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

nathalie.pillet@cea.fr

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 :

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

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)

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

Mixing coefficients

Using Wick’s theorem, one can extract a mean field part and a residual part

Rearrangement termsSecular equation problem

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

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

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

•

Correlation energy evolution according to proton and neutron valence spaces

Ground state, β=0-Ecor (BCS) =0.124 MeV

-TrΔΚ ~ 2.1 MeV

-TrΔΚ

Correlation energy evolution according to neutron valence space and the harmonic oscillator basis size

-TrΔΚ

-TrΔΚ

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

Occupation probabilities

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

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

• 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

Outlook

•more general correlations than the pairing-type ones

•connection with RPA

•excited states

•axially deformed nuclei

.........

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

Rearrangement terms

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

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

Ground state Correlation energy

R.W. Richardson, Phys.Rev. 141 (1966) 949 Picket fence model

Ground state, β=0-Ecor (BCS) =0.588 MeV

-TrΔΚ ~ 6.7 MeV

Correlation energy evolution according to neutron and proton valence spaces

-TrΔΚ

Correlation energy evolution according to neutron and proton valence spaces

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

Occupation probabilities

-Ecor (BCS) =0.588 MeV

-TrΔΚ ~ 6.7 MeV

(D1S Nsh=9 )

-TrΔΚ

Correlation energy evolution according to neutron and proton valence spaces

Ground state, β=0-Ecor (BCS) =0.124 MeV

-TrΔΚ ~ 2.1 MeV

Ground state, β=0

(without self-consistency)

-Ecor (BCS) =0.588 MeV

-TrΔΚ ~ 2.1 MeV