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
Hi  (T  VNN )i  Ei i
Model-space Schroedinger equation
PH eff Pi  P( H 0  Veff ) Pi  Ei Pi ,
d
where
P

i
i
defines the model space
1
and H 0  T  U with U an auxiliary SP 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
d5/2f7/2
g7/2f7/2
= 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
g7/2f7/2
V3p1h
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