View
10
Download
0
Category
Preview:
Citation preview
Skyrme EDF
and tensor terms
K. Bennaceur
Introduction
Skyrme
functional
Parametrizations
Spherical nuclei
Deformed nuclei
Instabilities
Conclusion
Tensor couplings in the Skyrme Energy DensityFunctional
K. Bennaceur
Université de Lyon, Institut de Physique Nucléaire de Lyon,CNRS–IN2P3 / Université Claude Bernard Lyon 1
November 23-24, 2009
Skyrme EDF
and tensor terms
K. Bennaceur
Introduction
Skyrme
functional
Parametrizations
Spherical nuclei
Deformed nuclei
Instabilities
Conclusion
Outline
The (standard) Skyrme functional (SLyX)
Tensor interaction, tensor couplings
Results for spherical nuclei
Effects on deformation
Spin instabilities
Conclusion
Skyrme EDF
and tensor terms
K. Bennaceur
Introduction
Skyrme
functional
Parametrizations
Spherical nuclei
Deformed nuclei
Instabilities
Conclusion
The standard Skyrme functional
Skyrme effective force
Veff = t0(
1+x0Pσ)
δ local
+t1
2
(
1+x1Pσ)(
k′2δ +δk2)
+ t2(
1+x2Pσ)
k′ ·δk non local
+t3
6
(
1+x3Pσ)
ρα δ dens. dep.
+ iW0 σ ·[
k′×δk]
spin-orbit
Skyrme Energy Density Functional :
H = T +Veff → → E = 〈Φ|H |Φ〉 =∫
E [ρ,τ,J]dr
Functional Energydensity
ρ(rσ ,r′σ ′)+∇+σ +zero range ⇒ ρ(r) , τ(r) , J(r) : local densities
9 or 10 parameters to fit
Skyrme EDF
and tensor terms
K. Bennaceur
Introduction
Skyrme
functional
Parametrizations
Spherical nuclei
Deformed nuclei
Instabilities
Conclusion
Effective force Vs. Functional
Time even part of the energy density (witout isospin index)
E = C ρ (ρ)ρ2 +C τ ρτ +C ∆ρ ρ∆ρ +CJJ
2 +C ∇J ρ∇ ·J
Force:
The coupling constants C are entirely determined by theparameters ti , xi and W0
Generates the “Hartree” and “exchange” part of the energy Systematic ways to go beyond (correlation energy) with
symmetry breaking(+restoration) and/or GCM The time odd part of the functional is determined... ... but not well constrained, part of it can be problematic
Functional:
More flexible Problematic terms can be dropped Time odd part ? Not so easy to generate the “exchange and correlation”
part of the energy
Skyrme EDF
and tensor terms
K. Bennaceur
Introduction
Skyrme
functional
Parametrizations
Spherical nuclei
Deformed nuclei
Instabilities
Conclusion
Effective tensor interaction
Why a tensor interaction ?
vT = 12 te
[
3(
σ1 ·k′)(
σ2 ·k′)
− (σ1 ·σ 2)k′2]
δ +c.c.
+ to
[
3(
σ1 ·k′)
δ (σ 2 ·k)− (σ1 ·σ 2)k′δ ·k]
⇒ Simplest term quadratic in k and σ⇒ Important part of the NN interaction⇒ Two additional parameters
Without tensor terms
E = C ρ (ρ)ρ2 +C τ ρτ +C ∆ρ ρ∆ρ +CJJ
2 +C ∇J ρ∇ ·J
+ time odd part ∝ s ·T , s ·F , s ·∆s , (∇ ·s)2
C τ , C ∆ρ and CJ are determined by t1, x1, t2 and x2
With tensor terms
C τ and C ∆ρ are determined by t1, x1, t2 and x2
CJ is determined by t1, x1, t2, x2, te, to
No new terms in the functional... at least for spherical nuclei.
Skyrme EDF
and tensor terms
K. Bennaceur
Introduction
Skyrme
functional
Parametrizations
Spherical nuclei
Deformed nuclei
Instabilities
Conclusion
Effective interaction with tensor terms
Spherical nuclei: J ≡ J
v = vloc. + vnonloc. + vs.o. + vtens.
E ∝ ρ2 ρτ ρ∇ ·J J2
ρ∆ρ
• Spin-orbit field: W =δE
δJ
Deformed nuclei: J ≡ Jµν , with µ , ν = x , y , z
vnonloc. vtens
J2 ∝ ∑µν
JµνJµν ∑µ
J2µµ ∑
µνJµνJν µ
Skyrme EDF
and tensor terms
K. Bennaceur
Introduction
Skyrme
functional
Parametrizations
Spherical nuclei
Deformed nuclei
Instabilities
Conclusion
Local densities – Spin saturation
ρ and τ change rather smoothly with N and Z
J depends on the spin saturation
= occupation of spin partner states j = ℓ+1/2 and j = ℓ−1/2
Spin-orbit current density (radial)
Jq(r) =1
4πr3 ∑n,j,ℓ
(2j +1)v2njℓ
[
j(j +1)− ℓ(ℓ+1)− 34
]
ψ2njℓ(r)
expl:
40Ca (spin saturated), 48Ca (spin unsaturated)
J2n acts on the ν s.p.e. with the ν filling
Jn ·Jp acts on the π s.p.e. with the ν filling
Skyrme EDF
and tensor terms
K. Bennaceur
Introduction
Skyrme
functional
Parametrizations
Spherical nuclei
Deformed nuclei
Instabilities
Conclusion
Fitting procedure
Saturation point of infinite nuclear matter(ρsat, E/A, K∞, m∗, aI , κv)
E.o.S. of infinite neutron matter
Masses: 40−48Ca, 56Ni, 90Zr, 100−132Sn, 208Pb.
Radii: 40−48Ca, 56Ni, 90Zr, 132Sn, 208Pb.
Spin-orbit splitting: 3p neutron in 208Pb.
Constraint x2 = −1 was released
almost the same recipe as for the SLy forces.
Phenomenology + microscopic inputs ⇒ some predictive power
(at least we hope...)
Skyrme EDF
and tensor terms
K. Bennaceur
Introduction
Skyrme
functional
Parametrizations
Spherical nuclei
Deformed nuclei
Instabilities
Conclusion
Coupling constants for the tensor terms
EJ ∝ CJ0 J2 +CJ
1 (Jn −Jp)2 ∝1
2α ∑
q
J2q +β Jn ·Jp
. ...
. .
.
Zone of « reasonable » parameters
(Stancu, Brink, Flocard ’77)
Existing forces
(from central parts)
CJ0 = 1
2 (α +β )
CJ1 = 1
2 (α −β )
Skyrme EDF
and tensor terms
K. Bennaceur
Introduction
Skyrme
functional
Parametrizations
Spherical nuclei
Deformed nuclei
Instabilities
Conclusion
Parametrizations
-150
-120
-90
-60
-30
0
30
60
90
120
150
-60 -30 0 30 60 90 120 150 180 210 240 270
CJ 1
[Mev
fm5 ]
CJ0 [Mev fm5]
T11
T12
T13
T14
T15
T16
T21
T22
T23
T24
T25
T26
T31
T32
T33
T34
T35
T36
T41
T42
T43
T44
T45
T46
T51
T52
T53
T54
T55
T56
T61
T62
T63
T64
T65
T66SLy4
SLy5
SkP
SkO’
BSk9T6Zσ
Skxta
Skxtb
Colo
Brink
T22: no tensor terms
( spherical nuclei )
∼ SLy4
36 parametrizations TIJ with
α = 60(J −2) MeV fm5
β = 60(I −2) MeV fm5
Skyrme EDF
and tensor terms
K. Bennaceur
Introduction
Skyrme
functional
Parametrizations
Spherical nuclei
Deformed nuclei
Instabilities
Conclusion
Spin-orbit splittings
0
0.2
0.4
0.6
0.8
(∆ε t
h - ∆
ε exp
) / ∆
ε exp
16O
ν1p π1p90Zr
π2p132Sn
ν2d π2d208Pb
ν3p π2d
T62T64T66
0
0.2
0.4
0.6
0.8
(∆ε t
h - ∆
ε exp
) / ∆
ε exp
T42T44T46
0
0.2
0.4
0.6
0.8
(∆ε t
h - ∆
ε exp
) / ∆
ε exp
T22T24T26
Splittings of doublets withℓ6 2 (n > 2) larger than theexperimental values
also true for the ν3p
states in 208Pb includedin the fit !
Splittings of states with ℓ ≥ 3(n = 1) underestimated
Skyrme EDF
and tensor terms
K. Bennaceur
Introduction
Skyrme
functional
Parametrizations
Spherical nuclei
Deformed nuclei
Instabilities
Conclusion
Spin-orbit splittings
-0.4
-0.2
0
0.2
0.4
(∆ε t
h - ∆
ε exp
) / ∆
ε exp
56Ni
ν1f π1f132Sn
ν1h π1g208Pb
ν1i π1h
T62T64T66
-0.4
-0.2
0
0.2
0.4
(∆ε t
h - ∆
ε exp
) / ∆
ε exp
T42T44T46
-0.4
-0.2
0
0.2
0.4
(∆ε t
h - ∆
ε exp
) / ∆
ε exp
T22T24T26
Splittings of doublets withℓ6 2 (n > 2) larger than theexperimental values
also true for the ν3p
states in 208Pb includedin the fit !
Splittings of states with ℓ ≥ 3(n = 1) underestimated
Skyrme EDF
and tensor terms
K. Bennaceur
Introduction
Skyrme
functional
Parametrizations
Spherical nuclei
Deformed nuclei
Instabilities
Conclusion
Centroids of the spin-orbit doublets – intruders
Are the spin-orbit splittings determining all the spectroscopy ?
132Sn: ν 1h, π 1g
−14
−12
−10
−8
−6
−4
−2
0
ε i [M
eV] 132Sn, ν
1h centroid
Exp. T22
82
T42 T62 T24 T44 T64 T26 T46 T66
(7/2+)(5/2+)(1/2+)
(11/2−)(3/2+)
(7/2−)(3/2−)(9/2−)(1/2−)
1g7/2
2d5/2
3s1/2
2d3/2
1h11/2
2f7/2
3p3/2
1h9/2
3p1/2
2f5/2
208Pb: ν 1i , π 1h
Centroids are too high compared to the positions estimated fromdata. Property related with the central part of the potential.
Skyrme EDF
and tensor terms
K. Bennaceur
Introduction
Skyrme
functional
Parametrizations
Spherical nuclei
Deformed nuclei
Instabilities
Conclusion
Centroids of the spin-orbit doublets – intruders
Are the spin-orbit splittings determining all the spectroscopy ?
132Sn: ν 1h, π 1g
−14
−12
−10
−8
−6
−4
−2
0
ε i [M
eV] 132Sn, ν
1h centroid
Exp. T22
82
T42 T62 T24 T44 T64 T26 T46 T66
(7/2+)(5/2+)(1/2+)
(11/2−)(3/2+)
(7/2−)(3/2−)(9/2−)(1/2−)
1g7/2
2d5/2
3s1/2
2d3/2
1h11/2
2f7/2
3p3/2
1h9/2
3p1/2
2f5/2
208Pb: ν 1i , π 1h
Centroids are too high compared to the positions estimated fromdata. Property related with the central part of the potential.
Skyrme EDF
and tensor terms
K. Bennaceur
Introduction
Skyrme
functional
Parametrizations
Spherical nuclei
Deformed nuclei
Instabilities
Conclusion
Centroids of the spin-orbit doublets – intruders
Are the spin-orbit splittings determining all the spectroscopy ?
132Sn: ν 1h, π 1g
−18
−16
−14
−12
−10
−8
−6
−4
ε i [M
eV]
132Sn, π1g centroid
Exp. T22
50
T42 T62 T24 T44 T64 T26 T46 T66
(1/2−)(9/2+)
(7/2+)(5/2+)
(3/2+)(11/2−)
2p1/2
1g9/2
1g7/2
2d5/2
2d3/2
3s1/2
1h11/2
208Pb: ν 1i , π 1h
Centroids are too high compared to the positions estimated fromdata. Property related with the central part of the potential.
Skyrme EDF
and tensor terms
K. Bennaceur
Introduction
Skyrme
functional
Parametrizations
Spherical nuclei
Deformed nuclei
Instabilities
Conclusion
Centroids of the spin-orbit doublets – intruders
Are the spin-orbit splittings determining all the spectroscopy ?
132Sn: ν 1h, π 1g
−10
−8
−6
−4
−2
0
ε i [M
eV]
208Pb, ν1i centroid
Exp. T22
126
T42 T62 T24 T44 T64 T26 T46 T66
13/2+3/2−5/2−1/2−
9/2+11/2+15/2−5/2+1/2+
3p3/2
2f5/2
1i13/2
3p1/2
2g9/2
1i11/2
3d5/2
4s1/2
1j15/2
2g7/2
3d3/2
208Pb: ν 1i , π 1h
Centroids are too high compared to the positions estimated fromdata. Property related with the central part of the potential.
Skyrme EDF
and tensor terms
K. Bennaceur
Introduction
Skyrme
functional
Parametrizations
Spherical nuclei
Deformed nuclei
Instabilities
Conclusion
Centroids of the spin-orbit doublets – intruders
Are the spin-orbit splittings determining all the spectroscopy ?
132Sn: ν 1h, π 1g
−12
−10
−8
−6
−4
−2
0
ε i [M
eV]
208Pb, π1h centroid
Exp. T22
82
T42 T62 T24 T44 T64 T26 T46 T66
5/2+11/2−3/2+1/2+
9/2−7/2−
2d5/2
2d3/2
3s1/2
1h11/2
1h9/2
2f7/2
1i13/2
2f5/2
208Pb: ν 1i , π 1h
Centroids are too high compared to the positions estimated fromdata. Property related with the central part of the potential.
Skyrme EDF
and tensor terms
K. Bennaceur
Introduction
Skyrme
functional
Parametrizations
Spherical nuclei
Deformed nuclei
Instabilities
Conclusion
Results for deformed nuclei
• T22, T26, T44, T62:refitted interactions
• T22:no tensor at sphericity
• SLy4:no tensor couplings
• SLy5:non local tensor couplings
• SLy5+T:tensor added, no refit
• SLy4T:tensor added, no refit
• SLy4Tmin:tensor added, refit
TIJ: Lesinski et al., PRC 76, 014312, SLy5+T: Colò et al., PLB 646, 227.
SLy4T, SLy4Tmin: Zalewski et al., PRC 77, 024316.
Skyrme EDF
and tensor terms
K. Bennaceur
Introduction
Skyrme
functional
Parametrizations
Spherical nuclei
Deformed nuclei
Instabilities
Conclusion
Linear response formalism
Several instabilities often experienced with the Skyrme forces
Ferromagnetic instabilities
* spin: polarization n ↑, p ↑* spin-isospin: polarization n ↑, p ↓
Isospin instabilities: neutrons-protons segregation
Response of the system to a perturbation described by:
Q(α) = ∑a eiq·ra Θ(α)a ,
Θssa = 1a , Θvs
a = σ a, Θsva =~τa , Θvv
a = σa~τa
The response fonctions are defined by(Cf. C. Garcia–Recio et al., Ann. of Phys. 214 (1992) 293–340)
χ(α)(ω,q)=1
Ω ∑n
|〈n|Q(α)|0〉|2[
1
ω −En0 + iη−
1
ω +En0 − iη
]
Skyrme EDF
and tensor terms
K. Bennaceur
Introduction
Skyrme
functional
Parametrizations
Spherical nuclei
Deformed nuclei
Instabilities
Conclusion
Instabilities and refit
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0 0.5 1 1.5 2 2.5 3 3.5 4
S=0,T=1sat. density
k[
fm−1
]
ρ[
fm−
3]
Skyrme EDF
and tensor terms
K. Bennaceur
Introduction
Skyrme
functional
Parametrizations
Spherical nuclei
Deformed nuclei
Instabilities
Conclusion
Instabilities and refit
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0 0.5 1 1.5 2 2.5 3 3.5 4
S=0,T=1sat. density
k[
fm−1
]
ρ[
fm−
3]
at ρ ∼ 0.3 fm−3
appearance of domains
(S =0,T =1) with size ∼ 2π2.7
Skyrme EDF
and tensor terms
K. Bennaceur
Introduction
Skyrme
functional
Parametrizations
Spherical nuclei
Deformed nuclei
Instabilities
Conclusion
Instabilities and refit
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0 0.5 1 1.5 2 2.5 3 3.5 4
S=0,T=0S=0,T=1
S=1, T=0, M=0S=1, T=0, M=1S=1, T=1, M=0S=1, T=1, M=1
sat. density
T22
k[
fm−1
]
ρ[
fm−
3]
“Anything that can go wrong will go wrong”, Murphy’s law.
Skyrme EDF
and tensor terms
K. Bennaceur
Introduction
Skyrme
functional
Parametrizations
Spherical nuclei
Deformed nuclei
Instabilities
Conclusion
Instabilities and refit
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0 0.5 1 1.5 2 2.5 3 3.5 4
S=0,T=0S=0,T=1
S=1, T=0, M=0S=1, T=0, M=1S=1, T=1, M=0S=1, T=1, M=1
sat. density
Refit
k[
fm−1
]
ρ[
fm−
3]
“Anything that can go wrong will go wrong”, Murphy’s law.
Skyrme EDF
and tensor terms
K. Bennaceur
Introduction
Skyrme
functional
Parametrizations
Spherical nuclei
Deformed nuclei
Instabilities
Conclusion
How large is the parameters space ?
SLy5t0 t1 t2 t3 W0
-2484.880 483.130 -549.400 13763.000 126.000x0 x1 x2 x3 γ
0.778 -0.328 -1.000 1.267 1/6
Refit with C∆s0 < 20
t0 t1 t2 t3 W0
-2593.544 432.018 -374.898 15004.204 118.221x0 x1 x2 x3 γ
0.879 0.135 -0.826 1.202 1/6
“Dangerous” coefficients:
C∆ρ0 = −76.5 → −70.7, C
∆ρ1 = 16.4 → 29.6
C ∆s0 = 46.1 → 18.6, C ∆s
1 = 14.1 → 14.4
ρsat = 0.160 → 0.161 fm−3 aV = 16.0 → 15.9 MeV K∞ = 230 → 222 MeV
m∗/m = 0.697 → 0.757 aI = 32.0 → 28.4 MeV κv = 0.25 → 0.47
Perfectly stableCorrect masses for (spherical) nuclei
Skyrme EDF
and tensor terms
K. Bennaceur
Introduction
Skyrme
functional
Parametrizations
Spherical nuclei
Deformed nuclei
Instabilities
Conclusion
Conclusion
Evolution of spin-orbit splittings with mass is wrong and cannot be corrected by a tensor interaction
⇒ Different spin-orbit term ?
Spin-orbit doublet centroids do not evolve correctly with ℓ
⇒ Central part of the interaction ? D wave ?
Strong constraints on the tensor interaction at sphericity canlead to unwanted behaviors with deformation
Spin instabilities
⇒ Must be taken into account during the fit procedure
Skyrme EDF
and tensor terms
K. Bennaceur
Introduction
Skyrme
functional
Parametrizations
Spherical nuclei
Deformed nuclei
Instabilities
Conclusion
Work done in collaboration with
• M. Bender CENBG
• D. Davesne IPNL
• T. Duguet IRFU/SPhN
• P.-H. Heenen ULB
• T. Lesinski ORNL/UTK
• M. Martini IPNL/CEA-DIF
• J. Meyer IPNL
• Spherical nuclei, PRC 76, 014312 (2007)
• Linear response, PRC 80, 024314 (2009)
• Deformed nuclei, PRC in print
Recommended