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

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

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

B(E2;0+  2+) = 0.103(15) e2b2 A. Gargano – Napoli Pisa 2005

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

A. Gargano – Napoli Pisa 2005

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

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

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

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, 051301(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

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

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

134Sn  in 82-126 shell 86% (f7/2)2 81% (f7/2)2 BEExpt =6.365 ± 0.104 MeV PRL 1999 BECalc=6.082 ± 0.064 MeV  = 70 keV A. Gargano – Napoli Pisa 2005

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

Proton-particle neutron-hole multiplets  in the 50-82 shell -1 in the 50-82 shell 132Sb L. Coraggio et al., PRC 66, 064311 (2002) A. Gargano – Napoli Pisa 2005

Proton-particle neutron-particle multiplets  in the 50-82 shell  in the 82-126 shell BEExpt =12.952 ± 0.052 MeV PRL 1999 BECalc=12.849 ± 0.058 MeV 134Sb d5/2f7/2 g7/2f7/2  = 42 keV A. Gargano – Napoli Pisa 2005

135Sb  in 50-82 shell  in 82-126 shell BEExpt =16.575 ± 0.104 MeV PRL 1999 BECalc=16.411 ± 0.074 MeV  = 72 keV A. Gargano – Napoli Pisa 2005

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

135Sb 75% g7/2 (f7/2)2 +... 45% d5/2 (f7/2)2 + 23% g7/2 (f7/2)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

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 10-3 ( 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. 3.0  A. Gargano – Napoli Pisa 2005

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 =28.564 ± 0.050 MeV PRL 1999 BECalc=28.656 ± 0.082 MeV  =100 keV A. Gargano – Napoli Pisa 2005

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

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

Two-body  matrix elements V3p1h g7/2f7/2 V4p2h V2p Veff Vlow-k A. Gargano – Napoli Pisa 2005

Two-body  matrix elements d5/2f7/2 V3p1h V4p2h V2p Veff ● Vlow-k A. Gargano – Napoli Pisa 2005

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