Shell model calculations for exotic nuclei with realistic potentials

Shell model calculations for exotic nuclei with
realistic potentials: reliability and
predictiveness
Luigi Coraggio
Istituto Nazionale di Fisica Nucleare - Sezione di Napoli
From Rare Isotopes to Neutron Stars
September 17th, 2015
ECT∗ , Trento
Luigi Coraggio
ECT∗ -APCTP Joint Workshop
A. Covello (Università “Federico II” di Napoli)
A. Gargano (INFN)
N. Itaco (Università “Federico II” di Napoli and INFN)
T. T. S. Kuo (SUNY at Stony Brook, USA)
L. C. (INFN)
Luigi Coraggio
ECT∗ -APCTP Joint Workshop
Part I
The theoretical framework
Luigi Coraggio
ECT∗ -APCTP Joint Workshop
Introductory remark
What is a realistic effective shell-model hamiltonian ?
Luigi Coraggio
ECT∗ -APCTP Joint Workshop
19
An example:
F
19
F
9 protons & 10 neutrons
interacting
spherically symmetric
mean field (e.g. harmonic
oscillator)
protons
neutrons
1 valence proton & 2
valence neutrons
interacting in a truncated
model space
The degrees of freedom of the core nucleons and the
excitations of the valence ones above the model space are not
considered explicitly.
Luigi Coraggio
ECT∗ -APCTP Joint Workshop
19
An example:
F
19
F
9 protons & 10 neutrons
interacting
d3/2
d3/2
d5/2
d5/2
s1/2
s1/2
p1/2
p1/2
p3/2
p3/2
s1/2
s1/2
protons
neutrons
spherically symmetric
mean field (e.g. harmonic
oscillator)
1 valence proton & 2
valence neutrons
interacting in a truncated
model space
The degrees of freedom of the core nucleons and the
excitations of the valence ones above the model space are not
considered explicitly.
Luigi Coraggio
ECT∗ -APCTP Joint Workshop
19
An example:
F
19
F
9 protons & 10 neutrons
interacting
d3/2
d3/2
d5/2
d5/2
s1/2
s1/2
model space
p1/2
p1/2
p3/2
p3/2
16
O
s1/2
s1/2
protons
neutrons
spherically symmetric
mean field (e.g. harmonic
oscillator)
1 valence proton & 2
valence neutrons
interacting in a truncated
model space
The degrees of freedom of the core nucleons and the
excitations of the valence ones above the model space are not
considered explicitly.
Luigi Coraggio
ECT∗ -APCTP Joint Workshop
Effective shell-model hamiltonian
The shell-model hamiltonian has to take into account in an
effective way all the degrees of freedom not explicitly
considered
Two alternative approaches
phenomenological
microscopic
VNN (+VNNN ) ⇒ many-body theory ⇒ Heff
Definition
The eigenvalues of Heff belong to the set of eigenvalues of the
full nuclear hamiltonian
Luigi Coraggio
ECT∗ -APCTP Joint Workshop
Workflow for a realistic shell-model calculation
1
Choose a realistic NN potential (NNN)
2
Pin down the model space better tailored to study the
system under investigation
3
Derive the effective shell-model hamiltonian by way of the
many-body theory
4
Calculate the physical observables (energies, e.m.
transition probabilities, ...)
Luigi Coraggio
ECT∗ -APCTP Joint Workshop
Realistic nucleon-nucleon potential: VNN
Strong short-range
repulsion
Several realistic potentials χ2 /datum ' 1:
CD-Bonn, Argonne V18, Nijmegen, ...
How to handle the short-range repulsion ?
Brueckner G matrix
EFT inspired approaches
Vlow−k , SRG
chiral potentials
Luigi Coraggio
ECT∗ -APCTP Joint Workshop
Realistic nucleon-nucleon potential: VNN
Strong short-range
repulsion
Several realistic potentials χ2 /datum ' 1:
CD-Bonn, Argonne V18, Nijmegen, ...
How to handle the short-range repulsion ?
Brueckner G matrix
EFT inspired approaches
=
Vlow−k , SRG
chiral potentials
Luigi Coraggio
ECT∗ -APCTP Joint Workshop
+
+
+
Realistic nucleon-nucleon potential: VNN
Strong short-range
repulsion
Several realistic potentials χ2 /datum ' 1:
CD-Bonn, Argonne V18, Nijmegen, ...
How to handle the short-range repulsion ?
Brueckner G matrix
EFT inspired approaches
Vlow−k , SRG
chiral potentials
Luigi Coraggio
ECT∗ -APCTP Joint Workshop
Realistic nucleon-nucleon potential: VNN
Strong short-range
repulsion
Several realistic potentials χ2 /datum ' 1:
CD-Bonn, Argonne V18, Nijmegen, ...
k’
How to handle the short-range repulsion ?
k
Brueckner G matrix
EFT inspired approaches
2
1
Vlow−k , SRG
chiral potentials
Luigi Coraggio
0
ECT∗ -APCTP Joint Workshop
Realistic nucleon-nucleon potential: VNN
Strong short-range
repulsion
Several realistic potentials χ2 /datum ' 1:
CD-Bonn, Argonne V18, Nijmegen, ...
k’
How to handle the short-range repulsion ?
k
Brueckner G matrix
EFT inspired approaches
Vlow−k , SRG
chiral potentials
Luigi Coraggio
!0
ECT∗ -APCTP Joint Workshop
!1
!2
Realistic nucleon-nucleon potential: VNN
Strong short-range
repulsion
Several realistic potentials χ2 /datum ' 1:
CD-Bonn, Argonne V18, Nijmegen, ...
How to handle the short-range repulsion ?
Brueckner G matrix
EFT inspired approaches
Vlow−k , SRG
chiral potentials
Luigi Coraggio
ECT∗ -APCTP Joint Workshop
The shell-model effective hamiltonian
A-nucleon system Schrödinger equation
H|Ψν i = Eν |Ψν i
with
A
X
X
H = H0 + H1 =
(Ti + Ui ) +
(VijNN − Ui )
i=1
i<j
Model space
|Φi i =
[a1† a2†
...
an† ]i |ci
⇒P=
d
X
|Φi ihΦi |
i=1
Model-space eigenvalue problem
Heff P|Ψα i = Eα P|Ψα i
Luigi Coraggio
ECT∗ -APCTP Joint Workshop
The shell-model effective hamiltonian


−1 HX
H
=
X
 PHP
PHQ 










 0
QHQ 
QHP = 0

 PHP





 QHP
⇒
Heff = PHP
0
Suzuki & Lee ⇒ X = eω with ω =
QωP
0
0

PHQ 





QHQ 
1
QH1 P−
− QHQ
1
−PH1 Q
ωH1eff (ω)
− QHQ
H1eff (ω) = PH1 P + PH1 Q
Luigi Coraggio
ECT∗ -APCTP Joint Workshop
The shell-model effective hamiltonian
Folded-diagram expansion
Q̂-box vertex function
Q̂() = PH1 P + PH1 Q
1
QH1 P
− QHQ
⇒ Recursive equation for Heff ⇒ iterative techniques
(Krenciglowa-Kuo, Lee-Suzuki, ...)
Heff = Q̂ − Q̂
0
Z
Q̂ + Q̂
0
Z
Z
Q̂
Luigi Coraggio
Q̂ − Q̂
0
Z
Z
Z
Q̂
Q̂
ECT∗ -APCTP Joint Workshop
Q̂ · · · ,
The perturbative approach to the shell-model H eff
Q̂() = PH1 P + PH1 Q
1
QH1 P
− QHQ
The Q̂-box can be calculated perturbatively
∞
X
1
(QH1 Q)n
=
− QHQ
( − QH0 Q)n+1
n=0
The diagrammatic expansion of the Q̂-box
a
b
a
b
p1
a
p2
b
p
a
a
b
p
h
h1
h
c
d
c
d
1
c
2
a
b
d
c
c
3
a
b
h2
c
4
a
5
b
a
b
p
c
d
6
c
h
h
p
d
7
Luigi Coraggio
c
d
8
c
d
9
ECT∗ -APCTP Joint Workshop
The perturbative approach to the shell-model H eff
H eff for systems with one and two valence nucleons
Q̂-box ⇒ Goldstone diagrams up to third order in VNN (up
to 2p-2h core excitations )
Padè approximant [2|1] of the Q̂-box
0
1
2
2
3
[2|1] = VQbox
+ VQbox
+ VQbox
(1 − (VQbox
)−1 VQbox
)−1 ,
Luigi Coraggio
ECT∗ -APCTP Joint Workshop
Test case: p-shell nuclei
VNN ⇒ chiral N3 LO potential by Entem & Machleidt
(smooth cutoff ' 2.5 fm−1 )
Heff for two valence nucleons outside 4 He
Single-particle energies and residual two-body interaction
are derived from the theory. No empirical input
First, some convergence checks !
L.C., A. Covello, A. Gargano, N. Itaco, and T. T. S. Kuo, Ann.
Phys. 327 , 2125-2151 (2012)
Luigi Coraggio
ECT∗ -APCTP Joint Workshop
Convergence checks
The intermediate-state space Q
Q-space is truncated: intermediate states whose unperturbed
excitation energy is greater than a fixed value Emax are
disregarded
|0 − QH0 Q| ≤ Emax = Nmax ~ω
6 Li
yrast states
21+
31+
01+
11+
12
10
8
E [MeV]
6
4
results stable for Nmax ≥ 20
2
0
–2
–4
–6
2
4
6
8
10
12
14
16
18
20
22
Nmax
Luigi Coraggio
ECT∗ -APCTP Joint Workshop
Convergence checks
Order-by-order convergence
eff , H eff , H eff and H eff
Compare results from H1st
Padè
2nd
3rd
24
21+
31+
01+
11+
20
6.0
4.0
16
2.0
E [MeV]
12
0.0
8
–2.0
4
–4.0
0
–6.0
–4
–8
–8.0
1st
2nd
3rd
Pade‘
Luigi Coraggio
2nd
3rd
Pade‘
ECT∗ -APCTP Joint Workshop
Convergence checks
Dependence on ~ω
Auxiliary potential U ⇒ harmonic oscillator potential
HF-insertions
7.0
6.0
21+
31+
01+
11+
(b)
(c)
j
j
j
h
5.0
j
E [MeV]
4.0
j
j
(a)
(b)
3.0
2.0
zero in a self-consistent basis
1.0
neglected in most applications
0.0
18
19
h– ! [MeV]
20
18
19
disregard of HF-insertions
introduces relevant dependence
on ~ω
20
h– ! [MeV]
Luigi Coraggio
ECT∗ -APCTP Joint Workshop
Benchmark calculation
Approximations are under control ... and what about the
accuracy of the results ?
Compare the results with the “exact” ones
ab initio no-core shell model (NCSM)
P. Navrátil, E. Caurier, Phys. Rev. C 69, 014311 (2004)
P. Navrátil et al., Phys. Rev. Lett. 99, 042501 (2007)
Luigi Coraggio
ECT∗ -APCTP Joint Workshop
Benchmark calculation
To compare our results with NCSM we need to start from a
translationally invariant Hamiltonian PA p 2
PA
NN − pi ·pj =
i
Hint = 1 − A1
+
V
i=1 2m
i<j=1
ij
mA
pi2
A
i=1 ( 2m
hP
=
i hP
A
NN − U −
+ Ui ) +
i
i<j=1 (Vij
pi2
2mA
−
i
pi ·pj
mA )
8.0
21+
31+
01+
11+
6.0
4.0
E [MeV]
2.0
(a) not translationally invariant
Hamiltonian
(b) purely intrinsic hamiltonian
0.0
–2.0
–4.0
–6.0
–8.0
(a)
(b)
NCSM
Luigi Coraggio
ECT∗ -APCTP Joint Workshop
Benchmark calculation
Remark
H eff derived for 2 valence nucleon systems ⇒ 3-, 4-, .. n-body
components are neglected
0
–8
–16
Egs [MeV]
–24
ground-state energies for
N = Z nuclei
–32
–40
discrepancy grows with the
number of valence
nucleons
–48
–56
–64
Expt.
3
N LO realistic shell model
3
N LO NCSM
–72
6
8
10
12
A
Luigi Coraggio
ECT∗ -APCTP Joint Workshop
Benchmark calculation
10 B
relative spectrum
10
B
7
+
+
2
+
4
+
2
2
E (MeV)
6
5
4+
2+
3
4
2
3
+
2
+
3
2
+
+
discrepancy ≤ 1 MeV
+
2
+
+
2
0+
1
1
3
0
1
1
+
+
0
+
+
1
3
RSM
NCSM
minor role of many-body
correlations
+
+
Expt
Luigi Coraggio
ECT∗ -APCTP Joint Workshop
Part II
Reliability
Luigi Coraggio
ECT∗ -APCTP Joint Workshop
Large-scale realistic shell-model calculations
Neutron-rich isotopic chains
Approaching neutron drip line:
Shell-model study of the onset of collectivity at N = 40
L.C., A. Covello, A. Gargano, and N. Itaco, Phys. Rev. C 89, 024319 (2014)
Proton-rich isotopic chains
Approaching proton drip line:
Enhanced quadrupole collectivity of neutron-deficient tin isotopes
L.C., A. Covello, A. Gargano, N. Itaco, and T. T. S. Kuo, Phys. Rev. C 91, 041301
(2015)
Luigi Coraggio
ECT∗ -APCTP Joint Workshop
Collectivity at N = 40
E(2+) (MeV)
SM
2.5 EXP
2
B(E2;2+ -> 0+) (e2 fm4)
3
(a)
N=40
1.5
1
0.5
0
20 22 24 26 28
Z
500
(b)
400
300
200
100
0
SM
EXP
20 22 24 26 28
Z
⇒ shell-model study of neutron-rich isotopic chains outside
48 Ca
⇒ Collective behavior framed within the quasi-SU(3)
approximate symmetry
⇒ Two model spaces with 48 Ca inert core, including or not the
neutron 1d5/2 orbital
Luigi Coraggio
ECT∗ -APCTP Joint Workshop
The collectivity at N = 40
RAPID COMMUNICATIONS
PHYSICAL REVIEW C 81, 051304(R) (2010)
Collectivity at N = 40 in neutron-rich 64 Cr
A. Gade,1,2 R. V. F. Janssens,3 T. Baugher,1,2 D. Bazin,1 B. A. Brown,1,2 M. P. Carpenter,3 C. J. Chiara,3,4 A. N. Deacon,5
S. J. Freeman,5 G. F. Grinyer,1 C. R. Hoffman,3 B. P. Kay,3 F. G. Kondev,6 T. Lauritsen,3 S. McDaniel,1,2 K. Meierbachtol,1,7
A. Ratkiewicz,1,2 S. R. Stroberg,1,2 K. A. Walsh,1,2 D. Weisshaar,1 R. Winkler,1 and S. Zhu3
PHYSICAL REVIEW C 88, 024326 (2013)
1
National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, Michigan 48824, USA
2
Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 48824, USA
3
Physics Division, Argonne National Laboratory, Argonne, Illinois 60439, USA
4
Department of Chemistry and Biochemistry, University of Maryland, College Park, Maryland 20742, USA
5
School of Physics and Astronomy, Schuster Laboratory, University of Manchester, Manchester M13 9PL, United Kingdom
6
Nuclear Engineering Division, Argonne National Laboratory, Argonne, Illinois 60439, USA
7
Department of Chemistry, Michigan State University, East Lansing, Michigan 48824, USA
(Received 19 March 2010; published 28 May 2010)
9
Be-induced inelastic scattering of 62,64,66 Fe and 60,62,64 Cr was performed at intermediate beam energies.
Excited states in 64 Cr were measured for the first time. Energies and population patterns of excited states in
these neutron-rich Fe and Cr nuclei are compared and interpreted in the framework of large-scale shell-model
calculations in different model spaces. Evidence for increased collectivity and for distinct structural changes
between the neighboring Fe and Cr isotopic chains near N = 40 is presented.
DOI: 10.1103/PhysRevC.81.051304
COMMUNICATIONS
PACS number(s):RAPID
21.10.Re,
23.20.Lv, 25.60.−t, 27.50.+e
PHYSICAL REVIEW C 81, 061301(R) (2010)
Onset of collectivity in neutron-rich Fe isotopes: Toward a new island of inversion?
The shell structure of atomic nuclei is one of the central
For the N = 40 isotones, the most extreme proton-neutron
64
1,2,3 blocks 1for a comprehensive
2
2
4
5
6
building
description
of these
asymmetry
reachable
today
occurs in Cr, a nucleus where
J. Ljungvall,
A. Görgen, A. Obertelli,1 W. Korten,1 E. Clément,
G. de France,
A. Bürger,
J.-P. Delaroche,
A. Dewald,
6
9
10
A.strongly
Gadea,7 L.correlated
Gaudefroy,5 many-body
M. Girod,5 M. Hackstein,
J.systems.
Libert,8 D. Shell
Mengoni,
F. Nowacki,
T.onset
Pissulla,
Poves,11
the
of6 A.collectivity
has been conjectured to be most
quantum
strucF. Recchia,12 M. Rejmund,2 W. Rother,6 E. Sahin,12 C. Schmitt,2 A. Shrivastava,2 K. Sieja,10 J. J. Valiente-Dobón,12 +
pronounced and the 21 energy calculated to be the lowest
ture is fairly well understood K.for
stable nuclei. However,
O. Zell,6 and M. Zielińska13
1
significant modifications
have
been
encountered
for shortthe region [8,12]. In this rapid communication, we report
CEA Saclay, IRFU,
Service
de Physique
Nucléaire, F-91191
Gif-sur-Yvette,inFrance
+
2
GANIL, CEA/DSM-CNRS/IN2P3,
Bd Henri
Becquerel,
BP 55027,
F-14076 Caen,
lived, exotic species
with extreme
ratios
of proton
and
neutron
on France
the first measurement of the 2+
1 and 41 states in this
3
CNRS/IN2P3, F-91405 Orsay, France
numbers. Much
progress hasCSNSM,
been made
in recent years toward
key nucleus and the resulting first experimental signatures
4
Department of Physics, University of Oslo, PO Box 1048 Blindern, N-0316 Oslo, Norway
understanding these changes5 CEA,
as being
driven
toArpajon,
a largeFrance
extent
of collectivity from 9 Be(64 Cr,64 Cr + γ )X inelastic scattering.
DAM, DIF,
F-91297
6
Institut
für Kernphysik,
Universität zu Köln,
D-50937 Köln,
Germany
by spin-isospin
parts
of
the
nucleon-nucleon
interaction,
in
We
also
present
marked
structural
differences
between the
7
Instituto de Fisica Corpuscular, CSIC-Universidad de Valencia, E-46071 Valencia,
64 Spain 66
particular by8the
monopole parts of the tensor interaction [1–3].
Cr and Fe isotones. This confirms the occurrence of a
Institut de Physique Nucléaire, CNRS/IN2P3-Université Paris-Sud, F-91406 Orsay, France
66
Experimental9 Dipartimento
information
atdell’Università
the extremes
of proton-neutron
significant
Fe and 64 Cr
di Fisica
and INFN
Sezione di Padova, I-35131 Padova,
Italy change in nuclear structure between
10
IPHC, CNRS/IN2P3
and Université
Pasteur,
F-67037 Strasbourg, as
France
asymmetry11 is essential
to benchmark
the Louis
isospin
dependence
suggested recently from the small cross section for the
Departamento de Fı́sica Teórica, IFT-AM/CSIC, Universidad Autónoma, E-28049 Madrid, Spain
9
66
64
of this shell evolution.12 INFN, Laboratori Nazionali di Legnaro, I-35020 Legnaro, Italy two-proton knockout reaction, Be( Fe, Cr)X [10].
The region of 13neutron-rich
nuclei
above
doubly
Heavy Ion Laboratory,
Warsaw
University,
Warsaw, magic
PL-02097, Poland The measurement was performed at the National Super48
(Received
8
March
2010;
published
15
June
2010)
Ca has provided much insight into the nature of the forces
conducting Cyclotron Laboratory (NSCL) using 60,62,64 Cr
responsible for this modified shell structure. Evidence for a
and 62,64,66 Fe secondary beams produced by fragmentation
64
76
in 62
and has
Fe been
have been
measured for
using
The lifetimes
first excited
2+ states N
new subshell
gapofattheneutron
number
=Fe32
found
ofthea first
130timeMeV/u
Ge primary beam. Six different settings
the recoil-distance Doppler shift method after multinucleon transfer reactions in inverse kinematics. A sudden
[4], while
neutron-rich
Cr and
Fe64 Fe
nuclei
around
= 40 exhibit
of the
62
Fe to
is observed.
The N
experimental
results are compared
withA1900
new large-fragment separator [13] were used to deliver
increase
of collectivity from
2
9
scale shell-model
and Hartree-Fock-Bogolyubov–based
configuration-mixing
calculations
using beams onto a 370 mg/cm thick Be foil
collective
behavior.calculations
The description
of nuclei with N
= 40
these
secondary
the Gogny
D1S interaction.
a deeper
of the
mechanism located
leading to anatonset
challenges
theory
as theyThe
areresults
the give
subject
ofunderstanding
particularly
rapid
theof target position of the large-acceptance S800
68
32
collectivity near Ni, which is compared with the situation in the so-called island of inversion around Mg.
structural evolution. Driven by the deformation-driving neuspectrograph [14]. The A1900 separator was operated at a
DOI:intruder
10.1103/PhysRevC.81.061301
PACS
number(s):in21.10.Re,
21.10.Tg,
tron g9/2
orbital, they exhibit a remarkable
variation
3%
total 27.50.+e
momentum acceptance, except for the 62 Fe and 62 Cr
80
collectivity as a function of proton number: N = Z = 40 Zr
settings, where smaller momentum widths were transmitted.
+
68
It isisastrongly
common feature
of systems
of interacting
fermions
to a strong 2variation
of collectivity for
for 64
nuclei
deformed
[5] while
Z = 28
Ni haslead
a high-lying
Cr, awith
399 mg/cm2 9 Be production target was
1 68 Ni Specifically,
to form a shell structure. In atomic nuclei the spin-orbit
N ≈ 40. The nucleus
40 has many features of a doubly
28
2
used with
150
statelowers
and athereduced
collectivity
With
justwith
twohigh
Al wedge located at the midpoint of
interaction
energy ofquadrupole
the orbitals with
the highest [6].
magic
nucleus
excitation
energya of
the mg/cm
2+
1 state [3]
66 oscillator shell with
+
angular
momentum
the68
next
) value
[4]. Mass measurements,
and a smallN
B(E2;
0+
protons
lessinto
than
Ni,lower
Fe was the most neutron-rich
= 40
separator
to purify the secondary beam. The respective
1 → 21the
opposite
parity, leading
to the well-known
sequence of magic
on the prior
other hand,
show that
the N = 40
gap is weak
for 9 Be reaction target were 73.0, 67.5,
nucleus
with measured
spectroscopic
information
to
the
midtarget
energies
in
the
numbers. While the shell structure and the resulting energy+ 68 Ni [5,6], and it has been argued that the small B(E2)62,64,66
value
state
indicates
yet
and
82.6
MeV/u
for
present
work.
Its
low-lying
first
excited
2
gaps between the orbitals explain many general properties
does not necessarily indicate a shell gap at N = 40, but that Fe and 80.6, 74.6, and 87.0 MeV/u
of nuclei
across sudden
the nuclear
chart, itinhasnuclear
become evident
reflectsincreased
the character offor
the 60,62,64
2+
as respectively.
a predominant Particle identification was achieved
another
change
structureit with
Cr,
1 state
that the
shell structure [7].
and magic
numbers
change for
nuclei
neutron
[7]. The event-by-event
fragility of the N =with
40 subshell
collectivity
In its
vicinity,
indications
for excitation
collective
the S800 focal-plane detection system
with large neutron excess. As protons and
neutrons in such
gap is further evidenced by the developing collectivity in the
have
come
from
2+
in the Fe
isotopicZnchain
[14].
The the
energy
measured by the S800 ionization
exoticbehavior
nuclei occupy
different
orbitals
compared
to their stable
neutron-rich
and Ge isotopes,
for which
energy ofloss
the 2+
1 energies
1
68 effective single-particle energies are shifted. 62 state drops sharply from N = 38 to N = 42 while the B(E2)
counterparts,
Fe42 and in the Cr isotopes out to Cr38 [7–10]. For
chamber and the time-of-flight information between plastic
out tothe
As a 60,62
consequence,
some
shell gapsisvanish
new onesby the
values
increase [8,9]. The removal
of protons from the f7/2for the angle and momentum of
Cr direct
evidence
also and
provided
measurement
scintillators—corrected
emerge [1]. It is important to find experimental signatures for
orbital has a similar effect. The 21+ energies in the Fe isotopes
of theshell
quadrupole
deformation
inelastic
ion—were
used
to identify unambiguously the reaction
the changing
structure in order
to advance the lengths
theoretical in drop
sharply proton
above N = 36.each
The removal
of only two
protons
+
68
description
of exotic[11].
nuclei, e.g., by finding better shell-model
changes the energy of the 2residues
2033 keV infrom
Ni tothe target. The particle-identification
scattering
1 state fromemerging
interactions or better energy functionals for mean-field based
models.
The0556-2813/2010/81(5)/051304(4)
region of neutron-rich nuclei between Z = 20 (Ca)
and Z = 28 (Ni) is of particular interest for understanding
the evolution of the shell structure for nuclei with large
neutron excess. Adding protons to the 1f7/2 orbital changes
the relative energies of the neutron 2p3/2 , 2p1/2 , and 1f5/2
orbitals due to the strongly attractive proton-neutron spin-flip
573 keV in 66 Fe [10]. The 2+
1 energy drops even further to
517 keV in 68 Fe [11]. A similar behavior is observed in the
051304-1
©2010 The American Physical Society
Cr isotopes, where
the 2+
1 energies decrease gradually beyond
the N = 32 subshell closure [12]. Inelastic proton-scattering
experiments [13] and shell-model studies [14] corroborate the
development of substantial collectivity and deformation in the
neutron-rich Cr isotopes.
The role of the νg9/2 orbital for the onset of collectivity
Luigi Coraggio
68
Collectivity of neutron-rich Ti isotopes
H. Suzuki,1,2 N. Aoi,1,3 E. Takeshita,1,4 S. Takeuchi,1 S. Ota,5 H. Baba,1 S. Bishop,1 T. Fukui,5 Y. Hashimoto,6 E. Ideguchi,7
K. Ieki,4 N. Imai,8 M. Ishihara,1 H. Iwasaki,2,9,10 S. Kanno,4 Y. Kondo,6 T. Kubo,1 K. Kurita,4 K. Kusaka,1 T. Minemura,8
T. Motobayashi,1 T. Nakabayashi,6 T. Nakamura,6 T. Nakao,2 M. Niikura,2,7 T. Okumura,6 T. K. Ohnishi,2 H. J. Ong,2,3
H. Sakurai,2 S. Shimoura,7 R. Sugo,4 D. Suzuki,2,11 M. K. Suzuki,2 M. Tamaki,7 K. Tanaka,1 Y. Togano,4,6 and K. Yamada1
1
RIKEN Nishina Center, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan
2
Department of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan
3
RCNP, Osaka University, 10-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan
Department of Physics, Rikkyo University, 3-34-1 Nishi-Ikebukuro, Toshima, Tokyo 172-8501, Japan
5
Department of Physics, Kyoto University, Kitashirakawa, Kyoto 606-8502, Japan
6
Department of Physics, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro, Tokyo 152-8551, Japan
7
Center for Nuclear Study, University of Tokyo, RIKEN campus, 2-1 Hirosawa, Wako, Saitama 351-0298, Japan
8
Institute of Particle and Nuclear Study, KEK, 1-1 Oho, Tsukuba 305-0801, Japan
9
National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, Michigan 48824, USA
10
Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 48824, USA
11
Institut de Physique Nucléaire, IN2P3-CNRS, Université de Paris-Sud, F-91406 Orsay Cedex, France
(Received 13 April 2012; revised manuscript received 22 July 2013; published 30 August 2013)
4
The structure of the neutron-rich nucleus 58 Ti was investigated via proton inelastic scattering in inverse
kinematics at a mean energy of 42.0 MeV/nucleon. By measuring the deexcitation γ rays, three transitions
with the energies of 1046(11) keV, 1376(18) keV, and 1835(27) keV were identified. The angle-integrated cross
section for the 1046-keV excitation, which corresponds to the decay from the first 2+ state, was determined to
be 13(7) mb. The deformation length δp,p! was extracted from the cross section to be 0.83+0.22
−0.30 fm. The energy
of the first 2+ state and the δp,p! value are comparable to the ones of 56 Ti, which indicates that the collectivity of
the Ti isotopes does not increase significantly with neutron number until N = 36. This fact indicates that 58 Ti is
outside of the region of the deformation known in the neutron-rich nuclei around N = 40.
DOI: 10.1103/PhysRevC.88.024326
PACS number(s): 21.10.−k, 27.50.+e, 25.60.−t, 25.40.Ep
I. INTRODUCTION
Fe isotopes (Z = 26), the Ex (2+
1 ) values start decreasing at
N = 38, and this decrease continues monotonically beyond
The shell structure as the basis of our understanding of the
N = 40 [10,11]. This indicates that N = 40 does not exhibit
atomic nucleus has recently been found to change drastically
a magic character in Fe anymore [12,13]. In the Cr isotopes
due to the rearrangement of the single particle levels as
(Z = 24), the enhancement of the collectivity is seen already
one moves away from the stability line. In unstable nuclei
in the
nuclei
with (2010)
N = 36, where the Ex (2+
REVIEW
C 82,
054301
1 ) values drop
with very asymmetric proton and neutron PHYSICAL
numbers, various
much below the typical value of the open neutron-shell nuclei
exotic properties have been revealed such as the anomalous
along the Cr isotopic
[7,14]. The decrease continues
64 chain
deformation emerging in the very neutron-rich
regions around
+
Island
of inversion
around
until 64 Cr,
in which ECr
x (21 ) value shows the smallest among
N = 8 [1–3] and 20 [4,5]. These neutron numbers, which
the known Ex (2+
1 ) values in this region [15]. The enhancement
are considered to be magic in the traditional nuclear
1 theory,
3
of2collectivity
40 in2,*
the Cr isotopes was confirmed
S.region,
M. Lenzi,
F.nuclei
Nowacki,
A. Poves,toward
and N
K.=Sieja
lose the magicity in
the
neutron-rich
and
the
by
the Sezione
rapid increase
of deformation lengths (δ) and the
1
Dipartimento
Fisica dell’Università
INFN,
+
+ di Padova, I-35131 Padova, Italy
with these neutron numbers
becomediaccordingly
deformed. In and E
x (41 )/Ex (21 ) ratios (R4/2 ) [14,16]. A question arises as to
2
IPHC, IN2P3-CNRS
et Université dewhether
Strasbourg,
F-67037 deformation
Strasbourg,region
Francefound in Fe and Cr
order to account for such anomalous
deformation phenomena,
the anomalous
3
several theoretical
modelsde
have
beenTeórica
proposed,
although the Universidad
Departamento
Fı́sica
e IFT-UAM/CSIC,
Autónoma
E-28049 Madrid,
Spain
isotopes further
developsde
inMadrid,
more proton-deficient
isotopes,
in
mechanism is still unclear. The new
region of10deformation
(Received
September 2010;
published
particular
Ti. 2 November 2010)
recently found in the neutron-rich pf -shell region around
Theoretically, a deformation region is predicted to appear
N = 40 [6,7] may provide us a clue to clarify the mechanism.
around 62 Ti [6]. Indeed, the enhancement of collectivity
thenumber
development
collectivity
in the neutron-rich
nuclei around
N Ti
= isotopes
40, where
N = 40 We
is astudy
magic
for a of
harmonic
oscillator
has been suggested
for the
in the
thisexperimental
region from
58
potential,and
but theoretical
its energy evidence
gap is narrowed
the shape
intruderchange
suggest abyrapid
from
the spherical
to Approaching
the rotationalNregime,
in analogy
Ti to
their
β-decay
half-lives.
= 40 from
60 31
g9/2 orbital
lowered
by theat spin-orbit
Ti,Na.
theTheoretical
β-decay half-lives
become
gradually longer
calculations
are performed
withinthan
the
to what
happens
the islandinteraction.
of inversionBecause
surrounding
of the fragile
energyshell-model
gap, the nuclei
with Nin=a40
exhibit
shellbased
model
pf shell, which
is
whichtheencompasses
the full
interacting
framework
large
valence the
space,
on calculation
a 48 Ca core,within
a variety of characters depending on the proton number.
beingorbits
due toforthe
theeffective
neutron
1p=3/228,
, 1p1/2interpreted
, 0g9/2 , andas 1d
theadmixture
neutrons.ofThe
pf isotopes
shell forwith
the aprotons
thenumber
0f5/2 , Z
5/2
For the Ni
proton and
magic
νg9/2 configuration [17]. Considering the deformation-driving
68
interaction
is based on
a G matrix
obtained
from a realistic
nucleon-nucleon
potential
whose
part is
Ni shows
small collectivity
reflecting
the magic
character
nature of
the νg9/2 orbital, the
increase
of the monopole
νg9/2 contribution
corrected
empirically
produce
single-particle
withTothe
experimental
data.
of N = 40
as indicated
by its tolarge
valueeffective
of the energy
wouldenergies
enhancecompatible
the collectivity.
locate
the boundary
of We
the
+
excited
state [Ebetween
valueresults
of
of the first
find2a+ good
agreement
the small
theoretical
and
the available
experimental
data.
predict the
x (21 )] and
anomalous
deformation
region in
the We
Ti isotopes,
the onset
study of
quadrupole
transition probability
However,
theisotopic
collectivity
of TiThe
isotopes
is of great
importance.
deformation
at different[B(E2)]
neutron[8,9].
numbers
for theinvarious
chains.
maximum
collectivity
occurs in the
chromium isotopes where the large deformation regime already starts at N = 38. The shell evolution responsible
0556-2813/2013/88(2)/024326(5)
Physical Society
024326-1
for the observed shape changes is discussed in detail,
in parallel to the situation in ©2013
the N American
= 20 region.
DOI: 10.1103/PhysRevC.82.054301
PACS number(s): 21.60.Cs, 21.10.Re, 21.30.Fe, 27.50.+e
I. INTRODUCTION
In the last decades, more and more experimental evidence
has been accumulated establishing the breaking of the shell
closures known at the stability valley when approaching the
drip lines, mainly at the very neutron-rich side. The first
example of an unexpected disappearance of a shell closure
(N = 8) was found in 11 Be, whose ground state is an intruder
1/2+ located 320 keV below the “natural” 0h̄ω 1/2− state [1].
However, the true relevance of this finding was not recognized
till many years later, and even now it is shadowed by the
fame of its more neutron-rich isobar 11 Li. In the sd shell, the
ECT∗ -APCTP Joint Workshop
very low-lying 2+ state. Along the iron chain, indications
for a collective behavior come from the systematics of the
2+ states [8] as well as from the recent measurement of the
B(E2) values in 64,66 Fe [9,10]. The evolution of the B(E2)
values in iron isotopes points to a sudden increase of collectivity when approaching N = 40. Only very recently has the first
measurement of the excited levels in 64 Cr been reported [11].
This is the nucleus where the theoretical calculations predicted
the lowest-lying 2+ level in the region [12–14]. The measured
2+ state energy agrees within 100 keV with those theoretical
predictions. The sudden shape change at N = 40 challenges
Expt.
Model space (I)
Model space (II)
(a)
2.0
1.6
1.2
(b)
160
1.6
1.2
0.8
0.4
0.0
ENSDF
Rother 2011
Crawford 2013
(b)
2
80
0
2.0
28
30
32
34
36
38
+
240
Expt.
Model space (I)
Model space (II)
(a)
2.4
300
+
+
320
+
2
400
4
0.4
0.0
400
2.8
+
0.8
21 excitation energy [MeV]
2.4
B(E2 ; 2 → 0 ) [e fm ]
2.8
4
B(E2 ; 2 → 0 ) [e fm ]
+
21 excitation energy [MeV]
Collectivity at N = 40
200
100
0
40
28
30
32
N
Expt.
Model space (I)
Model space (II)
(a)
3.0
21 excitation energy [MeV]
3.6
2.4
1.8
1.2
0.0
38
40
Expt.
Model space (I)
Model space (II)
(a)
2.8
2.4
2.0
1.6
1.2
0.8
0.4
(b)
60
0
28 30 32 34 36 38 40 42 44 46 48 50
2
4
0.0
400
320
+
120
ENSDF
Marchi 2013
3.2
240
+
+
180
(b)
B(E2 ; 2 → 0 ) [e fm ]
2
240
+
4
B(E2 ; 2 → 0 ) [e fm ]
+
0.6
300
36
N
+
21 excitation energy [MeV]
4.2
34
160
ENSDF
Rother 2011
Crawford 2013
80
0
20
N
22
24
26
Z
Luigi Coraggio
ECT∗ -APCTP Joint Workshop
28
0.25
This Work
GSI-DSA
Adopted
Riken
REX-ISOLDE
GSI
NSCL
ORNL
IUAC
HRIBF
(a)
0.2
0.15
0.1
0.05
E x( M e V )
2 2
B(E2;0+1 2+1) (e b )
Enhanced quadrupole collectivity in light tin isotopes
4
Sb isotopes
2p-1h
2
0
(b)
50
55
60
65
70
Neutron number
75
80
⇒ shell-model study of neutron-deficient tin isotopes using 88 Sr
as a core
⇒ Quadrupole collectivity enhanced by the Z = 50 cross-shell
excitations
⇒ Model space spanned by proton 1p1/2 , 0g9/2 , 0g7/2 , 1d5/2
and 0g7/2 , 1d5/2 orbitals
⇒ Theoretical single-particle energies, two-body matrix
elements, and effective charges have been employed
Luigi Coraggio
ECT∗ -APCTP Joint Workshop
Calculation of the effective charges
na la ja
0g9/2
0g9/2
0g9/2
0g7/2
0g7/2
0g7/2
1d5/2
1d5/2
1d5/2
1d3/2
1d3/2
0h11/2
nb lb jb
0g9/2
0g7/2
1d5/2
0g7/2
1d5/2
1d3/2
1d5/2
1d3/2
2s1/2
1d3/2
2s1/2
0h11/2
ha|ep |bi
1.62
1.67
1.60
1.73
1.74
1.76
1.73
1.72
1.76
1.74
1.76
1.72
na la ja
0g7/2
0g7/2
0g7/2
1d5/2
1d5/2
1d5/2
1d3/2
1d3/2
0h11/2
Luigi Coraggio
nb lb jb
0g7/2
1d5/2
1d3/2
1d5/2
1d3/2
2s1/2
1d3/2
2s1/2
0h11/2
ha|en |bi
0.94
0.96
0.95
0.94
0.97
0.79
0.96
0.79
0.87
ECT∗ -APCTP Joint Workshop
4.8
a)
Expt.
88
Shell model – Sr core
4.0
3.2
2.4
1.6
0.8
+
21 excitation energy [MeV]
Enhanced quadrupole collectivity in light tin isotopes
4
2
2400
B(E2 ; 0
+
2 ) [e fm ]
3000
+
0.0
3600
b)
NSCL
Riken
REX–ISOLDE
GSI
1800
1200
600
0
50
52
54
56
58
N
Luigi Coraggio
ECT∗ -APCTP Joint Workshop
Part III
Predictiveness
Luigi Coraggio
ECT∗ -APCTP Joint Workshop
Nuclear models and predictive power
RIBs & advances in detection techniques ⇒ unknown structure
of nuclei towards the drip lines
Luigi Coraggio
ECT∗ -APCTP Joint Workshop
Realistic shell-model calculations
realistic shell-model calculations in different mass regions
⇓
results in good agreement with experimental data
Can realistic shell-model calculations be predictive ?
few selected examples
Luigi Coraggio
ECT∗ -APCTP Joint Workshop
Few selected physics cases
Sn isotopes beyond N = 82
heavy calcium isotopes
neutron-rich titanium and nickel isotopes
Single-particle energies from the experiment ⇒ reduced role of
3N force
Luigi Coraggio
ECT∗ -APCTP Joint Workshop
Sn isotopes beyond N = 82
⇒ shell-model study of Sn isotopes beyond N = 82
⇒ Vlow−k from CD-Bonn NN potential
⇒ h9/2 fpi13/2 model space with 132 Sn inert core
⇒ SP energies from 133 Sn
Luigi Coraggio
ECT∗ -APCTP Joint Workshop
Sn isotopes beyond N = 82
⇒ shell-model study of Sn isotopes beyond N = 82
... It is the aim of our study to compare the results of our
calculations with the available experimental data and to make
predictions for the neighboring heavier isotopes ...
Luigi Coraggio
ECT∗ -APCTP Joint Workshop
Sn isotopes beyond N = 82
+
+
Excitation energies of the 2+
1 , 41 , and 61 states in Sn isotopes
6
+
E (MeV)
1,5
4
+
1
2
+
0,5
134
138
136
140
A
Luigi Coraggio
ECT∗ -APCTP Joint Workshop
Sn isotopes beyond N = 82
+
+
Excitation energies of the 2+
1 , 41 , and 61 states in Sn isotopes
6
+
E (MeV)
1.5
4
+
1
2
+
0.5
134
138
136
140
A
Luigi Coraggio
ECT∗ -APCTP Joint Workshop
Heavy calcium isotopes
⇒ first mass measurements of 53 Ca and 54 Ca
⇒ new method of precision mass spectroscopy with ISOLTRAP
Luigi Coraggio
ECT∗ -APCTP Joint Workshop
Heavy calcium isotopes
“ ... pronounced decrease in S2n revealed by the new 53 Ca and
54 Ca ISOLTRAP masses ...”
Luigi Coraggio
ECT∗ -APCTP Joint Workshop
Heavy calcium isotopes
⇒ spectroscopic study of 54 Ca
⇒ proton knockout reactions involving 55 Sc and 56 Ti projectiles
Luigi Coraggio
ECT∗ -APCTP Joint Workshop
Heavy calcium isotopes
⇒ shell-model study of neutron-rich calcium isotopes
⇒ fp model space with 40 Ca inert core
⇒ predictions for the (at that time) unknown spectra of 53−56 Ca
Luigi Coraggio
ECT∗ -APCTP Joint Workshop
Heavy calcium isotopes: shell-model results
22
20
4.0
16
14
12
10
21 excitation energy [MeV]
Expt.
Vlow–k (2009)
GXPF1A (2005)
8
6
4
+
S2n [MeV]
18
3.5
Expt.
Vlow–k (2009)
GXPF1A (2005)
3.0
2.5
2.0
1.5
1.0
22
24
26
28
30
32
34
22
24
26
N
28
N
Luigi Coraggio
ECT∗ -APCTP Joint Workshop
30
32
34
Heavy calcium isotopes: shell-model results
22
20
14
12
10
Expt.
Vlow–k (2009)
GXPF1A (2005)
8
+
6
4
Expt.
Vlow–k (2009)
GXPF1A (2005)
4.0
16
21 excitation energy [MeV]
S2n [MeV]
18
3.5
3.0
2.5
2.0
1.5
1.0
22
24
26
28
30
32
34
22
24
26
N
28
30
32
34
N
6.0
different monopole properties
ESPE [MeV]
5.0
f5/2
4.0
Present calculations
3.0
GXPF1A
p1/2
2.0
1.0
0.0
28
30
32
34
Neutron number
Luigi Coraggio
ECT∗ -APCTP Joint Workshop
36
38
Isotopic chains “north-east” of 48 Ca
PHYSICAL REVIEW C 89, 024319 (2014)
Realistic shell-model calculations for isotopic chains “north-east” of 48 Ca in the (N,Z) plane
L. Coraggio,1 A. Covello,2 A. Gargano,1 and N. Itaco1,2
2
1
Istituto Nazionale di Fisica Nucleare, Complesso Universitario di Monte S. Angelo, Via Cintia - I-80126 Napoli, Italy
Dipartimento di Fisica, Università di Napoli Federico II, Complesso Universitario di Monte S. Angelo, Via Cintia - I-80126 Napoli, Italy
(Received 16 October 2013; revised manuscript received 9 December 2013; published 26 February 2014)
We perform realistic shell-model calculations for nuclei with valence nucleons outside 48 Ca, employing
two different model spaces. The matrix elements of the effective two-body interaction and electromagnetic
multipole operators have been calculated within the framework of many-body perturbation theory, starting from
a low-momentum potential derived from the high-precision CD-Bonn free nucleon-nucleon potential. The role
played by the neutron orbital 1d5/2 has been investigated by comparing experimental data on yrast quadrupole
excitations of isotopic chains north-east of 48 Ca with the results of calculations including or not including this
single-particle state in the model space.
DOI: 10.1103/PhysRevC.89.024319
PACS number(s): 21.60.Cs, 23.20.Lv, 27.40.+z, 27.50.+e
I. INTRODUCTION
An interesting aspect of the physics of nuclei approaching
the neutron drip line is the evolution of their shell structure.
This topic is currently investigated in different mass regions,
and the modern advances in the detection techniques and
new experimental devices provide data that drive to a better
understanding of the microscopic mechanism underlying
modifications of the “magic numbers.”
In this context, in the last decade the key role played by
the tensor component of the residual two-body interaction
between spin-orbit partner single-particle states has been
recognized [1–3]. This may give rise to a breaking of shell
closures leading to the possible appearance of the so-called
“island of inversion;” namely, a region of nuclei where a
rapid development of collectivity is observed. The best-known
example of this phenomenon is given by neutron-rich nuclei
around 32 Mg [4].
The region of nuclei with valence protons outside doubly
closed 48 Ca may be considered an interesting laboratory to
study the shell evolution when adding neutrons, since there
are long isotopic chains, such as those of iron and nickel
isotopes, whose exoticity reaches an N/Z value of 1.79 in
78
Ni.
Coraggio
As a matter of fact, it can Luigi
be observed
that the shell
It is worth pointing out that there is a general belief
[15] that shell-model effective interactions derived from
realistic nucleon-nucleon (NN) potentials are defective in their
monopole component, so that they are not able to provide a
good description of the evolution of spectroscopic properties
along the isotopic chains, unless including contributions from
three-nucleon forces [16,17].
On these grounds, we have found it challenging to perform
realistic shell-model calculations [18] for some isotopic
chains north-east of 48 Ca, starting from the high-precision
CD-Bonn NN potential renormalized by way of the so-called
Vlow-k approach [19,20]. In particular, in order to explore the
microscopic processes leading to the disappearance of the
N = 40 shell closure and the onset of collectivity, we have
derived two effective shell-model interactions for two different
model spaces. The first one, which is spanned by the proton
0f7/2 and 1p3/2 orbitals and by the neutron 1p3/2 , 1p1/2 ,
0f5/2 , 0g9/2 orbitals, is able, as we show in the following, to
reproduce the dropping of the N = 40 magic number in iron
and chromium isotopes as well as many other spectroscopic
properties of nuclei north-east of 48 Ca. The second one is
spanned by the same orbitals plus the neutron 1d5/2 . The
calculations within the latter model space are able to reproduce
the onset of∗collectivity in heavy iron and chromium isotopes,
ECT
-APCTP
Workshop
at the
same time
preservingJoint
the results
obtained with the first
⇒ shell-model study of neutron-rich isotopic chains outside
48 Ca
⇒ fpgd model space with 48 Ca inert core
⇒ predictions for the (at that time) unknown spectra exotic Ti
isotopes and of 78 Ni shell closure
4.2
(a)
21 excitation energy [MeV]
2.4
Expt.
Theory
2.0
1.6
1.2
0.8
(b)
Expt.
Theory
(a)
3.0
2.4
1.8
1.2
0.6
40
0
26
28
30
32
34
36
38
40
42
2
240
+
+
80
0.0
300
180
+
120
+
2
4
B(E2 ; 2 → 0 ) [e fm ]
0.0
160
4
B(E2 ; 2 → 0 ) [e fm ]
3.6
+
0.4
+
21 excitation energy [MeV]
Isotopic chains “north-east” of 48 Ca: shell-model
results
120
(b)
ENSDF
Marchi 2013
60
0
26 28 30 32 34 36 38 40 42 44 46 48 50 52
N
N
Luigi Coraggio
ECT∗ -APCTP Joint Workshop
4.2
21 excitation energy [MeV]
2.4
Expt.
Theory
Gade 2014
(a)
2.0
1.6
1.2
0.8
(b)
Expt.
Theory
RIKEN 2015
(a)
3.0
2.4
1.8
1.2
0.6
40
0
26
28
30
32
34
36
38
40
42
2
240
+
+
80
0.0
300
180
+
120
+
2
4
B(E2 ; 2 → 0 ) [e fm ]
0.0
160
4
B(E2 ; 2 → 0 ) [e fm ]
3.6
+
0.4
+
21 excitation energy [MeV]
Isotopic chains “north-east” of 48 Ca: shell-model
results
120
(b)
ENSDF
Marchi 2013
60
0
26 28 30 32 34 36 38 40 42 44 46 48 50 52
N
N
Luigi Coraggio
ECT∗ -APCTP Joint Workshop
Conclusions and outlook
The agreement of our results with the experimental data
testifies the reliability of a microscopic shell-model
calculation with realistic potentials.
We have now evidence of the predictive power of realistic
shell model
Role of real three-body forces and three-body correlations
should be investigated.
Perspectives: benchmark calculations with other
many-body approaches.
Luigi Coraggio
ECT∗ -APCTP Joint Workshop
These terms introduce density dependence into the effective
shell-model hamiltonian
Luigi Coraggio
ECT∗ -APCTP Joint Workshop
Conclusions and outlook
The agreement of our results with the experimental data
testifies the reliability of a microscopic shell-model
calculation with realistic potentials.
We have now evidence of the predictive power of realistic
shell model
Role of real three-body forces and three-body correlations
should be investigated.
Perspectives: benchmark calculations with other
many-body approaches.
Luigi Coraggio
ECT∗ -APCTP Joint Workshop

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