Nuclear Pairing and Pairing Vibrations in Stable and

Neutron-Rich Nuclei

Marcella Grasso

FIDIPRO-EFES Workshop, 20-24 April 2009

• Model and pairing interaction (different surface/volume mixing). Fitting procedure (two-neutron separation energies)

• Results for nuclear matter? Validity of local density approximation?

• New constraints? Pairing Vibrations

• How the response function and the transition densities are affected by the choice of the pairing interaction

• Conclusions and perspectives

Pairing in HFB mean field framework

• Gogny interaction (same interaction in the particle-hole and in the pairing channel)

• With Skyrme interactions: different strategies

• Non empirical pairing energy density functional derived at lowest order in the two-nucleon vacuum interaction including Coulomb (many-body perturbation theory) (Lesinski, et al., arXiv:0809.2895 [nucl-th]

• Derived from a microscopic nucleon-nucleon interaction. If we assume the validity of Local Density Approximation: fit to reproduce the gap in symmetric and neutron matter (Margueron, et al. Phys. Rev. C 77, 054309 (2008))

• Also dependence on the isovector density (Margueron)

• Empirical pairing energy density functional with constraints from nuclei: odd-even mass staggering, separation energies

• Also dependence on the isovector density (Yamagami, et al., arXiv:0812.3197 [nucl-th])

• In the context of empirical pairing functional: New constraints?

Model: Skyrme-HFB (SLy4) with zero-range pairing interaction

and dependence on the isoscalar density

210

021 1 rrrxVrrV

0 = 0.16 fm-3

= 1

Ecutoff = 60 MeV

Values for x: 0.35, 0.5, 0.65 (MIXED INTERACTIONS)

1 (SURFACE INTERACTION)

V0 is adjusted to reproduce the two- neutron separation energy

-1200

-1150

-1100

-1050

-1000

-950

E (

MeV

) ExpTh.

Sn isotopes

112 120 128 136 144 152 160 168 176A

02468

1012141618

S2n

(Mev

)

Last spherical with HFB Gogny : A=146Last bound with HFB Gogny : A=160

x V0 (MeV fm -1)

0.35 -285

0.5 -340

0.65 -390

1 -670

Some results for 124Sn and 136Sn

0,04

0,08

0,12

0,16

Den

siti

es

124Sn

136Sn

0 1 2 3 4 5 6 7 8 9 10r (fm)

-4

-3,2

-2,4

-1,6

-0,8

0

Pai

ring f

ield

Surface interactionMixed interaction

Khan, Grasso, and Margueron, in preparation

124Sn

Surface: x = 1

Mixed: x = 0.35

Pairing gap in symmetric nuclear matter for three values of x

Khan, Grasso, and Margueron, in preparation

0 0,1 0,2

ρ (fm-3

)

0

2

4

6

8

pair

ing

gap

Δ (M

eV)

0 0,1 0,2

ρ (fm-3

)

-600

-400

-200

0

V0g(

ρ)

(MeV

.fm

3 )

η = 0.35η = 0.65η = 1

x = 0.35

x = 0.65

x = 1

0 0,04 0,08 0,12

ρ (fm-3

)

0

2

4

6

8

pair

ing

gap

Δ (M

eV)

SurfaceMixed

Symmetric nuclear matter

Check of the validity of Local Density Approximation

124Sn

matter

matter

124Sn

In the case of a mixed pairing interaction the LDA is a good approximation at the surface

region (low density)

RNrRrrRW /,),( 22

22 ,)( rRdrrRN

This is qualitatively confirmed by

Pillet et al. results obtained with Gogny (locally normalized pairing tensor)

Pillet et al. PRC 76 024310 (2007)

2121 rrR

21 rrr

How to disentangle between surface and mixed interactions

in nuclei?

If we could answer this question we would also know if and in which cases LDA is a

reasonable approximation for pairing

Try pairing vibrations as additional constraints?

QRPA response function for 136Sn in the two- neutron 0+

addition mode

Pure surface case: solid line

X = 0.65 ->dotted line

x = 0.35 -> dashed-dotted line

Khan, Grasso, and Margueron, in preparation

More neutron-rich case: 136SnSurface interaction

Mixed interaction

Solid line: surface

Dotted line: x= 0.65

Dashed-dotted line: x=0.35Khan, Grasso, and Margueron, in preparation

Neutron transition density for the first two peaks for the mixed interaction

Khan, Grasso, and Margueron, in preparation

Neutron transition density for the first three peaks for the surface interaction

Conclusions• Choice of the model: empirical pairing

energy density functional. Fit on the two- neutron separation energy (Sn isotopes)

• Check on nuclear matter: validity of LDA?

• Additional constraint for the pairing interaction? Pairing vibrations?

• Response functions and transition densities in 124Sn and 136Sn