Neutron-Rich Nuclei
within
a realistic shell-model approach
Angela Gargano
Napoli
L. Coraggio (Napoli)
A. Covello (Napoli)
N. Itaco
(Napoli)
T.T.S. Kuo (Stony Brook)
A. Gargano
Napoli
JAPEN-ITALY EFES Workshop
Plan of the talk
 Theoretical framework:
Renormalization of the bare NN potential by means of the Vlow-k
approach
Derivation of the model space effective interaction by means of
the
plus folded diagram method
 Outline of calculations
 Results:
neutron-rich nuclei northeast of 132Sn
and comparison with conterpart nuclei in the stable
neutron-rich Ca isotopes
208Pb
region
neutron-rich C isotopes
 Summary
A. Gargano
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Shell-model calculations
H  i H 0(i)   j  j Vij
 
H 0(i)  Ti  U c(ri )  U so(ri )si  li
defined within a reduced model space
and acting only between valence nucleons
1. Model space
2. Single-particle energies (taken from the experimental
spectra of nuclei with one-valence nucleon or treated as free
parameters)
3. Two-body matrix elements
4. Construction and diagonalization of the energy matrices
Traditional approach:
Two-body matrix elements treated as free parameters
Empirical effective interactions containing adjustable parameters
[e.g., s-d shell nuclei, B. A. Brown and B. H. Wildenthal, Ann. Rev. Nucl. Part. Sci.38, 28(1988)]
A. Gargano
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JAPEN-ITALY EFES Workshop
Challenge for nuclear physics: understand the properties of
nuclei starting from the forces among nucleons
H  T  VNN  VNNN  ...
Hψα  Eαψα
Ab-initio calculations:
nuclear properties, such as binding and excitation energies, are
calculated directly from first principles of quantum mechanics, using
an appropriate computational scheme
Green’s function Montecarlo method, no-core shell model,
coupled cluster method, UCOM
(three-nucleon interactions have been also taken into account)
A. Gargano
Napoli
JAPEN-ITALY EFES Workshop
Realistic shell-model calculations:
We start from
H  T  VNN  (T  U)  (VNN  U)  H0  H1
Hψα  Eαψα
where U is a one-body auxiliary potential introduced
to define a convenient single-paticle basis
and define the effective shell-model Hamiltonian
H  H0  Veff
through the model-space Schrödinger equation
PHef fPΨ  P(H
0
 Vef f )P Ψ  E PΨ
where the E and the corresponding  are required to be a subset of the
eigenvalues and eigenvectors of the original Hamiltonian
The P operator projects into the chosen model space
A. Gargano
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JAPEN-ITALY EFES Workshop
Derivation of Veff
● Choice of the nucleon-nucleon potential
CD-Bonn, Argonne V18 , Chiral potentials,…
all modern NN potentials fit equally well the deuteron properties
and the NN scattering data up the inelastic threshold
2/Ndata ~ 1
Note these potentials cannot be used directly in the derivation of Veff
due to their strong short-range repulsion, but a
• Renormalization procedure is needed
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JAPEN-ITALY EFES Workshop
Renormalization of the NN interaction
Traditional approach: Brueckner G-matrix method
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).
L. Coraggio et al, Prog. Part. Nucl. Phys. 62 (2009) 135
A. Gargano
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JAPEN-ITALY EFES Workshop
Vlow-k approach
 Derived from the original VNN by integrating out the high-momentum
components of the original VNN potential down to the cutoff momentum 
Vlow-k decouples high- and low-energy degrees of freedom
preserves the physics of the original NN interaction
the deuteron binding energy
scattering phase-shifts up to the cutoff momentum Λ
How to contruct Vlow-k?
Effective interaction technique based on the the Lee-Suzuki similariry transformation
(Prog. Theor Phys 64 (1980) 2091)
  X 1HX
Q  0
Vlow k  -T
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 low-momentum space
Q complementary space
X
similarity transformation
Decoupling equation solved by the iterative procedure
proposed by Andreozzi (Phys Rev. C 54 (1996) 684)
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Features of Vlow-k

real effective potential in the momentum space (indipendent from the starting
energy or from the model space, as instead the case of the G matrix defined in the
nuclear medium)
eliminates sources of non-perturbative behavior
 can be used directly in
nuclear structure calculations
gives an approximately unique representation of the NN potential for
  2 fm-1
ELab 350 MeV - Vlow-k’s extracted from various phase-shift equivalent potentials are
very similar to each other
Note
Vlow-k is developed for the two-body system
for A>2 the low-energy observables are not the same
of the original NN potential
and depend (to a certain extent) on the value of 
This may be removed complementing the two-body
Vlow-k with three- and higher-body components
A. Gargano
Napoli
JAPEN-ITALY EFES Workshop
+ folded diagram method
developed within the framework of the time-dependent perturbative
approach by Kuo and co-workers
T.T.S. Kuo and E. Osnes, Lecture Notes in Physics, vol 364 (1990)
L. Coraggio et al, Prog. Part. Nucl. Phys. 62 (2009) 135

ˆ F
Vef f  Q
i
i 1
collection of irreducible valence-linked diagrams with Vlow-k
replacing VNN in the interaction vertices
Fi
i-folded diagrams (expressed in terms of
derivatives)
Veff ,constructed for two valence particles, is defined
-in the nuclear medium
-in a subspace of the Hilbert space
 accounts perturbatively for
• configurations excluded from the chosen model space
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• excitations of the core nucleons
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Diagramatic expression of the
1-body diagrams up to 2nd
order
S-box
2-body diagrams up to 2nd order:
V
V1p1h
+
V2p
V2p2h
+
…
Construction of Veff
Calculation of
:
• inclusion of diagrams up to a finite order in the interaction
• truncation of the intermediate-state summation
Sum of the folded series by the Lee-Suzuki method [Prog. Theor. Phys. 64, 2091 (1980)]
(2)
Note Vef f  V(1)

V
ef f
ef f
H0  V(1)
ef f
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TB component of the shell-model Hamiltoniam
 single-particle energies
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Results
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JAPEN-ITALY EFES Workshop
protons
n-rich
nuclei
nuclei beyond the N=82 shell
magic nature of 132Sn?
n-rich Ca isotopes
N=34 shell closure?
n-rich C isotopes
location of the neutron drip line?
A. Gargano
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neutrons
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132Sn
•
region
132Sn
Input of our calculations
core
Valence neutron levels: 1f7/2, 2p3/2, 0h9/2, 2p1/2, 1f5/2, 0i13/2
Valence proton levels: 0g7/2, 1d5/2, 1d3/2, 0h11/2, 2s1/2
CD-Bonn + Vlow-k
: second-order calculation
Single-particle energies from the experimental spectra of
133Sb
and
133Sn
n-rich Ca isotopes
•
40Ca
core
Valence neutron levels: 0f7/2, 0f5/2, 1p3/2, 1p1/2
CD-Bonn + Vlow-k
: third-order calculation
Single-neutron energies from a fit to exp energies of
47Ca
and
49Ca
n-rich C isotopes
•
14C
core
Valence neutron levels: 0d5/2, 0d3/2, 1s1/2
N3LOW [chiral potential with a sharp momentum cutoff at 2.1 fm-1]
: third-order calculation
Theoretical single-neutron energies
A. Gargano
Napoli
JAPEN-ITALY EFES Workshop
134S
from the f7/2p1/2 configuration
n
their location below the 8+
due to the new position of the
p1/2 level measured @ ORNL
p1/2=1.36 MeV
(old value: 1.66 MeV)
[Nature 465 (2010)]
0.726
Expt.
134Sn
Theory
Coulex (Oak Ridge)
B(E2;0+ 2+) = 0.029(4) e2b2
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Theory
B(E2;0+ 2+) = 0.033 e2b2
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134Sn

132Sn
+ 2 (f7/2)2 multiplet
134Sn Expt
134Sn Calc
1,5
1
0,5
0
0+
2+
J
4+
6+
210Pb Expt
210Pb Calc
2
210Pb
 208Pb + 2
(g9/2)2 multiplet
E(MeV)
E(MeV)
2
1,5
1
0,5
0
0+
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2+
J 4+
6+
8+
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136Sn
132Sn
212Pb
+ 4
136Sn Calc
212Pb Calc
208Pb
+ 4
212Pb Expt
E(MeV)
2
1,5
1
0,5
0
0+
2+
4+
6+
8+
J
A. Gargano
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JAPEN-ITALY EFES Workshop
N/Z
A
BE Calc
relative to
1.70
1.72
134Sn
135Sn
136Sn
137Sn
8.30
11.86
14.18
8.23
8.20
8.15
5.91
1.74
132Sn
BE Expt
relative to
1.68
5.92*
132Sn
BE/A Calc
8.27
124Sn(stable)
with N/Z=1.48
BE/A=8.46
* M. Dworschak et al. Phys. Rev. Lett. 100, (2008) 072501
Old value (Fogelberg et al., 1999): 6.365 MeV
neutron shell gap at N= 82 restored
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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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JAPEN-ITALY EFES Workshop
136Sb
Expt
136Sb
Theory
is at present the most exotic open-shell nucleus beyond
information exists on excited states
A. Gargano
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132Sn
for which
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50Ca
Expt
52Ca
Theory
Expt
Theory
J.J. Valiente Dobón et al. PRL 102, 242502 (2009)
L. Coraggio et al. Phys. Rev. C 80, (2009) 044311
A. Gargano
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JAPEN-ITALY EFES Workshop
Ca isotopes - Ground-state energy per valence neutron
A
A. Gargano
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JAPEN-ITALY EFES Workshop
Ca isotopes
Excitation energies of the J =21+ states
*
no shell gap at N=34
A
A. Gargano
Napoli
* M. Honma et al. EPJ A 25, 499 (2005)
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ESPE(j)  ε   n V ;
j j j jj
(2 J1) jj V jj J
V 
jj J
(2 J1)
Effective single particle energies of the f5/2 and p1/2 levels
(relative to the p3/2 level )
0 f 5/ 2
1p1/ 2
N  34
1p3/ 2
1f 7/ 2
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JAPEN-ITALY EFES Workshop
C isotopes from 16C to 24C – ground-state energy
(relative to
•
•
22C
14C
)
is the last bound isotope
-427 keV* K. Tanaka et al PRL 104, 062701 (2010)
S2n(evaluation)=420 keV S2n(calc)=601 keV
•
•
•
•
*to reproduce the exp g.s. energy of 15C relative to 14C
Egs(calc)=-0.79 ; Egs(exp)= -1.22 MeV
A. Gargano
Napoli
L. Coraggio et al. PR C 81, 064303 (2010)
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C isotopes
no subshell closure at N=14
g.s. in 20C:
14% of (d5/2)6 configuration
ESPE (MeV)
Excitation energies of the J =21+ states
N
A. Gargano
Napoli
JAPEN-ITALY EFES Workshop
Summary
Reliability of “realistic shell-model calculations”
for light  heavy nuclei
This outcome gives confidence in its predictive power, and may
stimulate and be helpful to future experiments.
Three-body forces seem to contribute mainly to the absolute
energy of the single-particle. Role of three-body forces needs futher
investigations
It is of key importance to gain more experimental
information
A. Gargano
Napoli
JAPEN-ITALY EFES Workshop