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Structure of heavy neutron-rich nuclei Angela
Gargano 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 N82
shell closure
Realistic shell model and nuclei with several
valence nucleons Results for the N82 isotonic
chain
Summary
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Shell-model calculations
1. Model space

2. Single-particle energies
4. Construction and diagonalization of the
energy matrices

• 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

• 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

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Realistic shell-model calculations
Veff from the bare potential
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The role of the NNN potential haa been evidenced
by investigation on light nuclei
NCSM Navratil 2007
Tjon line
Nogga 2004
Inclusion of threebody forces in the shell
model approach not yet attempted

• 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)

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Realistic shell-model calculations
Veff from the bare potential
Nuclear many-body Schroedinger equation
Model-space Schroedinger equation
Model space defined by the operator
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Realistic shell-model calculations
Basic ingredients
Nucleon-nucleon potential
Many-body theory to derive the effective
interaction
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Modern (phase-shift equivalent) NN potentials
Nijmegen I - (PD 5.66) - 41 parameters -
?2/Ndata 1.03 Nijmegen II - (PD 5.64) - 47
parameters - ?2/Ndata 1.03 Argonne V18 - (PD
5.76) - 40 parameters - ?2/Ndata 1.09 CD Bonn
- (PD 4.85) - 43 parameters - ?2/Ndata
1.02
based upon the OBE model
(1999 NN Database 5990 pp and np scattering data)
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Chiral potentials

• NN potential derived from chiral effective field
  theory
• S. 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 pN 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 - ?2/Ndata 1.10 (np database
  below 290 MeV)
• ?2/Ndata 1.50 (pp database below 290 MeV)

• Important advantage of the chiral perturbation
  theory
• it generates NNN forces (starting from 3rd order)

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Matrix elements of VNN in the 1S0 channel
NN potentials are not completely constrained by
low-energy NN data
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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
Infinite sum of ladder diagrams
VNN
G
VNN



...
VNN
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Vlow-k Low-momentum potential confined within a
momentum-space cutoff
? 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 ? the deuteron
binding energy scattering
phase-shifts ? Iterative method
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).
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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

Matrix elements of Vlow-k with ?2.1 fm-1 in the
1S0 channel
k
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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
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Realistic effective interaction folded-diagram
expansion
auxliary 1b potential
? Calculation of the
collection of irreducible and valenced linked

diagrams at any order in V
2-body 2nd order diagrams V
V2p V1p1h V2p2h
? Sum of the folded-diagram series (by
Kuo-Krengiglowa or Lee-Suzuki iterative technique)
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Results
132Sn neighbors beyond N82
N82 isotonic chain
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126
harmonic-oscillator basis
i13/2 f5/2 p1/2 h9/2 p3/2 f7/2
. . .
133Sn
82
h11/2 s1/2 d3/2 d5/2 g7/2
? space
133Sb
. . .
50
132Sn
? space
VNN CD-Bonn Vlow-k with ? 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
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Z52 134Te 135Te 136Te 137Te 138Te 139Te
Z51 133Sb 134Sb 135Sb 136Sb
Z50 132Sn 133Sn 134Sn
N82 N83 N84 N85 N86 N87
N/Z1.67
N/Z1.68

• Across the N82 shell gap
• Anomalous behavior of the first 2 state in
  even Sn isotopes
• "
  in even Te isotopes
• Anomalous behavior of the B(E2 0?2) value in
  even Sn isotopes
• "
  in even Te isotopes
• Anomalous behavior of the first 5/2 in Sb
  isotopes
• Anomalously low position of the 1- state in 134Sb

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2 in Te isotopes
2 in Sn isotopes
odd Sb isotopes
Downshift of the d5/2 proton level relative to
the g7/2 one? Onset of a modification in
the shell structure?
7/2
5/2
N
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134Te
Expt
Cd-Bonn
NijmII
Argonne V18
s(keV)115
s(keV)143
s(keV)128
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134Sn
86 (?f7/2)2 81 (?f7/2)2
Expt
Calc
? 70 keV
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134Sb

• Calc.
• ? Expt.

?d5/2?f7/2
?g7/2?f7/2
? 42 keV
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134Sb

• Calc.
• ? Expt.

?g7/2?f7/2
Large space
?g7/2?f7/2
Small space intermediate states composed of
particle and hole states restricted to two major
shells above and below the FS
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Diagonal matrix elements of interaction for
the ?g7/?f7/2 configuration
Large space
J
V1p1h
V2p2h
V2p
Veff
Vlow-k
Small space
J
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Yrast states with Jp from 0- to 7- dominated by
the ?g7/2?(f7/2 )3
136Sb

• Calc.
• ? Expt.

G.S. Simpson et al. 2007 , ILL Grenoble
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The ?g7/2?f7/2 multiplet in
compared to that in
136Sb
134Sb
134Sb
136Sb
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135Sb
0.282
anomalously low position
Calc
Expt
? 72 keV
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135Sb wave functions
J0
75 ?g7/2 (?f7/2)2 ...
45 ?d5/2 (?f7/2)2 23 ?g7/2 (?f7/2)2 ...
J2
J0

• J?0 and J?2 ME in the (?f7/2)2
  configuration differ only by 300 keV
• J? 1- ME
• (?g7/2 ?f7/2) ? -600 keV
• (?d5/2 ?f7/2) ? -500 keV

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136Te
Expt
Calc
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Comparison of the experimental and
theoretical B(E2) e2fm4 and B(M1) µ2N
E2 eeff(?) 0.70e eeff(?) 1.55e
M1 effective operator 2nd order core
no meson-exchange
corrections
new measurements yield values which are higher ?
50 HRIBF-ORL ? 20
ISOLDE-CERN
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N82 isotones 134Te 136Xe 138Ba 140Cs
142Nd 144Sm 146Gd 148Dy 150Er 152Yb 154Hf
Zval from 4 to 22
Zval2 test of Veff
Role of three-body forces both genuine and
effective
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J2
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J4
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J6
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Ground-state binding energy per valence proton
Zval
The Expt and Theor behaviors of the g.s. binding
energy per valence proton diverge ? contribution
of many-body effective forces
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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.
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Cutoff momentum ?
Choice of ?

• Low-momentum NN interaction in few-nucleon
  systems
• - S. Fujii, E. Epelbaum, H. Kamada, R. Okamoto,
  K. Suzuki, W. Glöckle,
• Phys. Rev. C 70, 024003 (2004).
• A. Nogga, S. K. Bogner, A. Schwenk, Phys. Rev. C
  70, 061002(R) (2004).
• L. Coraggio, A. Covello, A. Gargano, N. Itaco,
  T.T.S. Kuo, Phys. Rev. C 73, 014304 (2006).

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? 2.1 fm-1
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Correlation plots fra gli e.m. di NijmII e
Cd-Bonn nella base di HO e le corrispondenti
matrici GT per la shell sd
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and the multiplet ?h9/2?g9/2 in 210Bi?
L
?h9/2?g9/2
S
?h9/2?g9/2
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structure of states
unkown
Mixed-symmetry state (antisimmetric with respect
to interchanges between proton and neutron pairs
)
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? Second step - sum of the folded - diagram
series (by Kuo-Krengiglowa or Lee-Suzuki
iterative technique)

H1eff contains both 1- and 2-body
contributions subtraction procedure to remove
from H1eff the 1-body terms and sp energies
from experiment

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