Aucun titre de diapositive

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Beyond the mean field with a multiparticle-multih
ole wave function and the Gogny force
N. Pillet J.-F. Berger M. Girod CEA,
Bruyères-le-Châtel
E.Caurier IReS, Strasbourg
01/07/2005
nathalie.pillet_at_cea.fr
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Nuclear Correlations
Pairing correlations (BCS-HFB)
(non conservation of particle number )
Correlations associated with collective
oscillations

• Small amplitude (RPA)

(Pauli principle not respected )

• Large amplitude (GCM)

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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 collective and non collective
states
?Description of pairing-type correlations in all
pairing regimes
?Description of particle-vibration coupling
? Will the D1S Gogny force be adapted to describe
correlations beyond the mean field in this
approach ?
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Trial wave function
Similar to the m-scheme
Superposition of Slater determinants
corresponding to multiparticle-multihole
excitations upon a given ground state of HF type
dn are axially deformed harmonic oscillator
states

• Description of the nucleus in an axially deformed
  basis
• (time-reversal symmetry conserved)

Simultaneous Excitations of protons and neutrons
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Some Properties of the mpmh wave function
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)

Importance of the different ph excitation
orders ?
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Richardson exact solution of Pairing hamiltonian
Picket fence model (for one type of particle)
ei1
g
ei
d
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 ...)
R.W. Richardson, Phys.Rev. 141 (1966) 949
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Ground state Correlation energy
Ecor E(g?0) - E(g0)
gc0.24
?Ecor(BCS) 20
N.Pillet, N.Sandulescu, Nguyen Van Giai and
J.-F.Berger , Phys.Rev. C71 , 044306 (2005)
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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
Hamiltonian
Total energy
One-body density
Correlation energy
Energy functional minimization
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Mixing coefficient determination
Rearrangement terms
Using Wicks theorem, one can extract the usual
mean field part and the residual part.
Use of the Shell Model technology !
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npnhlt Ft V Ft gtmpmh
n-m2
n-m0
h1
h2
p1
p2
h1
p1
p2
h1
p1
h2
h2
p2
p1
p2
h2
h1
h3
h1
p4
p2
n-m1
p1
h4
p3
h2
h1
p3
p1
h1
p2
p1
h3
h2
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Determination of optimized single particle states
In the general case, h and ? are no longer
simultaneously diagonal
Iterative resolution ? selfconsistent
procedure No inert core Shift of single
particle states with respect to those of the HF
solution
Use of the mean field technology !
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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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Correlation energy evolution according to
neutron and proton valence spaces
-Tr??
Ground state, ß0 (without self-consistency)
-Ecor (BCS) 0.124 MeV -Tr?? 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 (HFBCS) 22O
Nsh 9
11 13 15 17
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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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Wave function components (without
self-consistency)
Nsh9
Nsh11
T(0,0) 89.87 84.91
T(0,1) 7.50 10.98
T(1,0) 2.19 3.17
T(0,2) 0.24 0.51
T(1,1) 0.17 0.39
T(2,0) 0.03 0.04
T(3,0) T(0,3) T(2,1) T(1,2) 0.0003
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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
without
with
T(0,0) 89.87 84.04
T(0,1) 7.50 11.77
T(1,0) 2.19 3.17
T(0,2) 0.24 0.56
T(1,1) 0.17 0.42
T(2,0) 0.03 0.04
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Summary

• 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

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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
Neutron with without
1s1/2 1p3/2 1p1/2 1d5/2 2s1/2 1d3/2
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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Correlation energy evolution according to
neutron and proton valence spaces
-Tr??
Ground state, ß0 (without self-consistency)
-Ecor (BCS) 0.588 MeV -Tr?? 6.7 MeV
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Wave function components (without
self-consistency)
T(0,0) 82.65
T(0,1) 10.02
T(1,0) 5.98
T(0,2) 0.56
T(1,1) 0.54
T(0,2) 0.23
15 keV
T(3,0) T(0,3) T(2,1) T(1,2) 0.03
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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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Wave function components (without
self-consistency)
T(0,0) 90.84
T(0,1) 5.02
T(1,0) 3.72
T(0,2) 0.16
T(1,1) 0.18
T(0,2) 0.09
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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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Wave function components (without
self-consistency)
T(0,0) 94.77
T(0,1) 2.75
T(1,0) 2.35
T(0,2) 0.03
T(1,1) 0.07
T(0,2) 0.02
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Occupation probabilities (without
self-consistency)
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Ground state, ß0 (without self-consistency)
-Ecor (BCS) 0.588 MeV -Tr?? 2.1 MeV
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Self-consistency effect on the mean field energy
22O
E(?) Tr(K?) ½ Tr Tr(?V?)

• HF
• E(?HF) -168.786 Etot -168.786
• mpmh without rearrangement terms
• E(?cor) -166.488 Etot -171.820
• mpmh with rearrangement terms
• E(?cor) -164.830 Etot -171.960

E(?cor)
E(?HF)
Etot
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Ground state, ß0 (without self-consistency)
-Ecor (BCS) 0.588 MeV -Tr?? 6.7 MeV
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ea e
Two particles-two levels model
ea 0
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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