Beyond the mean field with a multiparticle-multihole wave function and the Gogny force

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

01/07/2005nathalie.pillet@cea.fr

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