1. Nuclear structure far from stability
Atelier de l’Espace de Structure Nucléaire Théorique, Saclay February 4 – 6, 2008
Nuclear structure far from stability
Marcella Grasso

2. General interest: Correlations in finite fermion many-body systems
Adopted approaches: Microscopic mean field approaches and extensions

3. Extensions of RPA (avoiding the quasi-boson approximation)
DIFFERENT TOPICS:
Nuclear structure. Exotic nuclei (properties of exotic nuclei, pairing, continuum coupling, shell structure evolution along isotopic chains,…)
Mean field, HF, HFB + QRPA
Collaborations: Elias Khan, Jerome Margueron, Nguyen Van Giai, IPN-Orsay
Nicu Sandulescu, Bucarest
Nuclear astrophysics. Neutron star crusts (pairing, excitation modes, specific heat,…)
Mean field, HFB + QRPA
Collaborations: Elias Khan, Jerome Margueron, Nguyen Van Giai, IPN-Orsay
Extensions of RPA (avoiding the quasi-boson approximation)
Collaborations: Francesco Catara, Danilo Gambacurta, Michelangelo Sambataro, Catania
Interdisciplinary activity: ultra-cold trapped Fermi gases
Mean field, finite temperature HFB and QRPA
Collaborations: Elias Khan, Michael Urban, IPN-Orsay

4. Second meeting:
May
Next meeting: to be fixed (2008)

5. Neutron drip line position?
Two-neutron separation energy
S2n(N,Z)=E(N,Z)-E(N-2,Z)
Exp. values
Isotopes of Ni
S2n (MeV)
Last observed isotope
Drip line A
Noyaux riches en neutrons – Approche self-consistante champ moyen + appariement Hartree – Fock – Bogoliubov (HFB)
Etats du continuum: comportement asymptotique (états de diffusion) et largeur des résonances
Microscopic mean field approach. Pairing is included in a self-consistent way (Bogoliubov quasiparticles): Hartree-Fock-Bogoliubov (HFB)
Boundary conditions of scattering states for the wave functions of continuum states
Grasso et al, PRC 64, (2001)

6. Pairing and continuum coupling in neutron-rich nuclei. What to look at?
Direct reaction studies: pair transfer? (LoI GASPARD for Spiral2)

7. Reduction of spin-orbit splitting for neutron p states in 47Ar
Transfer reaction 46Ar(d,p)47Ar: energies and spectroscopic factors of neutron states p3/2, p1/2 and f5/2 in 47Ar. Comparison with 49Ca: reduction the spin – orbit splitting for the f and p neutron states
Gaudefroy, et al. PRL 97, (2006)

8. Energy difference between the states 2s1/2 and 1d3/2
Grasso, Ma, Khan, Margueron, Van Giai, PRC 76, (2007)

9. Effect due to the tensor contribution with SLy5
Grasso, Ma, Khan, Margueron, Van Giai, PRC 76, (2007)

10. HF proton density in 46Ar with SkI5
INVERSION
Khan, Grasso, Margueron, Van Giai, NPA 800, 37 (2008)

11. Perspectives Particle-phonon coupling
Extensions of RPA (to include correlations that are not present in a standard mean field approach). Applications to nuclei

12. B(E2;0+ g.s. -> 21+) (e2fm4) Inv. No inv. B (E2) (e2 fm4) 256 24
SkI SLy4
Inv No inv.
B (E2) (e2 fm4)
218  31 e2 fm4
Khan, Grasso, Margueron, Van Giai,
NPA 800, 37 (2008)
Riley, et al. PRC 72, (2005)
Raman, et al., At. Data Nucl. Data Tables 36, 1 (2001)

13. Inversion of s and d proton states
Theoretical analysis. Relativistic mean field (RMF). 48Ca et 46Ar
Inversion of s and d proton states
48Ca Z=20
1d3/ s1/2
2s1/ d3/2
1d5/ d5/2
46Ar Z=18
1d3/ s1/2
2s1/ d3/2
1d5/ d5/2
Todd-Rutel, et al., PRC 69, (R) (2004)

14. Kinetic, central and spin – orbit contributions to the energy difference between the states 2s1/2 and 1d3/2
Grasso, Ma, Khan, Margueron, Van Giai, PRC 76, (2007)

15. Extension of RPA: starting from the Hamiltonian a boson image is introduced via a mapping procedure (Marumori type)
Approximation:
Degree of expansion of the boson Hamiltonian (quadratic -> standard RPA)
If higher-order terms are introduced the RPA equations are non linear (the matrices A and B depend on the amplitudes X and Y)
Grasso catara sambataro

16. Test on a 3-level Lipkin model
Grasso et al.
Diag. of HB in B
RPA
Extension

17. Spin – orbit potential
Non relativistic case and standard Skyrme forces
q, q’ -> proton or neutron
Relativistic case
The potential is proportional to

18. Hartree-Fock equations with the equivalent potential.
Central term

19. Veqcentr

20. Kinetic contribution
Central contribution
Spin-orbit contribution

21. Important contributions of the HF potential
Central term
It favors the inversion
Density-dependent term
Against the inversion

22. …and the tensor contribution?
Shell model : T. Otsuka, et al., PRL 95, (2005)
Relativistic mean field: RHFB : W. Long, et al., PLB 640, 150 (2006)
Non relativistic mean field:
Skyrme : G. Colò, et al., PLB 646, 227 (2007)
Gogny : T. Otsuka, et al., PRL 97, (2006)

23. Variation of the energy density (dependence on J)
J -> spin density
The spin – orbit potential is modified: