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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. Nuclear Correlations. Pairing correlations (BCS-HFB). (non conservation of particle number ).

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

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  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

  2. Nuclear Correlations Pairing correlations (BCS-HFB) (non conservation of particle number ) Correlations associated to 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 the pairing-type correlations in all pairing regimes Description of particle-vibration coupling Will the D1S 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 givenground state of HF type {d+n} are axially deformed harmonic oscillator states Description of the nucleus in a deformed basis Simultaneous Excitationsof protons and neutrons

  5. Some Properties of the mpmh wave function • Treatment of the proton-neutron residual partof 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 the 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 Ecorr=E(g≠0)-E(g=0) gc=0.24 ΔEcorr(BCS)~ 20% N.Pillet, N.Sandulescu, Nguyen Van Giai and J.-F.Berger , Phys.Rev. C71 , 044306 (2005)

  8. Ground state Occupation probabilities

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

  10. Variational Principle Determination of • the mixing coefficients • the optimized single particle states used in building the Slater determinants. Definition Hamiltonian Total energy One-body density Correlation energy Minimization of the energy functional

  11. Determination of the mixing coefficients Using Wick’s theorem, one can extract the usual mean field part and the residual part Use of the Shell Model technology !

  12. 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

  13. Determination of optimized single particle states In the general case, h and ρ are no longer diagonal simultaneously •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 !

  14. Preliminary results with the D1S Gogny force in the case of pairing-type correlations Nsh = 9 Nsh = 9 Ground state, β=0 (without self-consistency) -Ecor (BCS) =0.124 MeV -TrΔΚ ~ 2.1 MeV

  15. T(0,0)= 89.87% T(0,1)= 7.50% T(1,0)= 2.19% T(0,2)= 0.24% T(1,1)= 0.17% T(2,0)= 0.03% T(3,0) + T(0,3) + T(2,1) + T(1,2) = 0.0003%

  16. Nsh = 9 Nsh = 9 Ground state, β=0 (without self-consistency) -Ecor (BCS) =0.588 MeV -TrΔΚ ~ 6.7 MeV

  17. 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%

  18. Occupation probabilities (without self-consistency)

  19. Occupation probabilities (without self-consistency)

  20. Outlook •the effect of the selfconsistency •more general correlations than the pairing-type ones •connection with RPA •excited states •axially deformed nuclei •e-e, e-o, o-o nuclei •charge radii, bulk properties .........

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

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

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

  24. Numerical application 0.3750.1460.6250.854 0.4500.3790.5500.578 0.488 0.4220.5120.578

  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. Ground state Correlation energy

  27. Rearrangement terms •Polarization effect

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