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. 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

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

5. 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

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. 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

10. 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

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. 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

13. 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

14. 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

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

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

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

18. 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

19. 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

20. 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

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

22. 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

23. 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

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 of core polarization effects
More experimental information is needed
A. Gargano – Napoli
Pisa 2005