TrΔΚ ~ 2.1 MeV
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-Ecor (BCS) =0.588 MeV
-TrΔΚ ~ 6.7 MeV
Ground state, β=0
(without self-consistency)
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Numerical application
0.375 0.146 0.625 0.854
0.450 0.379 0.550 0.578
0.488 0.422 0.512 0.578
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Projected BCS wave function (PBCS) on particle number
BCS wave function
Notation
PBCS : • contains particular ph excitations
• specific mixing coefficients : particle coefficients x hole coefficients
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Ground state Correlation energy
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Rearrangement terms
•Polarization effect
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