CEA Bruyères-le-Châtel Kazimierz sept 200 5 , Poland
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CEA Bruyères-le-Châtel Kazimierz sept 2005, Poland
Variational Multiparticle-Multihole Mixing
with the D1S Gogny force
N. Pillet (a) , J.-F. Berger (a) , E.Caurier (b) and M. Girod (a)
(a) CEA, Bruyères-le-Châtel, France, (b) IReS, Strasbourg, France
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An unified treatment of correlations beyond the mean field
•conserving the particle number
•enforcing the Pauli principle
•using the Gogny interaction
•Description of pairing-type correlations in all pairing regimes
•Test of the interaction : Will the D1S Gogny force be adapted to describe all correlations beyond the mean field in this method ?
•Description of particle-vibration coupling
Aim of the Variational Multiparticle-Multihole Mixing
Examples of possible studies :
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Trial wave function
Superposition of Slater determinants corresponding to
multiparticle-multihole (mpmh) excitations upon a ground state of HF type
{d+n} are axially deformed harmonic oscillator states
Description of the nucleus in axial symmetry
K good quantum number, time-reversal symmetry conserved
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Some Properties of the mpmh wave function
• Simultaneous excitations of protons and neutrons
(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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Variational Principle
• the mixing coefficients
• the optimized single particle states used in building the Slater determinants
•Definitions
•Total energy
•One-body density
•Energy functional minimization
•Correlation energy
•Hamiltonian
•Determination of
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Mixing coefficients
Using Wick’s theorem, one can extract a mean field part and a residual part
Rearrangement termsSecular equation problem
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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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Optimized single particle states
•Iterative resolution selfconsistent procedure
h[ρ] (one-body hamiltonian) and ρ are no longer simultaneously
diagonal
•No inert core
•Shift of single particle states with respect to those of the HF-type solution
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Preliminary results with the D1S Gogny force in the case of pairing-type correlations
• Pairing-type correlations : mpmh wave function built with pair excitations
(pair : two nucleons coupled to KΠ = 0+ )
• No residual proton-neutron interaction
•
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Correlation energy evolution according to proton and neutron valence spaces
Ground state, β=0-Ecor (BCS) =0.124 MeV
-TrΔΚ ~ 2.1 MeV
-TrΔΚ
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Correlation energy evolution according to neutron valence space and the harmonic oscillator basis size
-TrΔΚ
-TrΔΚ
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T(0,0) 89.87% 84.91%
T(0,1) 7.50% 10.98%
T(0,2) 0.24% 0.51%
T(2,0) 0.03% 0.04%
T(1,1) 0.17% 0.39%
T(1,0) 2.19% 3.17%
T(3,0) + T(0,3) + T(2,1) + T(1,2) = 0.0003%
Wave function components
Nsh=9 Nsh=11
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Occupation probabilities
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Self-consistency (SC) effects
• Correlation energy gain
• Wave function components
• Single-particle spectrum
Up to 2p2h ~ 340 keV
Up to 4p4h ~ 530 keV
T(0,0) T(0,1) T(1,0) T(0,2) T(1,1) T(2,0)
With SC 84.04 11.77 3.17 0.56 0.42 0.04
Without SC 89.87 7.50 2.19 0.24 0.17 0.03
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Self-consistency effect on single-particle spectrum 22O
Δe (MeV)
HF mpmh
1s1/2
1p3/2
1p1/2
1d5/2
2s1/2
1d3/2
→ Single-particle spectrum compressed in comparison to the HF one
18.870 18.820
4.669 4.790
11.370 11.177
3.444 3.373
4.331 4.322
17.203 16.879
6.065 6.014
9.852 9.868
5.622 5.470
3.435 3.393
proton neutron
Δe (MeV)
HF mpmh
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• derivation of a self-consistent method that is able to treat correlations beyond the mean field in an unified way.
Summary
•treatment of pairing-type correlations
for 22O, Ecor~ 2.5 MeV
BCS → Ecor ~ 0.12 MeV
•Importance of the self-consistency
for 22O, correlation energy gain of 530 keV
Self-consistency effect on the single particle spectrum
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Outlook
•more general correlations than the pairing-type ones
•connection with RPA
•excited states
•axially deformed nuclei
.........
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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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Rearrangement terms
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Richardson exact solution of Pairing hamiltonian
Picket fence model
(for one type of particle)
g
The exact solution corresponds to the MC 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
ΔEcor(BCS) ~ 20%
Ecor = E(g0) - E(g=0)
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Ground state
Occupation probabilities
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Ground state Correlation energy
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R.W. Richardson, Phys.Rev. 141 (1966) 949 Picket fence model
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Ground state, β=0-Ecor (BCS) =0.588 MeV
-TrΔΚ ~ 6.7 MeV
Correlation energy evolution according to neutron and proton valence spaces
-TrΔΚ
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Correlation energy evolution according to neutron and proton valence spaces
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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
Wave function components
Nsh=9 Nsh=11
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Occupation probabilities
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-Ecor (BCS) =0.588 MeV
-TrΔΚ ~ 6.7 MeV
(D1S Nsh=9 )
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-TrΔΚ
Correlation energy evolution according to neutron and proton valence spaces
Ground state, β=0-Ecor (BCS) =0.124 MeV
-TrΔΚ ~ 2.1 MeV
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Ground state, β=0
(without self-consistency)
-Ecor (BCS) =0.588 MeV
-TrΔΚ ~ 2.1 MeV