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
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
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 )
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
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
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 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
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)
Ground state
Occupation probabilities
R.W. Richardson, Phys.Rev. 141 (1966) 949
Picket fence model
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
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
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 diagonal
simultaneously
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
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%
Ground state, β=0
(without self-consistency)
-Ecor (BCS) =0.588 MeV
-TrΔΚ ~ 6.7 MeV
Nsh = 9Nsh = 9
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
Occupation probabilities (without self-consistency)
Occupation probabilities (without self-consistency)
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
.........
Two particles-two levels model
εa= 0
εα= ε
BCS
mpmh
Ground state, β=0
(without self-consistency)
-Ecor (BCS) =0.588 MeV
-TrΔΚ ~ 2.1 MeV
-Ecor (BCS) =0.588 MeV
-TrΔΚ ~ 6.7 MeV
Ground state, β=0
(without self-consistency)
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
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
Ground state Correlation energy
Rearrangement terms
•Polarization effect