Skyrme EDF and tensor terms K. Bennaceur Introduction ...lea-colliga/public-docs/2009MeetingParis/bennaceur… · and tensor terms K. Bennaceur Introduction Skyrme functional Parametrizations

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