Testing shell model on nuclei across the N=82 shell gap Testing shell model on nuclei across the N=82 shell gap 1.Test nuclei 2.New experimental data 3.Realistic

Testing shell model on nuclei across the N=82 shell gap

Testing shell model on nuclei across the N=82 shell gap

1. Test nuclei

2. New experimental data

3. Realistic shell model calculations: basic ingredients

4. Results and comparison with experiment

5. Analysis of the two-body matrix elements

6. Summary

Angela Gargano Angela Gargano

INFN - NapoliINFN - Napoli

Napoli-Stony Brook Collaboration

Napoli-Stony Brook Collaboration

L. Coraggio

A. Covello

A. G.

N. Itaco

T.T.S. Kuo

A. Gargano – Napoli

Pisa 2005

130Sn

131Sn

132Sn

133Sn

134Sn

131Sb

132Sb

133Sb

134Sb

135Sb

132Te

134Te

136Te

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

Behavior of the first 2Behavior of the first 2+ + state in even Sn isotopesstate in even Sn isotopes

"" in even Te isotopes in even Te isotopes

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

"" in even Te in even Te isotopesisotopes

Behavior of the first 5/2Behavior of the first 5/2+ + in odd Sb isotopes in odd Sb isotopes

Multiplets in odd-odd Sb isotopesMultiplets in odd-odd Sb isotopes A. Gargano – Napoli

Pisa 2005

B(E2;0+ 2+) = 0.103(15) e2b2


Pisa 2005

D. Radford - ENAM04


Pisa 2005

132Sn and 134Sn results from J.R. Beene –ENAM04


Pisa 2005

● Many-body theory: derivation of the effective interaction

Realistic shell-model calculationsRealistic shell-model calculations

Two main ingredients Two main ingredients

● Nucleon-nucleon potential

No adjustable parameter in the calculation of two-body matrix elements


Pisa 2005

Two-body matrix elements of the Hamiltonian derived from the free nucleon-nucleon potential

Shell-model effective interaction

Shell-model effective interaction

Model-space Schroedinger equation

potential auxiliary an with and

space model thedefines where

,)(

0

1

0

SPUUTH

P

PEPVHPPPH

i

d

i

iiieffieff

Nuclear many-body Schroedinger equation

iiiNNi EVTH )(


Pisa 2005

Nucleon-nucleon potentialNucleon-nucleon potential

π ρ ω σ1σ2

● CD-Bonn potential

High-precision NN potential based upon the OBE model

2/Ndata= 1.02

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

43 parameters


Pisa 2005

Renormalization of the NN interactionRenormalization of the NN interaction

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

confined within a momentum-space cutoff

S. Bogner, T.T.S. Kuo, L. Coraggio, A. Covello, N. Itaco, Phys. Rev C 65, 051301(R) (2002).

Derived from the original VNN by integrating out the high-momentum components by means of an iterative method. Vlow-k preserves the physics of the original NN interaction up to the cut-off momentum Λ: the deuteron binding energy and low-energy scattering phase-shifts are reproduced.

k


Pisa 2005

3210 FFFFVeff

Derivation of the realistic effective interaction by means of the folded-diagram expansion

Derivation of the realistic effective interaction by means of the folded-diagram expansion

1. Calculation of boxQ

Vertex function composed of irreducibile and valence linkeddiagrams in Vlow-k

2. Sum of the folded-diagram expansion by Kreciglowa-Kuo or Lee-Suzuki method

boxQ

derivative 1

boxQst

boxQ

sderivative 2 & 1

boxQndst

boxQ


Pisa 2005

We include one and two-body diagrams up to second order in Vlow-k

“Bubble”

50

82

.

.

.

132Sn

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

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

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

space

space

-1space

NN-potential CD-Bonn

g7/2

d5/2

d3/2

s1/2

h11/2

-9.663

-8.701

-7.223

-6.870*

-6.836

SP energies

133Sb

p3/2

h9/2

p1/2

f5/2

i13/2

f7/2

-1.601

-0.894

-0.805

-0.450

0.239*

-2.455

SP energies

133Sn

7.325

7.425

7.657

8.980

9.759

d3/2

h11/2

s1/2

d5/2

g71/2

-1 SP energies

131Sn


Pisa 2005

126

134Sn

in 82-126 shell

= 70 keV

86% (f7/2)2

81% (f7/2)2

BEExpt =6.365 ± 0.104 MeV PRL 1999BECalc=6.082 ± 0.064 MeV


Pisa 2005

Sn isotopesSn isotopes

eeff=0.70e fromB(E2;6+ 4+) in 134Sn

eeff=0.75e fromB(E2;10+ 8+) in

134Sn

▲ Expt.

● Calc.


Pisa 2005

Proton-particle neutron-hole multiplets

Proton-particle neutron-hole multiplets Jjllj 1)''()(

in the 50-82 shell-1 in the 50-82 shell

132Sb


Pisa 2005

L. Coraggio et al., PRC 66, 064311 (2002)

Proton-particle neutron-particle multipletsProton-particle neutron-particle multiplets

Jjllj )''()( in the 50-82 shell in the 82-126 shell

= 42 keV

BEExpt =12.952 ± 0.052 MeV PRL 1999BECalc=12.849 ± 0.058 MeV134Sb


Pisa 2005

d5/2f7/2

g7/2f7/2

135Sb

in 50-82 shell

in 82-126 shell

= 72 keV



Pisa 2005

Sb isotopesSb isotopes

N

■ Splitting of the centroids of the g7/2 nd d5/2 SP strengths A. Gargano – Napoli

Pisa 2005

7/2+

5/2+

27

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

25

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

The low-energy 2+ state in 134Sn is responsible for the mixing in the 5/2+ state

The low position of the 5/2+ is strictly related to the two J = 1- matrix

elements: (g7/2 f7/2) -600 keV

(d5/2 f7/2) -500 keV (the two 1- in 134Sb)


Pisa 2005

135Sb

B(M1;5/2+ 7/2+) 2 x 10-3

Expt. Calc.

(with free g factors)

0.29▲ 25

M1 effective operator: including 2nd order core-polariazation effects

4.0 2 x 10-3 ( a factor 14)

a factor 90

Non-zero off diagonal matrix element between g7/2 and d5/2 is responsible for the B(M1) reduction

The magnetic moment of the g.s. state is 2.5 to be compared to 1.7 obtained wth free g factors - Expt. 3.0 A. Gargano – Napoli

Pisa 2005

135Sb

▲H. Mach, in Proc. of th 8th Inter. Spring Seminar on Nucl .Phys., Paestum 2004

136Te

=100 keV


in the 50-82 shell in the 82-126 shell

Dominant componentfrom 2+ state of 134Te

Dominant componentfrom 2+ state of 134Sn


Pisa 2005

Te isotopesTe isotopes

eeff() as Sn isotopes

eeff() = 1.55e fromB(E2;4+ 2+) in 134Te

J. Terasaki et al. PRC (2002)

N. Shimuzu et al. PRC (2004)

S. Sarkar et al. EPJA (2004)


Pisa 2005

Two-body effective matrix elements (in MeV) Two-body effective matrix elements (in MeV)

Config. Veff Vlow-k

(f7/2)2 -0.654 -0.403

(p3/2)2 -0.404 -0.101

diagonal matrix elements for J=0+

Config. Veff Vlow-k

(g7/2)2 -0.738 0.063

(d5/2)2 -0.486 -0.304

identical particles

diagonal matrix elements for J=2+ diagonal matrix elements for J=0+

diagonal matrix elements for J=2+

Config. Veff Vlow-k

(f7/2)2 -0.286 -0.289

Config. Veff Vlow-k

(g7/2)2 -0.037 -0.016

-1-1 diagonal matrix elements for J=0+

Config. Veff Vlow-k

(d3/2)2 -0.325 -0.184

(h11/2)2 -1.058 -0.417

(s1/2)2 -0.726 -0.869

-1-1 diagonal matrix elements for J=2+

Config. Veff Vlow-k

(d3/2)2 -0.036 -0.097

(h11/2)2 -0.507 -0.445


Pisa 2005

Vlow-k

Veff

V3p1h

V2p

V4p2h

Two-body matrix elementsTwo-body matrix elementsg7/2f7/2


Pisa 2005

Two-body matrix elementsTwo-body matrix elementsd5/2f7/2

Vlow-k

Veff

V3p1h

V2p

V4p2h

●


Pisa 2005

Summary Summary

Properties of exotic nuclei in 132Sn region below and above the N=82 shell closure are well reproduced by our realistic calculations

No evidence of shell structure modification in these neutron rich nuclei

Very relevant role of core polarization effects


Pisa 2005

More experimental information is needed

Documents

Testing shell model on nuclei across the N=82 shell gap Testing shell model on nuclei across the N=82 shell gap 1.Test nuclei 2.New experimental data 3.Realistic