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

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A. Gargano Cortona - 2008 Napoli V low-k Derived from the original V NN by integrating out the high-momentum components of the original V NN potential decouples low-energy physics from high-momentum details V low-k preserves the physics of the original NN interaction up to the cutoff momentum Λ: the deuteron binding energy scattering phase-shifts Features of V low-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 E Lab 350 MeV V low-k ( ) class of potentials all having cutoff independent NN observables 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). <~<~ Low-momentum potential confined within a momentum-space cutoff

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

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A. Gargano Cortona - 2008 Napoli TBME of V eff from V NN 1. renormalization of V NN through V low-k 2. V eff calculation by the folded-diagram perturbation theory

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A. Gargano Cortona - 2008 Napoli 2. V eff calculation by folded-diagram perturbation theory 2.1 -box calculation 2-body diagrams up to 2nd order: V V 1p1h V 2p V 2p2h 1-body diagrams up to 2nd order S-box collection of irreducible valenced-linked diagrams with at least 1 H 1 vertex We start from with ω energy variable and Q (intermediate-state space) =1 – P ^Q^Q

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A. Gargano Cortona - 2008 Napoli Sum through the Lee-Suzuki iterative technique [Suzuki-Lee Prog. Theor. Phys. 64, 2091 (1980)] H eff =(T+U)+ H 1 eff =H 0 + H 1 eff H 1 eff contains both 1- and 2-body contributions subtraction procedure to remove from H 1 eff the 1-body terms single-particle energies from experiment 2.2 Folded diagram series

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2. Two-body matrix elements from the CD-Bonn NN potential renormalized through the V low-k with =2.2 fm -1 1. Single-particle energies from expt data of nuclei with one-valence nucleon Calculations A. Gargano Cortona - 2008 Napoli U harmonic oscillator with ћ ω = 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

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A. Gargano Cortona - 2008 Napoli s 1/2 h 11/2 d 3/2 d 5/2 g 7/2 i 13/2 f 5/2 p 1/2 h 9/2 p 3/2 f 7/2 82 5050 5050 132 Sn 134 Sb 132 Sn + 1 + 1π ε j da 133 Sn ε j da 133 Sb π space space........................

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A. Gargano Cortona - 2008 Napoli Calc. Expt. 134 Sb Small intermediate-state space Large intermediate-state space g 7/2 f 7/2 0-0- 95 1-1- 94 2-2- 88 3-3- 100 4-4- 94 5-5- 95 6-6- 94 7-7- J π % g 7/2 f 7/2

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A. Gargano Cortona - 2008 Napoli V low-k Diagonal matrix elements of interaction for the g 7/2 f 7/2 configuration J V eff Matrix Elements (MeV) V 1p1h

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A. Gargano Cortona - 2008 Napoli V low-k for various values of g 7/2 f 7/2 Matrix Elements (MeV)

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A. Gargano Cortona - 2008 Napoli 210 Bi 134 Sb h 9/2 g 9/2 g 7/2 f 7/2 Experimental multiplets 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.

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A. Gargano Cortona - 2008 Napoli 210 Bi h 9/2 g 9/2 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 Calc. Expt.

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A. Gargano Cortona - 2008 Napoli 134 Te 132 Sn + 2π 134 Sn 132 Sn + 2 ( f 7/2 ) 2 multiplet ( g 7/2 ) 2 multiplet 210 Pb 208 Pb + 2 210 Po 208 Pb + 2π ( h 9/2 ) 2 multiplet ( g 9/2 ) 2 multiplet

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A. Gargano Cortona - 2008 Napoli Diagonal matrix elements of Interaction in 132 Sn region ( f 7/2 ) 2 ( g 7/2 ) 2

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

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

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A. Gargano Cortona - 2008 Napoli V 1p1h for various values of Λ g 7/2 f 7/2 Matrix Elements (MeV)

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A. Gargano Cortona - 2008 Napoli V low-k and V 1p-1h for various values of Λ ( g 7/2 ) 2 Matrix Elements (MeV) V low-k V 1p-1h J

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A. Gargano Cortona - 2008 Napoli V low-k and V 1p-1h for various values of Λ ( f 7/2 ) 2 Matrix Elements (MeV) V low-k V 1p-1h J

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