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. [email protected]. 01/07/2005. Nuclear Correlations. Pairing correlations (BCS-HFB). - 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
01/07/[email protected]
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E
E
Nuclear Correlations
Pairing correlations (BCS-HFB)
Correlations associated with 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 pairing-type correlations in all pairing regimes
→ Will the D1S Gogny 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 states
Description of the nucleus in an axially deformed basis(time-reversal symmetry conserved)
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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 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
ΔEcor(BCS) ~ 20%
Ecor = E(g0) - E(g=0)
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Ground state
Occupation probabilities
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Variational Principle
Determination of • 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 ijkl
kljiij
ji aaaaklVij4
1aajKiH
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Mixing coefficient determination
Use of the Shell Model technology !
Using Wick’s theorem, one can extract the usual mean field part and the residual part.
VHHH
Rearrangement terms
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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 simultaneously diagonal
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Preliminary results with the D1S Gogny force in the case of pairing-type correlations
Pairing-type correlations only pair excitations
No residual proton-neutron interaction
Ground state study
Without self-consistency HF calculation + one diagonalization of H in the multiconfiguration space
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-TrΔΚ
Correlation energy evolution according to neutron and proton valence spaces
Ground state, β=0
(without self-consistency)
-Ecor (BCS) =0.124 MeV
-TrΔΚ ~ 2.1 MeV
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Correlation energy evolution according to neutron and proton valence spaces
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Neutron single particle levels evolution according to the HO basis size (HF+BCS)
22O
Nsh = 9 11 13 15 17 19
1d 5/2
2s 1/2
1d 3/2
-7.133 -7.148 -7.157 -7.156 -7.159 -7.160
3.408 3.696 3.649 3.611 3.611 3.605
-3.725 -3.452 -3.498 -3.545 -3.548 -3.555
4.317 4.051 4.005 3.990 3.903 3.913
0.592 0.599 0.507 0.445 0.355 0.358
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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 (without self-consistency)
Nsh=9 Nsh=11
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Self-consistency effect on the correlation energy
With rearrangement terms
2p2h ~ 340 keV
4p4h ~ 530 keV
Without rearrangement terms
2p2h ~ 300 keV
4p4h ~ 390 keV
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Self-consistency effect on proton single particle levels 22O
HF BCS mpmh
1s1/2
1p3/2
1p1/2
1d5/2
2s1/2
1d3/2
-46.634 -46.402 -46.134
-29.431 -29.244 -29.255
-23.366 -23.161 -23.241
-13.514 -13.374 -13.373
- 7.892 -7.862 -7.903
- 4.457 -4.456 -4.510
→Single particle spectrum compressed in comparison to the HF and BCS ones.
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Self-consistency effect on neutron single particle levels 22O
HF BCS mpmh
1s1/2
1p3/2
1p1/2
1d5/2
2s1/2
1d3/2
-42.142 -41.894 -41.902
-23.172 -23.124 -23.082
-18.503 -18.179 -18.292
- 7.133 - 7.133 -7.115
- 3.689 -3.725 -3.742
0.642 0.592 0.580
→Single particle spectrum compressed in comparison to the HF and BCS ones.
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Self-consistency effect on the wave function components 22O
T(0,0) 89.87% 84.04%
T(0,1) 7.50% 11.77%
T(0,2) 0.24% 0.56%
T(2,0) 0.03% 0.04%
T(1,1) 0.17% 0.42%
T(1,0) 2.19% 3.17%
without with
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• derivation of a variational self-consistent method that is able to treat correlations beyond the mean field in an unified way.
•treatment of pairing-type correlations
for 22O, Ecor~ -2.5 MeV
BCS → Ecor ~ -0.12 MeV
•Importance of the self-consistency
(for 22O, gain of 530 keV )
•Importance of the rearrangement terms
(for 22O, contribution of 150 keV )
•Self-consistency effect on the single particle spectrum
Summary
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Outlook
•more general correlations than the pairing-type ones
•connection with RPA
•excited states
•axially deformed nuclei
•even-odd, odd-odd nuclei
•charge radii, bulk properties
.........
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Rearrangement terms
•Polarization effect
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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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Occupation probabilities (without self-consistency)
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Self-consistency effect on occupation probabilities 22O
Proton
with without
1s1/2
1p3/2
1p1/2
1d5/2
2s1/2
1d3/2
Neutron
with without
0.997 0.998
0.993 0.995
0.979 0.987
0.009 0.006
0.002 0.001
0.002 0.001
0.998 0.998
0.996 0.998
0.993 0.997
0.961 0.976
0.060 0.033
0.024 0.016
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Ground state, β=0
(without self-consistency)
-Ecor (BCS) =0.588 MeV
-TrΔΚ ~ 6.7 MeV
Correlation energy evolution according to neutron and proton valence spaces
-TrΔΚ
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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 (without self-consistency)
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Occupation probabilities (without self-consistency)
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Correlation energy evolution according to neutron and proton valence spaces
(without self-consistency)
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T(0,0)= 90.84%
T(0,1)= 5.02%
T(0,2)= 0.16%
T(0,2)= 0.09%
T(1,1)= 0.18%
T(1,0)= 3.72%
Wave function components (without self-consistency)
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Occupation probabilities (without self-consistency)
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Correlation energy evolution according to neutron and proton valence spaces
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T(0,0)= 94.77%
T(0,1)= 2.75%
T(0,2)= 0.03%
T(0,2)= 0.02%
T(1,1)= 0.07%
T(1,0)= 2.35%
Wave function components (without self-consistency)
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Occupation probabilities (without self-consistency)
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Ground state, β=0
(without self-consistency)
-Ecor (BCS) =0.588 MeV
-TrΔΚ ~ 2.1 MeV
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Self-consistency effect on the mean field energy 22O
E(ρHF)
E(ρcor)
Etot
E(ρ) = Tr(Kρ) + ½ Tr Tr(ρVρ)
• HF
E(ρHF) = -168.786 Etot= -168.786
• mpmh without rearrangement terms
E(ρcor) = -166.488 Etot= -171.820
• mpmh with rearrangement terms
E(ρcor) = -164.830 Etot= -171.960
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-Ecor (BCS) =0.588 MeV
-TrΔΚ ~ 6.7 MeV
Ground state, β=0
(without self-consistency)
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Two particles-two levels model
εa= 0
εα= ε
BCS
mpmh
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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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Ground state Correlation energy
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R.W. Richardson, Phys.Rev. 141 (1966) 949 Picket fence model
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