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**Testing shell model on nuclei**

across the N=82 shell gap Angela Gargano INFN - Napoli Test nuclei New experimental data Realistic shell model calculations: basic ingredients Results and comparison with experiment Analysis of the two-body matrix elements Summary

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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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**Behavior of the first 2+ state in even Sn isotopes **

131Sb 132Sb 133Sb 134Sb 135Sb 132Te 134Te 136Te 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) e2b2 A. Gargano – Napoli Pisa 2005

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**132Sn and 134Sn results from J.R. Beene –ENAM04**

D. Radford - ENAM04 132Sn and 134Sn results from J.R. Beene –ENAM04 A. Gargano – Napoli Pisa 2005

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A. Gargano – Napoli Pisa 2005

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**Realistic shell-model calculations**

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

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**Shell-model effective interaction**

Nuclear many-body Schroedinger equation Model-space Schroedinger equation A. Gargano – Napoli Pisa 2005

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**Nucleon-nucleon potential**

● CD-Bonn potential High-precision NN potential based upon the OBE model π ρ ω σ1σ2 43 parameters 2/Ndata= 1.02 (1999 NN Database: 5990 pp and np scattering data) A. Gargano – Napoli Pisa 2005

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**Renormalization of the NN interaction**

Difficulty in the derivation of Veff 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 Vlow-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 VNN by integrating out the high-momentum components by means of an iterative method. Vlow-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 Calculation of Vertex function composed of irreducibile and valence linked diagrams in Vlow-k We include one and two-body diagrams up to second order in Vlow-k “Bubble” Sum of the folded-diagram expansion by Kreciglowa-Kuo or Lee-Suzuki method A. Gargano – Napoli Pisa 2005

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**NN-potential CD-Bonn . 132Sn space -1space space 133Sn**

126 NN-potential CD-Bonn i13/2 f5/2 p1/2 h9/2 p3/2 f7/2 . space 82 h11/2 s1/2 d3/2 d5/2 g7/2 d3/2 h11/2 s1/2 g7/2 d5/2 -1space 50 132Sn space 133Sn SP energies 133Sb 131Sn SP energies f7/2 -2.455 -1 SP energies g7/2 d5/2 d3/2 s1/2 h11/2 -9.663 -8.701 -7.223 -6.870* -6.836 d3/2 h11/2 s1/2 d5/2 g71/2 7.325 7.425 7.657 8.980 9.759 p3/2 h9/2 p1/2 f5/2 i13/2 -1.601 -0.894 -0.805 -0.450 0.239* A. Gargano – Napoli Pisa 2005

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**134Sn in 82-126 shell 86% (f7/2)2 81% (f7/2)2**

BEExpt =6.365 ± MeV PRL 1999 BECalc=6.082 ± MeV = 70 keV A. Gargano – Napoli Pisa 2005

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**Sn isotopes ▲ Expt. ● Calc. eeff=0.75e from eeff=0.70e from**

B(E2;10+ 8+) in 134Sn eeff=0.70e from B(E2;6+ 4+) in 134Sn A. Gargano – Napoli Pisa 2005

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**Proton-particle neutron-hole multiplets**

in the shell in the shell 132Sb L. Coraggio et al., PRC 66, (2002) A. Gargano – Napoli Pisa 2005

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**Proton-particle neutron-particle multiplets**

in the shell in the shell BEExpt = ± MeV PRL 1999 BECalc= ± MeV 134Sb d5/2f7/2 g7/2f7/2 = 42 keV A. Gargano – Napoli Pisa 2005

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**135Sb in 50-82 shell in 82-126 shell**

BEExpt = ± MeV PRL 1999 BECalc= ± MeV = 72 keV A. Gargano – Napoli Pisa 2005

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**Sb isotopes ■ Splitting of the centroids of**

7/2+ 5/2+ N ■ Splitting of the centroids of the g7/2 nd d5/2 SP strengths A. Gargano – Napoli Pisa 2005

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135Sb 75% g7/2 (f7/2)2 +... 45% d5/2 (f7/2)2 + 23% g7/2 (f7/2) The low-energy 2+ state in 134Sn 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 134Sb) A. Gargano – Napoli Pisa 2005

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**135Sb 0.29▲ 25 Expt. Calc. B(M1;5/2+ 7/2+) 2 x 10-3**

(with free g factors) 0.29▲ 25 a factor 90 ▲H. Mach, in Proc. of th 8th Inter. Spring Seminar on Nucl .Phys., Paestum 2004 M1 effective operator: including 2nd order core-polariazation effects 4.0 2 x ( a factor 14) 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 A. Gargano – Napoli Pisa 2005

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**136Te in the 50-82 shell in the 82-126 shell**

Dominant component from 2+ state of 134Te Dominant component from 2+ state of 134Sn BEExpt = ± MeV PRL 1999 BECalc= ± MeV =100 keV A. Gargano – Napoli Pisa 2005

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**Te isotopes eeff() as Sn isotopes eeff() = 1.55e from**

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

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**Two-body effective matrix elements (in MeV)**

identical particles diagonal matrix elements for J=0+ diagonal matrix elements for J=2+ Config. Veff Vlow-k (f7/2)2 -0.654 -0.403 (p3/2)2 -0.404 -0.101 Config. Veff Vlow-k (f7/2)2 -0.286 -0.289 diagonal matrix elements for J=0+ diagonal matrix elements for J=2+ Config. Veff Vlow-k (g7/2)2 -0.738 0.063 (d5/2)2 -0.486 -0.304 Config. Veff Vlow-k (g7/2)2 -0.037 -0.016 -1-1 diagonal matrix elements for J=0+ -1-1 diagonal matrix elements for J=2+ Config. Veff Vlow-k (d3/2)2 -0.325 -0.184 (h11/2)2 -1.058 -0.417 (s1/2)2 -0.726 -0.869 Config. Veff Vlow-k (d3/2)2 -0.036 -0.097 (h11/2)2 -0.507 -0.445 A. Gargano – Napoli Pisa 2005

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**Two-body matrix elements**

V3p1h g7/2f7/2 V4p2h V2p Veff Vlow-k A. Gargano – Napoli Pisa 2005

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**Two-body matrix elements**

d5/2f7/2 V3p1h V4p2h V2p Veff ● Vlow-k A. Gargano – Napoli Pisa 2005

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Summary Properties of exotic nuclei in 132Sn 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 More experimental information is needed A. Gargano – Napoli Pisa 2005

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