V low-k and nuclear structure Angela Gargano Napoli

Vlow-k and nuclear structure

Angela Gargano

Napoli

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).

<~

Low-momentum potential confined within a momentum-space cutoff


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


TBME of Veff from VNN

1. renormalization of VNN through Vlow-k

2. Veff calculation by the folded-diagram perturbation

theory


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


)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


• 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


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

.

.

.

.

.

.

.

.

....


• 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


Vlow-k

Diagonal matrix elements of interaction for the

g7/2f7/2 configuration

J

Veff

Matr

ix E

lem

en

ts (

MeV

) V1p1h


Vlow-k for various values of Vlow-k for various values of g7/2f7/2

Matr

ix E

lem

en

ts (

MeV

)


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.


(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.


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


-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


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

V low-k and nuclear structure Angela Gargano Napoli

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