A. Gargano Eurisol User Group Wokshop – Firenze 2008 Structure of heavy neutron-rich nuclei Angela Gargano Napoli Napoli alistic shell-model calculations: where do we stand? alistic effective interaction: Renormalization of the bare NN potential - the V low-k approach Many body theory to construct V eff – folding expansion ealistic shell model and 132 Sn neighbors: esults for nuclei beyond the N=82 shell closure listic shell model and nuclei with several valence nucleons: ults for the N=82 isotonic chain Summary
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A. Gargano Eurisol User Group Wokshop – Firenze 2008 Structure of heavy neutron-rich nuclei Angela Gargano Napoli Realistic shell-model calculations: where.
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A. Gargano Eurisol User Group Wokshop – Firenze 2008
Structure of heavy neutron-rich nuclei
Angela Gargano
Napoli
Napoli
Realistic shell-model calculations: where do we stand?
Realistic effective interaction:• Renormalization of the bare NN potential - the Vlow-k approach;• Many body theory to construct Veff – folding expansion
Realistic shell model and 132Sn neighbors:Results for nuclei beyond the N=82 shell closure
Realistic shell model and nuclei with several valence nucleons:Results for the N=82 isotonic chain
Summary
A. Gargano Eurisol User Group Wokshop – Firenze 2008
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3. Two-body matrix elements
1. Model space 2. Single-particle energies
4. Construction and diagonalization of the energy matrices
Shell-model calculations
Shell-model TB effective interaction:•TBME based on “simple potentials” •TBME treated as parameters Cohen-Kurath 1965 - p shell nuclei A. Brown et al. 2006 - sd shell nuclei•Realistic TBME derived from the bare potential among nucleons Kuo-Brown 1966•Semi-realistic TBME realistic with some adjustements
Modern codes:•Oxbash - widely distributed and used A. Brown …•Oslo - m-scheme; dimension ~106 T. England•Antoine - m-scheme; dimension ~109 E. Caurier…•Nathan - coupled scheme; ~106 •Redstick - for 2-and 3- body ME P. Navratil, W.E. Ormand
A. Gargano Eurisol User Group Wokshop – Firenze 2008
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Realistic shell-model calculations
NNNNN VVTH
Veff from the bare potential
A. Gargano Eurisol User Group Wokshop – Firenze 2008
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The role of the NNN potential haa been evidenced by investigation on light nuclei
• effective interaction derived from the NN potential without any adjustement (to assess the quality and reliability of realistic effective interactions and the possible need for improvement)
• effective interaction with some modified ME (for instance monopole changes; Caurier et al. 2005)
Tjon line
Inclusion of three–body forces in the “shell model” approach not yet attempted
NCSM Navratil 2007
Nogga 2004
Model space defined by the operator
1 111
being complement the id
ii
d
i PQP
A. Gargano Eurisol User Group Wokshop – Firenze 2008
(1999 NN Database: 5990 pp and np scattering data)
π ρ ω σ1σ2 based upon the OBE model
A. Gargano Eurisol User Group Wokshop – Firenze 2008
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NN potential derived from chiral effective field theoryS. Weinberg (1990) “Nuclear forces from chiral lagrangians”D. R. Entem and R. Machleidt (2001- 2003): Idaho potential, N3LO potential E. Epelbaum, W. Glöckle, and U.-G. Meissner, 2005: N3LO potential
N3LO potential (Entem & Machleidt):
• Effective chiral πN Lagrangian • One and two-pion exchange contributions. TPE contributions up to fourth order of chiral perturbation theory• Short-range force parametrized in terms of 24 contact terms• Total number of parameters: 29• PD = 4.51% - χ2/Ndata
Important advantage of the chiral perturbation theory: it generates NNN forces (starting from 3rd order)
A. Gargano Eurisol User Group Wokshop – Firenze 2008
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Matrix elements of VNN
in the 1S0 channel
NN potentials are not completely constrained by low-energy NN data
A. Gargano Eurisol User Group Wokshop – Firenze 2008
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Renormalization of the NN potential
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
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).
)(1
)(
GQTQQ
QVVG NNNN
= + + ...~~~G VNN
VNN
VNN
Infinite sum of ladder diagrams
A. Gargano Eurisol User Group Wokshop – Firenze 2008
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● Derived from the original VNN by integrating out the high- momentum components of the original VNN potential
● Vlow-k preserves the physics of the original NN interaction up to the cutoff momentum Λ: the deuteron binding energy scattering phase-shifts
● Iterative method
Λk
Vlow-k: Low-momentum potential confined within a momentum-space cutoff
S. K. Bogner, T.T.S. Kuo, L. Coraggio, Nucl. Phys. A684, 432c (2001).S.K. Bogner, T.T.S. Kuo, L. Coraggio, A. Covello, N. Itaco, Phys. Rev. C 65, 051301(R) (2002).
Vlow-k approach
A. Gargano Eurisol User Group Wokshop – Firenze 2008
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Matrix elements of Vlow-k with Λ=2.1 fm-1
in the 1S0 channel
LabLab E
ME012.0
2 22
)(
)( 1
MeVE
fm
Lab
1.5 1.8 2.0 2.05 2.2 2.5
187 269 332 350 401 518-1fm 1.2
k
Vlow-k is free from high momentum modes
•Vlow-k is mooth potential suitable to be used directly as input for derive the effective interactionin
• Vlow-k gives an approximately unique representation of the NN potential
A. Gargano Eurisol User Group Wokshop – Firenze 2008
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G matrix vs Vlow-k
G matrix Vlow -- k
● Energy dependent ● NO energy dependent
● Model space dependent ● NO model space dependent
● No direct connection to the
original VNN potential
● It is a real effective potential -
in the k-space it reproduces all
the two body problem data
A. Gargano Eurisol User Group Wokshop – Firenze 2008
● Calculation of the : collection of irreducible and valenced linked diagrams at any order in V 2-body 2nd order diagrams:
V V2p V1p1h V2p2h
auxliary 1b potential
1 with :space Model111
id
ii
d
i PQP
● Sum of the folded-diagram series :(by Kuo-Krengiglowa or Lee-Suzuki iterative technique)
)ˆ(ˆ1
effQFQ
iiV
boxˆ Q
A. Gargano Eurisol User Group Wokshop – Firenze 2008
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Results
132Sn neighbors beyond N=82
N=82 isotonic chain
A. Gargano Eurisol User Group Wokshop – Firenze 2008
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50
82
.
.
.
132Sn
i13/2f5/2p1/2h9/2p3/2f7/2
h11/2s1/2d3/2d5/2g7/2
space
space
126
.
.
.
133Sb
133Sn
VNN : CD-Bonn
Vlow-k with Λ= 2.2 fm-1
Veff : ● 2nd order calculation and Lee-Suzuki method ● intermediate states composed of all possible hole states and particle states restricted to 5 shells above the Fermi surface (which guarantees the stability of results when increasing the number of intermediate states)
Coulomb force for protons added to Vlow-k
MeV 88.7harmonic-oscillator basis
A. Gargano Eurisol User Group Wokshop – Firenze 2008
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Across the N=82 shell gapAcross the N=82 shell gap
Anomalous behavior of the first 2Anomalous behavior of the first 2+ + state in even Sn isotopesstate in even Sn isotopes
"" in even Te isotopes in even Te isotopes
Anomalous behavior of the B(E2; 0Anomalous behavior of the B(E2; 0++22++) value in even Sn ) value in even Sn isotopesisotopes
"" in even Te in even Te isotopesisotopes
Anomalous behavior of the first 5/2Anomalous behavior of the first 5/2+ + in Sb isotopes in Sb isotopes
Anomalously low position of the 1Anomalously low position of the 1-- state in state in 134134SbSb
Role of three-body forces both genuine and effective
A. Gargano Eurisol User Group Wokshop – Firenze 2008
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J=2+
A. Gargano Eurisol User Group Wokshop – Firenze 2008
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J=4+
A. Gargano Eurisol User Group Wokshop – Firenze 2008
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J=6+
A. Gargano Eurisol User Group Wokshop – Firenze 2008
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Ground-state binding energyper valence proton
Zval
The Expt and Theor behaviors of the g.s. binding energy per valenceproton diverge contribution of many-body effective forces
A. Gargano Eurisol User Group Wokshop – Firenze 2008
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Summary
The Vlow-k approach to the renormalization of the bare NN potential is a valuable tool for nuclear structure calculations. This potential may be used directly in shell-model calculations without the need of first calculating the Brueckner G-matrix.
Effective interactions derived from modern NN potentials are able to describe with quantitative accuracy the spectroscopic properties of exotic nuclei near closed shells. This gives confidence in their predictive power in these regions.
At present no real evidence of shell modifications near 132Sn. It is of key importance to gain more experimental information.
Theoretical open problems: ▪ single-particle energies from the theory; ▪ role of genuine and effective three-body forces for heavy nuclei.