Shell-model study of exotic nuclei above
a realistic effective interaction
Exotic nuclei beyond
132Sn:
132Sn
with
where do we stand?
Shell-model calculations with realistic low-momentum
effective interactions: sketch of theoretical framework
Comparison of theory with available data and predictions
for future experiments
Summary and outlook
L. Coraggio
A. C.
Angela Gargano
N. Itaco
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Exoticity of nuclei beyond doubly magic 132Sn
124Sn (stable)
136Te
137Te
134Sb
135Sb 136Sb
134Sn
135Sn* 136Sn*
Z
50
52
52
51
51
51
50
50
50
N
74
84
85
83
84
85
84
85
86
1.48
1.61
1.63
1.63
1.65
1.67
1.68
1.70
1.72
N/Z
* Unknown
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Realistic shell-model calculations with two-body forces
Veff 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
L. Coraggio, A. Covello, A.Gargano, N.Itaco, T.T.S. Kuo, Prog. Part. Nucl. Phys. 62, 135 (2009)
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Veff calculated by a many-body perturbation
technique
In practical applications: diagrams first-, second-,
(and third-) order in the interaction
Veff should account for effects of the configurations
excluded from the model space: core polarization
effects
+…
“Bubble”
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Renormalization of the NN interaction
Traditional approach to this problem: Brueckner
G-matrix method.The G matrix is model-space
dependent as well as energy dependent
Vlow-k approach: construction of a low-momentum NN
potential Vlow-k confined within a momentum-space
cutoff k  Λ
S. Bogner,T.T.S. Kuo,L. Coraggio,A. Covello,N. Itaco, Phys. Rev C 65, 051301(R) (2002).
S. Bogner, T.T.S. Kuo, A. Schwenk, Phys. Rep. 386, 1 (2003).
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Vlow-k approach
Vlow-k preserves the physics of the original NN
interaction up to a certain cut-off momentum Λ:
the deuteron binding energy and low-energy
scattering phase-shifts of VNN are reproduced.
Vlow-k is a smooth NN potential
Low-momentum effective interactions
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1p3n
Z=54
1p1n
Z=52
N/Z
Z=51
4n
Z=50
N=82
2n
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3n
N=83
N=84
N=85
N=86
N/Z
Shell-model calculations with two-body effective interaction
derived from the CD-Bonn potential through the Vlow-k approach
132Sn
region
Λ = 2.2 fm-1
Model space & single-particle energies
Valence neutrons in the 1f7/2, 2p3/2, 0h9/2, 2p1/2,
1f5/2,0i13/2 levels of the 82-126 shell
Protons in the 0g7/2, 1d5/2, 1d3/2, 0h11/2, 2s1/2 of the
50-82 shell
Single-particle energies from the spectra of
133Sn
-
133Sb
and
L. Coraggio, A. Covello, A Gargano, N. Itaco, Phys. Rev. C 72, 057302 (2005)
L. Coraggio, A. Covello, A. Gargano, N. Itaco, Phys. Rev. C 73, 031302(R) (2006)
A.Covello, L. Coraggio, A. Gargano, N. Itaco, Prog. Part. Nucl. Phys. 59, 401 (2007)
A. Covello, L. Coraggio, A. Gargano, N. Itaco, Eur. Phys. J. ST 150, 93 (2007)
G.S. Simpson, J.C. Angelique, J. Genevey, J.A. Pinston, A. Covello, A. Gargano, U. Köster,
R. Orlandi, A. Scherillo, Phys. Rev. C 76, 041303(R) (2007).
- L. Coraggio, A. Covello, A. Gargano, N. Itaco, Phys. Rev. C 80, 061303(R) (2009).
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2n
Lowest first-excited 2+ level in
semi-magic even-even nuclei
over the whole chart of nuclei
0.726
Expt.
134Sn
Coulex (Oak Ridge)
B(E2;0+ 2+) = 0.029(4) e2b2
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Theory
Theory
B(E2;0+ 2+) = 0.033 e2b2
134Sn
(Theoretical predictions)
B(E2;42 ) = 1.64 W.u.
B(E2;64) = 0.81 W.u.
B(E2;222) = 0.34 W.u.
B(E2;224) = 0.22 W.u.
Q(2) = -1.3 efm2
µ(2) = -0.56 nm
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BE 134Sn (relative to 132Sn)
Old value (Fogelberg et al., 1999): 6.365 MeV
New expt. value 5.916 MeV
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Theory: 5.914 MeV
N/Z=1.72
Expt.
Theory
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Theory
Striking similarity of nuclear structure in the region of “exotic”
doubly magic 132Sn and in the region of stable doubly magic208Pb
136Sn Calc
212Pb Expt
212Pb Calc
E(MeV)
2
1,5
1
0,5
0
0+
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2+
4+
6+
8+
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134
51
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Sb 83
1p1n
Diagonal matrix elements of Vlow-k and contribution from two-body
second order diagrams for the πg7/2νf7/2 configuration.
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136Sb is at present the most exotic open-shell nucleus beyond 132Sn for which
information exists on excited states
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136
51
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Sb 85
1p3n
Summary and Outlook
The properties of exotic nuclei beyond 132Sn are
remarkably well described by a unique consistent
shell-model Hamiltonian derived from a realistic
free NN potential (CD-Bonn) renormalized through
the Vlow-k procedure. This outcome gives confidence
in its predictive power and may stimulate, and be
helpful to, future experiments.
At present no real evidence of shell modifications in
the 132Sn region.
It is a great challenge for RIBs to gain more
experimental information on exotic nuclei beyond
132Sn.
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Shell-model calculations
1.Model space
2.Single-particle energies
3.Two-body matrix elements
4.Construction and diagonalization of the energy matrices
which effective interaction?
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Derivation of Veff from the free NN potential
Nuclear many-body Schroedinger equation
ΗΨ i  (Η 0  Η1 )ΕiΨ i
H0  T  U
H1  VNN  U
Model-space Schroedinger equation
PH eff PΨ   P(H 0  Veff )PΨ   E PΨ  ,
  1...d ,
P
d
ψ
i 1
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i
ψi
Nucleon-nucleon potential
CD-Bonn potential
(R. Machleidt, 2001)
High-precision NN potential based upon the OBE model
π ρ ω σ1σ2
2/Ndata= 1.02
(1999 NN Database: 5990 pp and np scattering data)
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L. Coraggio
A. C.
Angela Gargano
Napoli
N. Itaco
T. T. S. Kuo
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Stony Brook