Testing shell model on nuclei across the N=82 shell gap

1. 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

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

3. 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

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

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

6. A. Gargano – Napoli Pisa 2005

7. 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

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

9. πρωσ1σ2 Nucleon-nucleon potential ● CD-Bonn potential High-precision NN potential based upon the OBE model 43 parameters 2/Ndata= 1.02 (1999 NN Database: 5990 pp and np scattering data) A. Gargano – Napoli Pisa 2005

10. Renormalization of the NN interaction Difficultyin the derivation of Veff from any modern NN potential: existence of a strong repulsive core which prevents its direct use in nuclear structure calculations. Traditionalapproach 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

11. 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

12. 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 -1SP energies A. Gargano – Napoli Pisa 2005

13.  in 82-126 shell 134Sn 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

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

15. 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

16. 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

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

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

19. 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

20. 135Sb  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

21. 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

22. Te isotopes 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

23. Two-body effective matrix elements (in MeV) identical particles  diagonal matrix elements for J=0+  diagonal matrix elements for J=2+  diagonal matrix elements for J=2+  diagonal matrix elements for J=0+ -1-1 diagonal matrix elements for J=0+ -1-1 diagonal matrix elements for J=2+ A. Gargano – Napoli Pisa 2005

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

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

26. 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 ofcore polarization effects • More experimental information is needed A. Gargano – Napoli Pisa 2005