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