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Testing shell model on nuclei across the N=82 shell gap Testing shell model on nuclei across the N=82 shell gap 1.Test nuclei 2.New experimental data 3.Realistic shell model calculations: basic ingredients 4.Results and comparison with experiment 5.Analysis of the two-body matrix elements 6.Summary Angela Gargano INFN - Napoli

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Napoli-Stony Brook Collaboration L. Coraggio A. Covello A. G. N. Itaco T.T.S. Kuo A. Gargano – Napoli Pisa 2005

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130 Sn 131 Sn 132 Sn 133 Sn 134 Sn 131 Sb 132 Sb 133 Sb 134 Sb 135 Sb 132 Te 134 Te 136 Te Across the N=82 shell gap Behavior of the first 2 + state in even Sn isotopes " in even Te isotopes Behavior of the B(E2; 0 + 2 + ) value in even Sn isotopes " in even Te isotopes Behavior of the first 5/2 + in odd Sb isotopes Multiplets in odd-odd Sb isotopes A. Gargano – Napoli Pisa 2005

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B(E2;0 + 2 + ) = 0.103(15) e 2 b 2 A. Gargano – Napoli Pisa 2005

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D. Radford - ENAM04 A. Gargano – Napoli Pisa Sn and 134 Sn results from J.R. Beene –ENAM04

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A. Gargano – Napoli Pisa 2005

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● Many-body theory: derivation of the effective interaction Realistic shell-model calculations Two main ingredients ● Nucleon-nucleon potential No adjustable parameter in the calculation of two-body matrix elements A. Gargano – Napoli Pisa 2005 Two-body matrix elements of the Hamiltonian derived from the free nucleon-nucleon potential

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Shell-model effective interaction Model-space Schroedinger equation Nuclear many-body Schroedinger equation A. Gargano – Napoli Pisa 2005

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Nucleon-nucleon potential π ρ ω σ 1 σ 2 ● CD-Bonn potential High-precision NN potential based upon the OBE model 2 /N data = 1.02 ( 1999 NN Database: 5990 pp and np scattering data) 43 parameters A. Gargano – Napoli Pisa 2005

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Renormalization of the NN interaction Difficulty in the derivation of V eff from any modern NN potential: existence of a strong repulsive core which prevents its direct use in nuclear structure calculations. Traditional approach to this problem: Brueckner G-matrix method New approach : construction of a low- momentum NN potential V low-k confined within a momentum-space cutoff S. Bogner, T.T.S. Kuo, L. Coraggio, A. Covello, N. Itaco, Phys. Rev C 65, (R) (2002). Derived from the original V NN by integrating out the high- momentum components by means of an iterative method. V low-k preserves the physics of the original NN interaction up to the cut-off momentum Λ: the deuteron binding energy and low- energy scattering phase-shifts are reproduced. A. Gargano – Napoli Pisa 2005

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Derivation of the realistic effective interaction by means of the folded-diagram expansion Derivation of the realistic effective interaction by means of the folded-diagram expansion 1.Calculation of Vertex function composed of irreducibile and valence linked diagrams in V low-k 2.Sum of the folded-diagram expansion by Kreciglowa-Kuo or Lee-Suzuki method A. Gargano – Napoli Pisa 2005 We include one and two-body diagrams up to second order in V low-k “Bubble”

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Sn i13/2 f5/2 p1/2 h9/2 p3/2 f7/2 h11/2 s1/2 d3/2 d5/2 g7/2 d3/2 h11/2 s1/2 g7/2 d5/2 space space - 1 space NN-potential CD-Bonn g7/2 d5/2 d3/2 s1/2 h11/ * SP energies 133 Sb p3/2 h9/2 p1/2 f5/2 i13/2 f7/ * SP energies 133 Sn d3/2 h11/2 s1/2 d5/2 g71/2 -1 SP energies 131 Sn A. Gargano – Napoli Pisa

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134 Sn in shell = 70 keV 86% ( f7/2) 2 81% ( f7/2) 2 BE Expt =6.365 ± MeV PRL 1999 BE Calc =6.082 ± MeV A. Gargano – Napoli Pisa 2005

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Sn isotopes e eff =0.70e from B(E2;6 + 4 + ) in 134 Sn e eff =0.75e from B(E2;10+ 8+) in 134 Sn ▲ Expt. ● Calc. A. Gargano – Napoli Pisa 2005

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Proton-particle neutron-hole multiplets in the shell -1 in the shell 132 Sb A. Gargano – Napoli Pisa 2005 L. Coraggio et al., PRC 66, (2002)

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Proton-particle neutron-particle multiplets in the shell in the shell = 42 keV BE Expt = ± MeV PRL 1999 BE Calc = ± MeV 134 Sb A. Gargano – Napoli Pisa 2005 d 5/2 f 7/2 g 7/2 f 7/2

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135 Sb in shell in shell = 72 keV BE Expt = ± MeV PRL 1999 BE Calc = ± MeV A. Gargano – Napoli Pisa 2005

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Sb isotopes N ■ Splitting of the centroids of the g 7/2 nd d 5/2 SP strengths A. Gargano – Napoli Pisa /2+ 5/2+

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75% g7/2 ( f7/2) % d5/2 ( f7/2) % g7/2 ( f7/2) The low-energy 2 + state in 134 Sn is responsible for the mixing in the 5/2 + state The low position of the 5/2 + is strictly related to the two J = 1 - matrix elements: ( g7/2 f7/2) -600 keV ( d5/2 f7/2) -500 keV (the two 1 - in 134 Sb) A. Gargano – Napoli Pisa Sb

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B(M1;5/2+ 7/2+) 2 x Expt. Calc. (with free g factors ) 0.29 ▲ 25 M1 effective operator: including 2nd order core-polariazation effects 4.0 2 x ( a factor 14) a factor 90 Non-zero off diagonal matrix element between g7/2 and d5/2 is responsible for the B(M1) reduction ®The magnetic moment of the g.s. state is 2.5 to be compared to 1.7 obtained wth free g factors - Expt. 3.0 A. Gargano – Napoli Pisa Sb ▲H. Mach, in Proc. of th 8 th Inter. Spring Seminar on Nucl.Phys., Paestum 2004

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136 Te =100 keV BE Expt = ± MeV PRL 1999 BE Calc = ± MeV in the shell in the shell Dominant component from 2+ state of 134 Te Dominant component from 2+ state of 134 Sn A. Gargano – Napoli Pisa 2005

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Te isotopes e eff ( ) as Sn isotopes e eff ( ) = 1.55e from B(E2;4 + 2 + ) in 134 Te J. Terasaki et al. PRC (2002 ) N. Shimuzu et al. PRC (2004) S. Sarkar et al. EPJA (2004) A. Gargano – Napoli Pisa 2005

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Two-body effective matrix elements (in MeV) Config. V eff V low-k (f7/2) (p3/2) diagonal matrix elements for J =0 + Config. V eff V low-k (g7/2) (d5/2) identical particles diagonal matrix elements for J =2 + diagonal matrix elements for J =0 + diagonal matrix elements for J =2 + Config. V eff V low-k (f7/2) Config. V eff V low-k (g7/2) diagonal matrix elements for J =0 + Config. V eff V low-k (d3/2) (h11/2) (s1/2) diagonal matrix elements for J =2 + Config. V eff V low-k (d3/2) (h11/2) A. Gargano – Napoli Pisa 2005

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V low-k V eff V 3p1h V 2p V 4p2h Two-body matrix elements g 7/2 f 7/2 A. Gargano – Napoli Pisa 2005

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Two-body matrix elements d 5/2 f 7/2 V low-k V eff V 3p1h V 2p V 4p2h ● A. Gargano – Napoli Pisa 2005

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Properties of exotic nuclei in 132 Sn region below and above the N=82 shell closure are well reproduced by our realistic calculations No evidence of shell structure modification in these neutron rich nuclei Very relevant role of core polarization effects A. Gargano – Napoli Pisa 2005 More experimental information is needed

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