Transcript Slide 1
Testing shell model on nuclei across the N=82 shell gap Angela Gargano INFN - Napoli 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 Napoli-Stony Brook Collaboration L. Coraggio A. Covello A. G. N. Itaco T.T.S. Kuo A. Gargano – Napoli Pisa 2005 130Sn 131Sb 131Sn 132Sb 132Te 132Sn 133Sb 133Sn 134Sn 134Sb 135Sb 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 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 Hi (T VNN )i Ei i Model-space Schroedinger equation PH eff Pi P ( H 0 Veff ) Pi Ei Pi , d where P i i defines the model space 1 and H 0 T U with U an auxiliarySP potential 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 k 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 highmomentum 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 lowenergy scattering phase-shifts are reproduced. A. Gargano – Napoli Pisa 2005 Derivation of the realistic effective interaction by means of the folded-diagram expansion 1. Calculation of Q box 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” 2. Sum of the folded-diagram expansion by Kreciglowa-Kuo or Lee-Suzuki method Veff F0 F1 F2 F3 Q box Q box 1st Q box derivative Q box 1st & 2nd Q box derivatives A. Gargano – Napoli Pisa 2005 NN-potential 126 space i13/2 f5/2 p1/2 h9/2 p3/2 f7/2 CD-Bonn . . . 82 -1space h11/2 s1/2 d3/2 d5/2 g7/2 d3/2 h11/2 s1/2 g7/2 d5/2 50 132Sn space 133Sn 133Sb SP energies 131Sn SP energies f7/2 -2.455 -1 SP energies g7/2 -9.663 p3/2 -1.601 d3/2 7.325 d5/2 -8.701 h9/2 -0.894 h11/2 7.425 d3/2 -7.223 p1/2 -0.805 s1/2 7.657 s1/2 -6.870* f5/2 -0.450 d5/2 8.980 h11/2 -6.836 i13/2 0.239* g71/2 9.759 A. Gargano – Napoli Pisa 2005 in 82-126 shell 134Sn 86% (f7/2)2 81% (f7/2)2 BEExpt =6.365 ± 0.104 MeV BECalc=6.082 ± 0.064 MeV PRL 1999 = 70 keV A. Gargano – Napoli Pisa 2005 Sn isotopes ▲ Expt. ● Calc. eeff=0.75e 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 (lj) (l ' j' ) 1 J 132Sb in the 50-82 shell -1 in the 50-82 shell L. Coraggio et al., PRC 66, 064311 (2002) A. Gargano – Napoli Pisa 2005 Proton-particle neutron-particle multiplets (lj) (l ' j' ) J 134Sb in the 50-82 shell in the 82-126 shell BEExpt =12.952 ± 0.052 MeV BECalc=12.849 ± 0.058 MeV PRL 1999 d5/2f7/2 g7/2f7/2 = 42 keV A. Gargano – Napoli Pisa 2005 in 50-82 shell in 82-126 shell 135Sb BEExpt =16.575 ± 0.104 MeV BECalc=16.411 ± 0.074 MeV PRL 1999 = 72 keV A. Gargano – Napoli Pisa 2005 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 135Sb 7 2 75% g7/2 (f7/2)2 +... 5 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 B(M1;5/2+ 7/2+) 2 x 10-3 Expt. Calc. a factor 90 (with free g factors) 0.29▲ 25 ▲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 BECalc=28.656 ± 0.082 MeV PRL 1999 =100 keV A. Gargano – Napoli Pisa 2005 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 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 Config. (f7/2)2 -0.654 -0.403 (f7/2)2 -0.286 -0.289 (p3/2)2 -0.404 -0.101 diagonal matrix elements for J=0+ Config. Veff (g7/2)2 -0.738 Vlow-k 0.063 Veff Vlow-k diagonal matrix elements for J=2+ Config. Veff Vlow-k (g7/2)2 -0.037 -0.016 (d5/2)2 -0.486 -0.304 -1-1 diagonal matrix elements for J=0+ Config. Veff (d3/2)2 -1-1 diagonal matrix elements for J=2+ Config. Veff -0.325 -0.184 (d3/2)2 -0.036 -0.097 (h11/2)2 -1.058 -0.417 (h11/2)2 -0.507 -0.445 (s1/2)2 -0.726 -0.869 Vlow-k Vlow-k A. Gargano – Napoli Pisa 2005 Two-body matrix elements g7/2f7/2 V3p1h V4p2h V2p Veff Vlow-k A. Gargano – Napoli Pisa 2005 Two-body matrix elements d5/2f7/2 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