A. Gargano Eurisol User Group Wokshop – Firenze 2008 Structure of heavy neutron-rich nuclei Angela...

A. Gargano Eurisol User Group Wokshop – Firenze 2008

Structure of heavy neutron-rich nuclei

Angela Gargano

Napoli

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 N=82 shell closure

Realistic shell model and nuclei with several valence nucleons:Results for the N=82 isotonic chain

Summary


Napoli

3. Two-body matrix elements

1. Model space 2. Single-particle energies

4. Construction and diagonalization of the energy matrices

Shell-model calculations

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

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


Napoli

Realistic shell-model calculations

NNNNN VVTH

Veff from the bare potential


Napoli

The role of the NNN potential haa been evidenced by investigation on light nuclei

• 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)

Tjon line

Inclusion of three–body forces in the “shell model” approach not yet attempted

NCSM Navratil 2007

Nogga 2004

Model space defined by the operator

1 111

being complement the id

ii

d

i PQP


Napoli

Model-space Schroedinger equation

iiici PEPVHPEPPH )( eff0eff

Nuclear many-body Schroedinger equation),,1()(),,1( AEVTAH iiNNi


NNNNN VVTH

Veff from the bare potential


Napoli

Many-body theory to derive the effective interaction

Nucleon-nucleon potential


Basic ingredients :


Napoli

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

(1999 NN Database: 5990 pp and np scattering data)

π ρ ω σ1σ2 based upon the OBE model


Napoli

NN potential derived from chiral effective field theoryS. 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 πN 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% - χ2/Ndata

= 1.10 (np database below 290 MeV); χ2/Ndata = 1.50 (pp database below 290 MeV)

Chiral potentials

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


Napoli

Matrix elements of VNN

in the 1S0 channel

NN potentials are not completely constrained by low-energy NN data


Napoli

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

New approach: construction of a low-momentum NN potential Vlow-k

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).

)(1

)(

GQTQQ

QVVG NNNN

= + + ...~~~G VNN

VNN

VNN

Infinite sum of ladder diagrams


Napoli

● 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 Λ: the deuteron binding energy scattering phase-shifts

● Iterative method

Λk

Vlow-k: Low-momentum potential confined within a momentum-space cutoff

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).

Vlow-k approach


Napoli

Matrix elements of Vlow-k with Λ=2.1 fm-1

in the 1S0 channel

LabLab E

ME012.0

2 22

)(

)( 1

MeVE

fm

Lab

1.5 1.8 2.0 2.05 2.2 2.5

187 269 332 350 401 518-1fm 1.2

k

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


Napoli

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


Napoli

Realistic effective interaction: “folded-diagram expansion”

1'0)()( HHUVUTVTH

● Calculation of the : collection of irreducible and valenced linked diagrams at any order in V 2-body 2nd order diagrams:

V V2p V1p1h V2p2h

auxliary 1b potential

1 with :space Model111

id

ii

d

i PQP

● Sum of the folded-diagram series :(by Kuo-Krengiglowa or Lee-Suzuki iterative technique)

)ˆ(ˆ1

effQFQ

iiV

boxˆ Q


Napoli

Results

132Sn neighbors beyond N=82

N=82 isotonic chain


Napoli

50

82

.

.

.

132Sn

i13/2f5/2p1/2h9/2p3/2f7/2

h11/2s1/2d3/2d5/2g7/2

space

space

126

.

.

.

133Sb

133Sn

VNN : CD-Bonn

Vlow-k with Λ= 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

MeV 88.7harmonic-oscillator basis


Napoli

Across the N=82 shell gapAcross the N=82 shell gap

Anomalous behavior of the first 2Anomalous behavior of the first 2+ + state in even Sn isotopesstate in even Sn isotopes

"" in even Te isotopes in even Te isotopes

Anomalous behavior of the B(E2; 0Anomalous behavior of the B(E2; 0++22++) value in even Sn ) value in even Sn isotopesisotopes

"" in even Te in even Te isotopesisotopes

Anomalous behavior of the first 5/2Anomalous behavior of the first 5/2+ + in Sb isotopes in Sb isotopes

Anomalously low position of the 1Anomalously low position of the 1-- state in state in 134134SbSb

Z=52 134Te 135Te 136Te 137Te 138Te 139TeZ=51 133Sb 134Sb 135Sb 136SbZ=50 132Sn 133Sn 134Sn

N=82 N=83 N=84 N=85 N=86 N=87

N/Z=1.68 N/Z=1.67


Napoli

7/2+

5/2+

odd Sb isotopes

2+ in Te isotopes

2+ in Sn isotopes

N

Downshift of the d5/2 proton level relative to the g7/2 one?Onset of a modification in theshell structure?


Napoli

134Te

σ(keV)=115 σ(keV)=143 σ(keV)=128

Expt Cd-Bonn

NijmII Argonne V18


Napoli

= 70 keV

86% (f7/2)2

81% (f7/2)2

Expt Calc

134Sn


Napoli

= 42 keV

134Sb• Calc.▲ Expt.

g7/2f7/2

d5/2f7/2


Napoli

• Calc.▲ Expt.

134Sb

Small space:intermediate states composed of particle and hole states restricted to two major shells above and below the FS

Large spaceg7/2f7/2

g7/2f7/2


Napoli

V1p1h

V2p2h

V2p

Veff

Vlow-k

Diagonal matrix elements of interaction for the

g7/f7/2 configuration

J

J

Large space

Small space


Napoli

136Sb

• Calc.▲ Expt.

Yrast states with Jπ from 0- to 7-

dominated by the g7/2(f7/2 )3

G.S. Simpson et al. 2007 , ILL Grenoble


Napoli

134Sb

136Sb

The “g7/2f7/2 multiplet” in compared to that in

136Sb134Sb


Napoli

135Sb

Expt Calc

= 72 keV

0.282anomalously low position


Napoli

27

75% g7/2 (f7/2)2 +...

25

45% d5/2 (f7/2)2 + 23% g7/2 (f7/2)2 + ...

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

135Sb: wave functions

J=0+ J=2+

J=0+


Napoli

Expt Calc

136Te


Napoli

E2 eeff() = 0.70e eeff() = 1.55e

M1 effective operator: 2nd order core & no meson-exchange corrections

Comparison of the experimental and theoreticalB(E2) [e2fm4] and B(M1) [μ2

N]

new measurements yield values which arehigher: ● 50% HRIBF-ORL ● 20% ISOLDE-CERN


Napoli

N=82 isotones

134Te

136Xe 138Ba 140Cs 142Nd 144Sm 146Gd 148Dy 150Er 152Yb 154Hf

Zval from 4 to 22

Zval=2 test of Veff

Role of three-body forces both genuine and effective


Napoli

J=2+


Napoli

J=4+


Napoli

J=6+


Napoli

Ground-state binding energyper valence proton

Zval

The Expt and Theor behaviors of the g.s. binding energy per valenceproton diverge contribution of many-body effective forces


Napoli

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.

A. Gargano Eurisol User Group Wokshop – Firenze 2008 Structure of heavy neutron-rich nuclei Angela...

Documents

Transcript of A. Gargano Eurisol User Group Wokshop – Firenze 2008 Structure of heavy neutron-rich nuclei Angela...