neutron-rich nuclei within a realistic shell-model approach angela gargano napoli

Neutron-Rich Nuclei within

a realistic shell-model approach

Angela Gargano

Napoli

A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010Napoli

L. Coraggio (Napoli)

A. Covello (Napoli)

N. Itaco (Napoli)

T.T.S. Kuo (Stony Brook)


Plan of the talk

Theoretical framework: Renormalization of the bare NN potential by means of

the Vlow-k approach

Derivation of the model space effective interaction by means of the plus folded diagram method

Outline of calculations

Results: neutron-rich nuclei northeast of 132Sn

and comparison with conterpart nuclei in the stable 208Pb regionneutron-rich Ca isotopes

neutron-rich C isotopes

Summary


Shell-model calculations

i jj ijV(i)HH 0iiisoici ls)(rU)(rUT(i)H

0

3. Two-body matrix elements

1. Model space

2. Single-particle energies (taken from the experimental spectra of nuclei with one-valence nucleon or treated as free parameters)

4. Construction and diagonalization of the energy matrices

Traditional approach:

Two-body matrix elements treated as free parametersTwo-body matrix elements treated as free parameters Empirical effective interactions containing adjustable Empirical effective interactions containing adjustable

parametersparameters [e.g., s-d shell nuclei, B. A. Brown and B. H. Wildenthal, Ann. Rev. Nucl. Part. Sci.38, 28(1988)]

defined within a reduced model space and acting only between valence nucleons


... VVTH NNNNN

ααα ψEHψ Ab-initio calculations:

Green’s function Montecarlo method, no-core shell model, coupled cluster method, UCOM

(three-nucleon interactions have been also taken into account)

nuclear properties, such as binding and excitation energies, are calculated directly from first principles of quantum mechanics, using an appropriate computational scheme

Challenge for nuclear physics: understand the properties of nuclei starting from the forces among nucleons


Realistic shell-model calculations:We start from

PΨE)P ΨVP(HPΨPH eff0eff

10NNNN HHU)(VU)(TVTH

ααα ψEHψ where U is a one-body auxiliary potential introduced to define a convenient single-paticle basis

and define the effective shell-model Hamiltonian

eff0 VHH

through the model-space Schrödinger equation

where the E and the corresponding are required to be a subset of the eigenvalues and eigenvectors of the original Hamiltonian

The P operator projects into the chosen model space


all modern NN potentials fit equally well the deuteron properties and the NN scattering data up the inelastic

threshold 2/Ndata ~ 1

● Choice of the nucleon-nucleon potential

Derivation of Veff

NoteNote these potentials cannot be used directly in the derivation of Vthese potentials cannot be used directly in the derivation of Veff eff

due to their strong short-range repulsion, but adue to their strong short-range repulsion, but a

• • Renormalization procedure is neededRenormalization procedure is needed

CD-Bonn, Argonne V18 , Chiral potentials,…


Renormalization of the NN interaction

Traditional approach: Brueckner G-matrix method

Vlow-k 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)S. Bogner, T.T.S. Kuo, A. Schwenk, Phys. Rep. 386, 1 (2003).L. Coraggio et al, Prog. Part. Nucl. Phys. 62 (2009) 135

Λk


Vlow-k approach

Derived from the original VNN by integrating out the high-momentum components of the original VNN potential down to the

cutoff momentum

Vlow-k decouples high- and low-energy degrees of freedom

preserves the physics of the original NN interaction

the deuteron binding energy

scattering phase-shifts up to the cutoff momentum ΛHow to contruct Vlow-k?

Effective interaction technique based on the the Lee-Suzuki similariry transformation (Prog. Theor Phys 64 (1980) 2091)

low-momentum spaceQ complementary space

X similarity transformation

Decoupling equation solved by the iterative procedure proposed by Andreozzi (Phys Rev. C 54 (1996) 684)

T- V

0Q

HXX

klow

1


Features of Vlow-k

real effective potential in the momentum space (indipendent from the starting energy or from the model space, as instead the case of the G matrix defined in the nuclear medium)

eliminates sources of non-perturbative behavior can be used directly in nuclear structure calculations

gives an approximately unique representation of the NN potential for 2 fm-1 ELab 350 MeV - Vlow-k’s extracted from various phase-shift equivalent potentials are very similar to each other

NoteNote VVlow-k low-k is developed for the two-body systemis developed for the two-body system for A>2 the low-energy observables are not the samefor A>2 the low-energy observables are not the same

of the original NN potential of the original NN potential and depend (to a certain extent) on the value of and depend (to a certain extent) on the value of

This may be removed complementing the two-body Vlow-k with three- and higher-body components


+ folded diagram method

1i

ieff FQV ˆ

collection of irreducible valence-linked diagrams with Vlow-k

replacing VNN in the interaction vertices

Fi

i-folded diagrams (expressed in terms of derivatives)

T.T.S. Kuo and E. Osnes, Lecture Notes in Physics, vol 364 (1990)L. Coraggio et al, Prog. Part. Nucl. Phys. 62 (2009) 135

Veff ,constructed for two valence particles, is defined

-in the nuclear medium

-in a subspace of the Hilbert space

accounts perturbatively for

• configurations excluded from the chosen model space

• excitations of the core nucleons

developed within the framework of the time-dependent perturbative approach by Kuo and co-workers


2-body diagrams up to 2nd order:V V1p1h V2p V2p2h

1-body diagrams up to 2nd order S-box

Calculation of : • inclusion of diagrams up to a finite order in the interaction • truncation of the intermediate-state summation

Sum of the folded series by the Lee-Suzuki method [Prog. Theor. Phys. 64, 2091 (1980)]

+ + …

Construction of Veff

NoteNote

NoteNote

(2)eff

(1)effeff VVV

Diagramatic expression of the

(1)eff0 VH single-particle energies

TB component of the shell-model Hamiltoniam


Results


neutrons

pro

tons

nuclei beyond the N=82 shell

n-rich nuclei

n-rich Ca isotopes

n-rich C isotopes

magic nature of 132Sn?

N=34 shell closure?

location of the neutron drip line?


132132Sn regionSn region132132Sn regionSn region

CD-Bonn + Vlow-k

: second-order calculationSingle-particle energies from the experimental spectra of 133Sb and 133Sn

• 132Sn coreValence neutron levels: 1f7/2, 2p3/2, 0h9/2, 2p1/2, 1f5/2, 0i13/2Valence proton levels: 0g7/2, 1d5/2, 1d3/2, 0h11/2, 2s1/2

n-rich Ca isotopesn-rich Ca isotopesn-rich Ca isotopesn-rich Ca isotopes

CD-Bonn + Vlow-k

: third-order calculationSingle-neutron energies from a fit to exp energies of 47Ca and 49Ca

• 40Ca coreValence neutron levels: 0f7/2, 0f5/2, 1p3/2, 1p1/2

n-rich C isotopesn-rich C isotopesn-rich C isotopesn-rich C isotopes

N3LOW [chiral potential with a sharp momentum cutoff at 2.1 fm-1] : third-order calculationTheoretical single-neutron energies

• 14C coreValence neutron levels: 0d5/2, 0d3/2, 1s1/2

Input of our calculations


Theory

B(E2;0+ 2+) = 0.033 e2b2

134Sn Coulex (Oak Ridge)

B(E2;0+ 2+) = 0.029(4) e2b2

Theory

0.726

134Sn

Expt.

from the f7/2p1/2 configurationtheir location below the 8+

due to the new position of

the p1/2 level measured @

ORNLp1/2=1.36 MeV

(old value: 1.66 MeV)[Nature 465 (2010)]


0

0,5

1

1,5

2

0+ 2+ 4+ 6+

E(M

eV

)

134Sn Expt 134Sn Calc

134Sn 132Sn + 2

0

0,5

1

1,5

2

0+ 2+ 4+ 6+ 8+

E(M

eV

)

210Pb Expt 210Pb Calc

(f7/2)2 multiplet

210Pb 208Pb + 2 (g9/2)2

multiplet

J

J


0

0,5

1

1,5

2

0+ 2+ 4+ 6+ 8+

E(M

eV

)136Sn Calc 212Pb Calc 212Pb Expt

J

136Sn 132Sn + 4

212Pb 208Pb + 4


A 134Sn 135Sn 136Sn 137SnBE Calcrelative to 132Sn

5.91 8.30 11.86 14.18

BE Exptrelative to 132Sn

5.92*

BE/A Calc 8.27 8.23 8.20 8.15

N/Z 1.70 1.72 1.74 1.68

124Sn(stable) with N/Z=1.48 BE/A=8.46

* M. Dworschak et al. Phys. Rev. Lett. 100, (2008) 072501Old value (Fogelberg et al., 1999): 6.365 MeV

neutron shell gap at N= 82 restored


B(E2;42 ) = 1.64 W.u.

B(E2;64) = 0.81 W.u.

B(E2;222) = 0.34 W.u.

B(E2;224) = 0.22 W.u.

Q(2) = -1.3 efm2

µ(2) = -0.56 nm

134Sn (Theoretical predictions)


136Sb is at present the most exotic open-shell nucleus beyond 132Sn for whichinformation exists on excited states

136Sb

Expt Theory


L. Coraggio et al. Phys. Rev. C 80, (2009) 044311

J.J. Valiente Dobón et al. PRL 102, 242502 (2009)

Expt Theory Expt Theory

50Ca 52Ca


Ca isotopes - Ground-state energy per valence neutron

A


A * M. Honma et al. EPJ A 25, 499 (2005)

*

no shell gap at N=34

Ca isotopesExcitation energies of the J =21

+ states

Effective single particle energies of the f5/2 and p1/2 levels (relative to the p3/2 level )


J 1)J(2JjjVjj1)J(2

jjV ;

jjV

j jn

jεESPE(j)

f0 5/2

34Np1 1/2

f1 7/2

p1 3/2

-427 keV*


C isotopes from 16C to 24C – ground-state energy (relative to 14C )

*to reproduce the exp g.s. energy of 15C relative to 14C Egs(calc)=-0.79 ; Egs(exp)= -1.22 MeV

22C is the last bound isotopeK. Tanaka et al PRL 104, 062701 (2010)

S2n(evaluation)=420 keV S2n(calc)=601 keV

L. Coraggio et al. PR C 81, 064303 (2010)

•

•

•

• •

•


C isotopesExcitation energies of the J =21

+ states

ES

PE (

MeV

)

N

no subshell closure at N=14g.s. in 20C:14% of (d5/2)6 configuration


Reliability of “realistic shell-model calculations” for light heavy nuclei

This outcome gives confidence in its predictive power, and may stimulate and be helpful to future experiments.

Three-body forces seem to contribute mainly to the absolute energy of the single-particle. Role of three-body forces needs futher investigations

It is of key importance to gain more experimental information

Summary

neutron-rich nuclei within a realistic shell-model approach angela gargano napoli

Documents

chosen model space

reduced model space

core shell model

properties of nuclei

conterpart nuclei

bare nn potential

neutronrich nuclei northeast

sd shell nuclei