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

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 terms Secular equation problem

h1h2p1p2 p1p2h2h1 p3p1 p2 p1h3h2 h1 h2 p1 p2 p1 p2 h2 h1 h4 h3 p2 p1 p3 p4 h2 h1 |n-m|=2 |n-m|=1 |n-m|=0 npnh 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 K Π = 0 + ) No residual proton-neutron interaction

Correlation energy evolution according to proton and neutron valence spaces Ground state, β=0 -E cor (BCS) =0.124 MeV -TrΔΚ ~ 2.1 MeV - TrΔ Κ

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

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) = % Wave function components N sh =9N sh =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 Without SC

Self-consistency effect on single-particle spectrum 22 O 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 protonneutron 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 22 O, E cor ~ 2.5 MeV BCS → E cor ~ 0.12 MeV Importance of the self-consistency for 22 O, 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...) εiεi ε i+1 d R.W. Richardson, Phys.Rev. 141 (1966) 949

N.Pillet, N.Sandulescu, Nguyen Van Giai and J.-F.Berger, Phys.Rev. C71, (2005) Ground state Correlation energy g c =0.24 ΔE cor (BCS) ~ 20% E cor = E(g  0) - 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 -E cor (BCS) =0.588 MeV -TrΔΚ ~ 6.7 MeV Correlation energy evolution according to neutron and proton valence spaces - TrΔ Κ

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 N sh =9N sh =11

Occupation probabilities

-E cor (BCS) =0.588 MeV -TrΔ Κ ~ 6.7 MeV (D1S N sh =9 )

- TrΔ Κ Correlation energy evolution according to neutron and proton valence spaces Ground state, β=0 -E cor (BCS) =0.124 MeV -TrΔΚ ~ 2.1 MeV

Ground state, β =0 (without self-consistency) -E cor (BCS) =0.588 MeV -TrΔ Κ ~ 2.1 MeV