V low-k and nuclear structure Angela Gargano Napoli
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Transcript of V low-k and nuclear structure Angela Gargano Napoli
Vlow-k and nuclear structure
Angela Gargano
Napoli
A. Gargano Cortona - 2008Napoli
A. Gargano Cortona - 2008Napoli
Vlow-k
● Derived from the original VNN by integrating out the high-momentum components of the original VNN potential decouples low-energy physics from high-momentum details
● Vlow-k preserves the physics of the original NN interaction up to the cutoff momentum Λ: the deuteron binding energy scattering phase-shifts
Features of Vlow-k
eliminates sources of non-perturbative behavior
real effective potential in the k-space gives an approximately unique representation of the NN potential for 2 fm-1 ELab 350 MeV
Vlow-k() class of potentials all having cutoff independent NN observables
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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Low-momentum potential confined within a momentum-space cutoff
A. Gargano Cortona - 2008Napoli
Realistic Shell model
EH
VTH NN
•A-nucleon system
•Hilbert space
PEPH
VHH
eff
eff'0eff
•N-valence nucleon system•Shell-model space
Ab-initio calculations (including also NNN forces):•GFMC calculations•no-core shell model•coupled-cluster method limited to small systems
Empirical shell-model calculationsno link with NN interaction
accounts for excitations above the model space
as well as for interactions with core particles
A. Gargano Cortona - 2008Napoli
TBME of Veff from VNN
1. renormalization of VNN through Vlow-k
2. Veff calculation by the folded-diagram perturbation
theory
A. Gargano Cortona - 2008Napoli
2. Veff calculation by folded-diagram perturbation theory
2.1 -box calculation
2-body diagrams up to 2nd order:
V V1p1h V2p V2p2h 1-body diagrams up to 2nd order S-box
collection of irreducible valenced-linked diagrams with at least 1 H1 vertex
bHH
QH
H
QHH
H
QHHabQa
1
01
011
011
ˆ
space model and )()( 10klowklow PHHUVUTVTH
We start from
with ω ≡ energy variable and Q (intermediate-state space) =1 – P
^Q
A. Gargano Cortona - 2008Napoli
)ˆ(ˆ1
eff
1QFQ
iiH
Sum through the Lee-Suzuki iterative technique [Suzuki-Lee Prog. Theor. Phys. 64, 2091 (1980)]
Heff =(T+U)+ H1eff =H0 + H1
eff
H1eff
contains both 1- and 2-body contributions “subtraction procedure” to remove from H1
eff the 1-body terms
single-particle energies from
experiment
VH HHHHHSFi
iH eff'0
eff1b
eff1
eff1b0eff
0
eff
1b )ˆ(
2.2 Folded diagram series
2
1)(
)(
000
0
2
2
2
1
Qd
Qd
d
QdQQ
d
QdQF
Qd
QdQF
2. Two-body matrix elements from the CD-Bonn NN potential renormalized through the Vlow-
k with =2.2 fm-1
1. Single-particle energies from expt data of nuclei with one-valence nucleon
Calculations
A. Gargano Cortona - 2008Napoli
• U harmonic oscillator with ћω = 45 A-1/3 - 25 A-2/3 ^• Q -box second-order calculation
• intermediate states composed of: hole and particle states restricted to 2 shells below and above the Fermi surface ↔ “small” intermediate-state space all hole states and particle states restricted to the five shells above the Fermi Surface ↔ “large” intermediate-state space
A. Gargano Cortona - 2008Napoli
s1/2
h11/2
d3/2
d5/2
g7/2
i13/2
f5/2
p1/2
h9/2
p3/2
f7/2
82
50 50
132Sn
134Sb 132Sn + 1 + 1π
134Sb 132Sn + 1 + 1π
εj da 133Sn
εj da 133Sb
π space space
.
.
.
.
.
.
.
.
....
A. Gargano Cortona - 2008Napoli
• Calc.▲ Expt.
134Sb “Small” intermediate-state space
“Large” intermediate-state space
g7/2f7/2
g7/2f7/2
0- 95
1- 94
2- 88
3- 100
4- 94
5- 95
6- 94
7- 94
Jπ %
g7/2f7/2
A. Gargano Cortona - 2008Napoli
Vlow-k
Diagonal matrix elements of interaction for the
g7/2f7/2 configuration
J
Veff
Matr
ix E
lem
en
ts (
MeV
) V1p1h
A. Gargano Cortona - 2008Napoli
Vlow-k for various values of Vlow-k for various values of g7/2f7/2
Matr
ix E
lem
en
ts (
MeV
)
A. Gargano Cortona - 2008Napoli
210Bi 134Sb
h9/2g9/2g7/2f7/
2
Experimental multiplets
Inversion of the 0- and 1- states ↔ long standing problem• role of tensor force evidenced in studies with empirical TBME• previous studies with realistic effective interactions fail to reproduce the g.s.
A. Gargano Cortona - 2008Napoli
(L)127210
83 Bi210Bi
h9/2g9/2
J
1p1h correlations produce the right effect to make the 1- the g.s. non central components arise from virtual interactions with the core nucleons
• Calc.▲Expt.
A. Gargano Cortona - 2008Napoli
0
0,5
1
1,5
2
0+ 2+ 4+ 6+
E(M
eV
)134Sn Expt 134Sn Calc
134Te Expt 134Te Calc
0
0,5
1
1,5
2
0+ 2+ 4+ 6+ 8+
E(M
eV
)
210Pb Expt 210Pb Calc
210Po Expt 210Po Calc
134Te 132Sn + 2π
134Sn 132Sn + 2(f7/2)2 multiplet
(g7/2)2 multiplet
210Pb 208Pb + 2
210Po 208Pb + 2π
(h9/2)2 multiplet
(g9/2)2 multiplet
A. Gargano Cortona - 2008Napoli
-0,8
-0,5
-0,2
0,1
0,4
0+ 2+ 4+ 6+
ME
(Me
V)
Vlow-k
1p1h
Veff
-0,8
-0,5
-0,2
0,1
0,4
0+ 2+ 4+ 6+
ME
(Me
V)
Vlow-k
1p1h
Veff
Diagonal matrix elements of
Interaction in
132Sn region
(f7/2)2
(g7/2)2
A. Gargano Cortona - 2008Napoli
Summary
Typical features of Veff originate from core polarization effects
π interaction0- and 1- spacing
in134Sb and 210Bi
ππ and interactionslow-energy 2+ state
in 134Sn and 210Pbwith respect to 134Te and 210Po
reasonable cutoff variations do not seem to change significantly two-body matrix elements
L. Coraggio (Napoli)
A. Covello (Napoli)
A. Gargano (Napoli)
N. Itaco (Napoli)
T.T.S. Kuo (Stony Brook)
A. Gargano
Napoli
Cortona - 2008