Testing shell model on nuclei

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Testing shell model on nuclei across the N82
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

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Napoli-Stony Brook Collaboration
L. Coraggio A. Covello A. G. N. Itaco T.T.S. Kuo
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• Across the N82 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

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D. Radford - ENAM04
132Sn and 134Sn results from J.R. Beene ENAM04
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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
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Shell-model effective interaction
Nuclear many-body Schroedinger equation
Model-space Schroedinger equation
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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)
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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 ? the deuteron binding energy and
low-energy scattering phase-shifts are reproduced.
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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

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NN-potential CD-Bonn
126
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
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?? in 82-126 shell
134Sn
86 (?f7/2)2 81 (?f7/2)2
BEExpt 6.365 0.104 MeV PRL
1999 BECalc6.082 0.064 MeV
? 70 keV
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Sn isotopes
? Expt. ? Calc.
eeff0.75e from
B(E210 ? 8) in 134Sn
eeff0.70e from
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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)
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Proton-particle neutron-particle multiplets
? in the 50-82 shell ? in the 82-126 shell
BEExpt 12.952 0.052 MeV PRL
1999 BECalc12.849 0.058 MeV
134Sb
?d5/2?f7/2
?g7/2?f7/2
? 42 keV
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? in 50-82 shell ?? in 82-126 shell
135Sb
BEExpt 16.575 0.104 MeV PRL
1999 BECalc16.411 0.074 MeV
? 72 keV
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Sb isotopes
7/2
5/2
N
Splitting of the centroids of the ?g7/2 nd
?d5/2 SP strengths
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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)

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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 ?

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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 BECalc28.656 0.082 MeV
? 100 keV
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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(E24 ? 2) in 134Te
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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
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Two-body ?? matrix elements
V3p1h
V4p2h
?g7/2?f7/2
V2p
Veff
Vlow-k
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Two-body ?? matrix elements
?d5/2?f7/2
V3p1h
V4p2h
V2p
Veff
?
Vlow-k
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Summary

• Properties of exotic nuclei in 132Sn region
  below and above the N82 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

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