CEA Bruyères-le-Châtel Kazimierz sept 200 5 , Poland

1. 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 CEA Bruyères-le-ChâtelKazimierz sept 2005, Poland

2. Aim of the Variational Multiparticle-Multihole Mixing An unified treatment of correlations beyond the mean field •conserving the particle number •enforcing the Pauli principle •using the Gogny interaction Examples of possible studies : • Description of pairing-type correlations in all pairing regimes • Description of particle-vibration coupling • Test of the interaction : Will the D1S Gogny force be adapted to describe all correlations beyond the mean field in this method ?

3. 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

4. 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)

5. Variational Principle • Determination of • the mixing coefficients • the optimized single particle statesused in building the Slater determinants • Definitions • Hamiltonian • Total energy • One-body density • Correlation energy • Energy functional minimization

6. Mixingcoefficients Rearrangement terms Secular equation problem Using Wick’s theorem, one can extract a mean field part and a residual part

7. npnh< Φτ |:V:| Φτ >mpmh |n-m|=2 |n-m|=0 p1 h1 h2 p2 p1 h1 h1 p2 h2 p1 p2 h2 h1 p1 p2 h2 h1 h3 p4 p2 |n-m|=1 p1 h4 p3 h2 p1 h1 p3 h1 p2 h2 p1 h3

8. Optimized single particle states h[ρ] (one-body hamiltonian) and ρ are no longer simultaneously diagonal •Iterative resolution selfconsistent procedure •No inert core •Shift of single particle states with respect to those of the HF-type solution

9. 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

10. Correlation energy evolution according to proton and neutron valence spaces -TrΔΚ -Ecor (BCS) =0.124 MeV -TrΔΚ ~ 2.1 MeV Ground state, β=0

11. Correlation energy evolution according to neutron valence space and the harmonic oscillator basis size -TrΔΚ -TrΔΚ

12. Wave function components Nsh=9 Nsh=11 T(0,0) 89.87% 84.91% T(0,1) 7.50% 10.98% T(1,0) 2.19% 3.17% T(0,2) 0.24% 0.51% T(1,1) 0.17% 0.39% T(2,0) 0.03% 0.04% T(3,0) + T(0,3) + T(2,1) + T(1,2) = 0.0003%

13. Occupation probabilities

14. Self-consistency (SC) effects • Correlation energy gain Up to 2p2h ~ 340 keV Up to 4p4h ~ 530 keV • Wave function components 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 • Single-particle spectrum

15. Self-consistency effect on single-particle spectrum 22O Δe (MeV) HF mpmh Δe (MeV) HF mpmh 1s1/2 1p3/2 1p1/2 1d5/2 2s1/2 1d3/2 17.20316.879 6.065 6.014 9.852 9.868 5.6225.470 3.435 3.393 18.87018.820 4.669 4.790 11.37011.177 3.4443.373 4.3314.322 proton neutron → Single-particle spectrum compressed in comparison to the HF one

16. Summary • derivation of a 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, correlation energy gain of 530 keV • Self-consistency effect on the single particle spectrum

17. Outlook •more general correlations than the pairing-type ones •connection with RPA •excited states •axially deformed nuclei .........

18. 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

19. Rearrangement terms

20. Richardson exact solution of Pairing hamiltonian Picket fence model (for one type of particle) εi+1 g εi d 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 ...) R.W. Richardson, Phys.Rev. 141 (1966) 949

21. Ground state Correlation energy Ecor = E(g0) - E(g=0) gc=0.24 ΔEcor(BCS) ~ 20% N.Pillet, N.Sandulescu, Nguyen Van Giai and J.-F.Berger , Phys.Rev. C71 , 044306 (2005)

22. Ground state Occupation probabilities

23. Ground state Correlation energy

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

25. Correlation energy evolution according to neutron and proton valence spaces -TrΔΚ -Ecor (BCS) =0.588 MeV -TrΔΚ ~ 6.7 MeV Ground state, β=0

26. Correlation energy evolution according to neutron and proton valence spaces

27. Wave function components Nsh=9 Nsh=11 T(0,0)= 82.65% T(0,1)= 10.02% T(1,0)= 5.98% T(0,2)= 0.56% T(1,1)= 0.54% T(0,2)= 0.23% ~ 15 keV T(3,0) + T(0,3) + T(2,1) + T(1,2) = 0.03%

28. Occupation probabilities

29. (D1S Nsh=9 ) -Ecor (BCS) =0.588 MeV -TrΔΚ ~ 6.7 MeV

30. Correlation energy evolution according to neutron and proton valence spaces -TrΔΚ -Ecor (BCS) =0.124 MeV -TrΔΚ ~ 2.1 MeV Ground state, β=0

31. Ground state, β=0 (without self-consistency) -Ecor (BCS) =0.588 MeV -TrΔΚ ~ 2.1 MeV