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

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

2. Nuclear Correlations Pairing correlations (BCS-HFB) (non conservation of particle number ) Correlations associated with collective oscillations • Small amplitude (RPA) (Pauli principle not respected ) • Large amplitude (GCM)

3. 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 collective and non collective states →Description of pairing-type correlations in all pairing regimes →Description of particle-vibration coupling → Will the D1S Gogny force be adapted to describe correlations beyond the mean field in this approach ?

4. Trial wave function Similar to the m-scheme Superposition of Slater determinants corresponding to multiparticle-multihole excitations upon a given ground state of HF type {d+n} are axially deformed harmonic oscillator states • Description of the nucleus in an axially deformed basis (time-reversal symmetry conserved) Simultaneous Excitations of protons and neutrons

5. Some Properties of the mpmh wave function • 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) • Importance of the different ph excitation orders ?

6. 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 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 ...) R.W. Richardson, Phys.Rev. 141 (1966) 949

7. 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)

8. Ground state Occupation probabilities

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

10. Mixing coefficient determination Rearrangement terms Using Wick’s theorem, one can extract the usual mean field part and the residual part. Use of the Shell Model technology !

11. 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

12. Determination of optimized single particle states In the general case, h 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 solution Use of the mean field technology !

13. 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

14. Correlation energy evolution according to neutron and proton valence spaces -TrΔΚ Ground state, β=0 (without self-consistency) -Ecor (BCS) =0.124 MeV -TrΔΚ ~ 2.1 MeV

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

16. 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

17. Wave function components (without self-consistency) 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%

18. 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

19. 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.

20. 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.

21. Self-consistency effect on the wave function components 22O without with T(0,0) 89.87% 84.04% T(0,1) 7.50% 11.77% T(1,0) 2.19% 3.17% T(0,2) 0.24% 0.56% T(1,1) 0.17% 0.42% T(2,0) 0.03% 0.04%

22. Summary • 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

23. 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 .........

24. Rearrangement terms •Polarization effect

25. 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

26. Occupation probabilities(without self-consistency)

27. Self-consistency effect on occupation probabilities 22O Proton with without Neutron with without 1s1/2 1p3/2 1p1/2 1d5/2 2s1/2 1d3/2 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

28. Correlation energy evolution according to neutron and proton valence spaces -TrΔΚ Ground state, β=0 (without self-consistency) -Ecor (BCS) =0.588 MeV -TrΔΚ ~ 6.7 MeV

29. Wave function components (without self-consistency) 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%

30. Occupation probabilities(without self-consistency)

31. Correlation energy evolution according to neutron and proton valence spaces (without self-consistency)

32. Wave function components (without self-consistency) T(0,0)= 90.84% T(0,1)= 5.02% T(1,0)= 3.72% T(0,2)= 0.16% T(1,1)= 0.18% T(0,2)= 0.09%

33. Occupation probabilities(without self-consistency)

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

36. Wave function components (without self-consistency) T(0,0)= 94.77% T(0,1)= 2.75% T(1,0)= 2.35% T(0,2)= 0.03% T(1,1)= 0.07% T(0,2)= 0.02%

37. Occupation probabilities(without self-consistency)

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

40. Self-consistency effect on the mean field energy 22O 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 E(ρcor) E(ρHF) Etot

41. Ground state, β=0 (without self-consistency) -Ecor (BCS) =0.588 MeV -TrΔΚ ~ 6.7 MeV

42. εα= ε Two particles-two levels model εa= 0 BCS mpmh

43. Numerical application 0.3750.1460.6250.854 0.4500.3790.5500.578 0.488 0.4220.5120.578

45. Ground state Correlation energy

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