1. Variational Multiparticle-Multihole Configuration Mixing Method with the D1S Gogny force INPC2007, Tokyo, 06/06/2007 Nathalie Pillet (CEA Bruyères-le-Châtel, France) nathalie.pillet@cea.fr Collaborators: J.-F. Berger, E. Caurier and H. Goutte

2. Nucleus = A interacting nucleons N-N interaction (QCD not yet usable) Many-body problem Bare forces INPC2007, Tokyo, 06/06/2007 Numerical solution of exact equations A ≤ 12-14 Approximations In medium forces ( Phenomenology) nathalie.pillet@cea.fr Shell model Mean field and beyond Independent particles

3. Variational mpmh configuration mixing o Unified description of correlations beyond the HF approximation {mainly Pairing + RPA + particle vibration} o Conservation of particle numbers and respect of the Pauli principle o Treatement on the same footing of even-even, odd and odd-odd nuclei o Description of both ground and excited states INPC2007, Tokyo, 06/06/2007 Beyond mean field approach to the many-body problem Theoretical motivations nathalie.pillet@cea.fr

4. o Mixing coefficients o Single particle orbitals Variational parameters Trial wave function: Superposition of Slater Determinants INPC2007, Tokyo, 06/06/2007 nathalie.pillet@cea.fr Formalism

5. => Simultaneous solution of both sets of equations (full self-consistency) => renormalization of HF field INPC2007, Tokyo, 06/06/2007 Variational principle oFunctional oDetermination of variational parameters One-body density matrix of the correlated state nathalie.pillet@cea.fr Mixing coefficients Secular equation Optimized single particle states Generalized HF equations +

6. Phenomenological effective D1S* Gogny force Central Density-dependent Spin-orbit Coulomb o The two ranges simulate a “molecular potential” o Density dependence necessary for saturation in nuclear matter o Spin-orbit necessary for magic numbers 14 parameters adjusted on nuclear matter properties and some stable nuclei *J.-F. Berger, M. Girod and D. Gogny, Comput. Phys. Commun. 63 (1991) 365. INPC2007, Tokyo, 06/06/2007 nathalie.pillet@cea.fr

7. Study of “usual” Pairing correlations nathalie.pillet@cea.fr INPC2007, Tokyo, 06/06/2007 o No proton-neutron residual interaction o Correlated wave function o Spin-Isospin components of the D1S Gogny force S=0 T=1S=1 T=1S=0 T=0S=1 T=0 Centralxxxx Densityx Spin-Orbitx Coulombxx Residual interaction “Usual” pairing S=0 T=1 A pair : two nucleons in time-reversed states

8. Usual Pairing in 116 Sn, 106 Sn and 100 Sn ground states INPC2007, Tokyo, 06/06/2007 nathalie.pillet@cea.fr o 116 Sn, 106 Sn and 100 Sn: spherical nuclei o Correlated wave function: up to 2 pair excitations (3 pair excitation negligibles) o Correlation energy: Example: 116 Sn (non-selfconsistent mpmh calculations) Proton valence space: 286 levels Number of neutron individual levels 1 pair 2 pairs BCS -E corr (MeV) => Majority of correlations comes from single particle levels closest to the Fermi level => Majority of correlations comes from configurations associated to 1 pair excitations => Convergence of correlation energy (finite ranges of the central term) => More correlations than in the BCS approach

9. INPC2007, Tokyo, 06/06/2007 nathalie.pillet@cea.fr (MeV) 116 Sn5.44 106 Sn4.62 100 Sn3.67 4.67 3.91 2.97 3.06 2.48 1.62 3.10 1.45 0.00 Without residual Coulomb interaction 4.68 3.92 2.98 o Residual Coulomb: non-negligible effect o mpmh induced correlations: S=0 => dominant pairing correlations S=1 => negligible contribution o BCS method is a better approximation in strong pairing regime ( 116 Sn) o Conservation of particle numbers: very important in weak and medium pairing regimes ( 100 Sn and 106 Sn) Usual Pairing in 116 Sn, 106 Sn and 100 Sn ground states

10. INPC2007, Tokyo, 06/06/2007 nathalie.pillet@cea.fr Correlated wave function components (%) 116 Sn 106 Sn 100 Sn HF65.3867.4490.85 1n pair26.0425.295.02 1p pair4.503.623.70 2n pairs2.682.530.16 1n+1p pairs1.230.900.18 2p pairs0.170.100.09 116 Sn occupation probabilities 2pairs BCS Neutron single particle states Neutron occupation probabilities 2pairs BCS Proton single particle statesProton occupation probabilities

11. Self-consistency effect - 116 Sn Preliminar results ([h[ρ],ρ]=0) o Correlation energy (MeV)1 pair No self-consistency4.47 Approximate self-consistency5.07 (%) HF1 pair No self-consistency87.2912.71 Approximate self-consistency82.6017.40 oCorrelated wave function components Energy gain INPC2007, Tokyo, 06/06/2007 nathalie.pillet@cea.fr

12. Summary and Perpectives INPC2007, Tokyo, 06/06/2007 nathalie.pillet@cea.fr o Self-consistent mpmh approach (new in nuclear physics) -unifies the description of important correlations beyond mean field in nuclei (Pairing, RPA, Particle vibration) -now tractable for medium-heavy nuclei with present computers (pairing hamiltonian) -still have to solve exactly the generalized HF equations o First applications to nuclear superfluidity quite encouraging o Future applications: collective vibrations, exotic light nuclei o Re-definition of effective N-N interaction needed in T=0 channel (based on the PhD thesis work of F.Chappert -> Gogny force with a finite range density-dependent term)

13. Single particle level spectrum INPC2007, Tokyo, 06/06/2007 nathalie.pillet@cea.fr