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
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]
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 )
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
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)
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)
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
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)
Ground state
Occupation probabilities
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
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
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
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
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
-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
Correlation energy evolution according to neutron and proton valence spaces
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
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
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
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.
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.
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
• 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
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
.........
Rearrangement terms
•Polarization effect
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
Occupation probabilities (without self-consistency)
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
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
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)
Occupation probabilities (without self-consistency)
Correlation energy evolution according to neutron and proton valence spaces
(without self-consistency)
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)
Occupation probabilities (without self-consistency)
Correlation energy evolution according to neutron and proton valence spaces
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)
Occupation probabilities (without self-consistency)
Ground state, β=0
(without self-consistency)
-Ecor (BCS) =0.588 MeV
-TrΔΚ ~ 2.1 MeV
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
-Ecor (BCS) =0.588 MeV
-TrΔΚ ~ 6.7 MeV
Ground state, β=0
(without self-consistency)
Two particles-two levels model
εa= 0
εα= ε
BCS
mpmh
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
Ground state Correlation energy
R.W. Richardson, Phys.Rev. 141 (1966) 949 Picket fence model