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V low-k and nuclear structure Angela Gargano Napoli

V low-k and nuclear structure Angela Gargano Napoli. A. Gargano Cortona - 2008. Napoli.

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V low-k and nuclear structure Angela Gargano Napoli

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  1. Vlow-kand nuclear structure Angela Gargano Napoli A. Gargano Cortona - 2008 Napoli

  2. Vlow-k Low-momentum potential confined within a momentum-space cutoff  S. K. Bogner, T.T.S. Kuo, L. Coraggio, Nucl. Phys. A684, 432c (2001). S.K. Bogner, T.T.S. Kuo, L. Coraggio, A. Covello, N. Itaco, Phys. Rev. C 65, 051301(R) (2002). ● Derived from the original VNN by integrating out the high-momentum components of the original VNN potential  decouples low-energy physics from high-momentum details ● Vlow-k preserves the physics of the original NNinteraction up to the cutoff momentum Λ: the deuteron binding energy scattering phase-shifts Features of Vlow-k • eliminates sources of non-perturbative behavior • real effective potential in the k-space • gives an approximately unique representation of the NN potential for  2 fm-1  ELab 350 MeV • Vlow-k()  class of potentials all having cutoff independent NN observables < ~ A. Gargano Cortona - 2008 Napoli

  3. Realistic Shell model A-nucleon system Hilbert space Ab-initio calculations (including also NNN forces): • GFMC calculations • no-core shell model • coupled-cluster method limited to small systems N-valence nucleon system Shell-model space accounts for excitations above the model space as well as for interactions with core particles Empirical shell-model calculations no link with NN interaction A. Gargano Cortona - 2008 Napoli

  4. TBME of Veff from VNN 1. renormalization of VNN through Vlow-k • Veff calculation by the folded-diagram perturbation theory A. Gargano Cortona - 2008 Napoli

  5. 2. Veff calculation by folded-diagram perturbation theory We start from ^ Q 2.1 -box calculation collection of irreducible valenced-linked diagrams with at least 1 H1 vertex with ω≡ energy variable and Q (intermediate-state space)=1 – P 1-body diagrams up to 2nd order 2-body diagrams up to 2nd order: V V1p1h V2p V2p2h S-box A. Gargano Cortona - 2008 Napoli

  6. 2.2 Folded diagram series Sum through the Lee-Suzuki iterative technique[Suzuki-Lee Prog. Theor. Phys. 64, 2091 (1980)] Heff =(T+U)+ H1eff =H0 + H1eff H1effcontains both 1- and 2-body contributions “subtraction procedure” to remove from H1eff the 1-body terms single-particle energies from experiment A. Gargano Cortona - 2008 Napoli

  7. 2. Two-body matrix elementsfrom the CD-Bonn NN potential renormalized through the Vlow-k with =2.2 fm-1 Calculations 1. Single-particle energies from expt data of nuclei with one-valence nucleon • U  harmonic oscillator with ћω = 45 A-1/3 - 25 A-2/3 ^ • Q -box  second-order calculation • intermediate states composed of: • hole and particle states restricted to 2 shells below and above the Fermi surface ↔ “small” intermediate-state space • all hole states and particle states restricted to the five shells above the Fermi Surface ↔ “large” intermediate-state space A. Gargano Cortona - 2008 Napoli

  8. . . . 134Sb 132Sn + 1 + 1π . . . i13/2 f5/2 p1/2 h9/2 p3/2 f7/2 εjda 133Sn s1/2 h11/2 d3/2 d5/2 g7/2 132Sn εj da 133Sb 82 50 . . . . . . πspace space A. Gargano Cortona - 2008 Napoli

  9. Calc. ▲Expt. “Small” intermediate-state space 134Sb g7/2f7/2 Jπ% g7/2f7/2 “Large” intermediate-state space g7/2f7/2 A. Gargano Cortona - 2008 Napoli

  10. Diagonal matrix elements of interaction for the g7/2f7/2 configuration Matrix Elements (MeV) J A. Gargano Cortona - 2008 Napoli

  11. Vlow-k for various values of  g7/2f7/2 Matrix Elements (MeV) A. Gargano Cortona - 2008 Napoli

  12. Experimental multiplets 134Sb 210Bi g7/2f7/2 h9/2g9/2 Inversion of the 0- and 1- states ↔ long standing problem • role of tensor force evidenced in studies with empirical TBME • previous studies with realistic effective interactions fail to reproduce the g.s. A. Gargano Cortona - 2008 Napoli

  13. 210Bi • Calc. ▲Expt. h9/2g9/2 J 1p1h correlations produce the right effect to make the 1- the g.s. • non central components arise from virtual interactions with the core nucleons A. Gargano Cortona - 2008 Napoli

  14. 134Sn 132Sn + 2 (f7/2)2 multiplet 134Te 132Sn + 2π (g7/2)2 multiplet 210Pb 208Pb + 2 (g9/2)2 multiplet 210Po 208Pb + 2π (h9/2)2 multiplet A. Gargano Cortona - 2008 Napoli

  15. Diagonal matrix elements of Interaction in 132Sn region (g7/2)2 (f7/2)2 A. Gargano Cortona - 2008 Napoli

  16. Summary • Typical features of Veff originate from core polarization effects ππ and  interactions low-energy 2+ state in 134Sn and 210Pb with respect to 134Te and 210Po π interaction 0- and 1-spacing in 134Sb and 210Bi • reasonable cutoff variations do not seem to change significantly two-body matrix elements A. Gargano Cortona - 2008 Napoli

  17. L. Coraggio (Napoli) A. Covello (Napoli) A. Gargano (Napoli) N. Itaco (Napoli) T.T.S. Kuo (Stony Brook) Cortona - 2008 A. Gargano Napoli

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