Tensor K. Bennaceur Introduction Skyrme EDF Constraints from nuclear matter Constraints from spectroscopy Hunting for instabilities Extentions Beyond mean-field Conclusion New terms and new constraints for the Skyrme Energy Density Functional K. Bennaceur 1 1 Université de Lyon, Institut de Physique Nucléaire de Lyon, CNRS–IN2P3 / Université Claude Bernard Lyon 1 Artic FIDIPRO-EFES Workshop, Finland, April 20–24, 2009
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Tensor
K. Bennaceur
Introduction
Skyrme EDF
Constraints from
nuclear matter
Constraints from
spectroscopy
Hunting for
instabilities
Extentions
Beyond
mean-field
Conclusion
New terms and new constraints for the Skyrme EnergyDensity Functional
K. Bennaceur1
1Université de Lyon, Institut de Physique Nucléaire de Lyon,CNRS–IN2P3 / Université Claude Bernard Lyon 1
Artic FIDIPRO-EFES Workshop, Finland, April 20–24, 2009
Tensor
K. Bennaceur
Introduction
Skyrme EDF
Constraints from
nuclear matter
Constraints from
spectroscopy
Hunting for
instabilities
Extentions
Beyond
mean-field
Conclusion
Outline
Introduction: the (standard) Skyrme (SLyX) EDF
Constraints from microscopic calculations:effective masses and spin-isospin content
Spectroscopic properties improvement with tensor couplings
Hunting for instabilities
Possible extensions for the functional
Practical constraints: EDF for beyond mean field calculations
Conclusion
Tensor
K. Bennaceur
Introduction
Skyrme EDF
Constraints from
nuclear matter
Constraints from
spectroscopy
Hunting for
instabilities
Extentions
Beyond
mean-field
Conclusion
Skyrme Hartree-Fock formalism
Skyrme effective force
Veff = t0(
1+x0Pσ)
δ local
+t1
2
(
1+x1Pσ)(
k′2 δ +δ k2)
non local
+ t2(
1+x2Pσ)
k′ ·δ k non local
+t3
6
(
1+x3Pσ)
ρα δ density dep.
+ iW0 σ ·[
k′×δ k]
spin-orbit
Skyrme Energy Density Functional :
E =
∫
E [ρ ,τ,J]dr
functional of the local density ρ(rσ ,rσ ′) ,
with ρ(rσq,r′σ ′q′) = ∑i6εF
ϕ∗i (rσq)ϕi(r
′σ ′q′)
9 or 10 parameters to fit
Tensor
K. Bennaceur
Introduction
Skyrme EDF
Constraints from
nuclear matter
Constraints from
spectroscopy
Hunting for
instabilities
Extentions
Beyond
mean-field
Conclusion
Skyrme Hartree-Fock formalism
Skyrme effective force
Veff = t0(
1+x0Pσ)
δ local
+t1
2
(
1+x1Pσ)(
k′2 δ +δ k2)
non local
+ t2(
1+x2Pσ)
k′ ·δ k non local
+t3
6
(
1+x3Pσ)
ρα δ density dep.
+ iW0 σ ·[
k′×δ k]
spin-orbit
Skyrme Energy Density Functional :
E =
∫
E [ρ ,τ,J]dr
functional of the local density ρ(rσ ,rσ ′) ,
with ρ(rσq,r′σ ′q′) = ∑i6εF
ϕ∗i (rσq)ϕi(r
′σ ′q′)
+ other terms if symmetry breaking
(deformation, rotation, pairing)
9 or 10 parameters to fit
Tensor
K. Bennaceur
Introduction
Skyrme EDF
Constraints from
nuclear matter
Constraints from
spectroscopy
Hunting for
instabilities
Extentions
Beyond
mean-field
Conclusion
Local Energy density functional (time even)
H = K +H0 +H3 +Heff +Hfin +Hso +Hsg +Hcoul
H0 =1
4t0
[
(2+x0)ρ20 − (2x0 +1)∑
q
ρ2q
]
= ∑T=0,1
CT [ρ0]ρ2T
H3 =1
24t3ρα
0
[
(2+x3)ρ20 − (2x3 +1)∑
q
ρ2q
]
Heff =1
8[t1(2+x1)+ t2(2+x2)]τ0ρ0 = ∑
T=0,1
C τT τT ρT
+1
8[t2(2x2 +1)− t1(2x1 +1)]∑
q
τqρq
Hfin =1
32[t2(2+x2)−3t1(2+x1)]ρ0∆ρ0 = ∑
T=0,1
C∆ρT ρT ∆ρT
+1
32[3t1(2x1 +1)+ t2(2x2 +1)]∑
q
ρq∆ρq
Hso = −W0
2
[
ρ0∇ ·J0 +∑q
ρq∇ ·Jq
]
= ∑T=0,1
C ∇JT ρT ∇ ·JT
Hsg = −t1x1 + t2x2
16J2
0 +t1 − t2
16 ∑q
J2q = ∑
T=0,1
CJT J2
T
Tensor
K. Bennaceur
Introduction
Skyrme EDF
Constraints from
nuclear matter
Constraints from
spectroscopy
Hunting for
instabilities
Extentions
Beyond
mean-field
Conclusion
Local Energy density functional (time even)
H = K +H0 +H3 +Heff +Hfin +Hso +Hsg +Hcoul
H0 =1
4t0
[
(2+x0)ρ20 − (2x0 +1)∑
q
ρ2q
]
= ∑T=0,1
CT [ρ0]ρ2T
H3 =1
24t3ρα
0
[
(2+x3)ρ20 − (2x3 +1)∑
q
ρ2q
]
Heff =1
8[t1(2+x1)+ t2(2+x2)]τ0ρ0 = ∑
T=0,1
C τT τT ρT
+1
8[t2(2x2 +1)− t1(2x1 +1)]∑
q
τqρq
Hfin =1
32[t2(2+x2)−3t1(2+x1)]ρ0∆ρ0 = ∑
T=0,1
C∆ρT ρT ∆ρT
+1
32[3t1(2x1 +1)+ t2(2x2 +1)]∑
q
ρq∆ρq
Hso = −W0
2
[
ρ0∇ ·J0 +∑q
ρq∇ ·Jq
]
= ∑T=0,1
C ∇JT ρT ∇ ·JT
Hsg = −t1x1 + t2x2
16J2
0 +t1 − t2
16 ∑q
J2q = ∑
T=0,1
CJT J2
T : SLy4
Tensor
K. Bennaceur
Introduction
Skyrme EDF
Constraints from
nuclear matter
Constraints from
spectroscopy
Hunting for
instabilities
Extentions
Beyond
mean-field
Conclusion
Local Energy density functional (time even)
H = K +H0 +H3 +Heff +Hfin +Hso +Hsg +Hcoul
H0 =1
4t0
[
(2+x0)ρ20 − (2x0 +1)∑
q
ρ2q
]
= ∑T=0,1
CT [ρ0]ρ2T
H3 =1
24t3ρα
0
[
(2+x3)ρ20 − (2x3 +1)∑
q
ρ2q
]
Heff =1
8[t1(2+x1)+ t2(2+x2)]τ0ρ0 = ∑
T=0,1
C τT τT ρT
+1
8[t2(2x2 +1)− t1(2x1 +1)]∑
q
τqρq
Hfin =1
32[t2(2+x2)−3t1(2+x1)]ρ0∆ρ0 = ∑
T=0,1
C∆ρT ρT ∆ρT
+1
32[3t1(2x1 +1)+ t2(2x2 +1)]∑
q
ρq∆ρq
Hso = −W0
2
[
ρ0∇ ·J0 +∑q
ρq∇ ·Jq
]
= ∑T=0,1
C ∇JT ρT ∇ ·JT
Hsg = −t1x1 + t2x2
16J2
0 +t1 − t2
16 ∑q
J2q = ∑
T=0,1
CJT J2
T : SLy4
C∇J0 6= 3C∇J
1 : SLy10
Tensor
K. Bennaceur
Introduction
Skyrme EDF
Constraints from
nuclear matter
Constraints from
spectroscopy
Hunting for
instabilities
Extentions
Beyond
mean-field
Conclusion
Cooking recipe
ρ0
EA
e0
ρ
ρ0, E/A
Tensor
K. Bennaceur
Introduction
Skyrme EDF
Constraints from
nuclear matter
Constraints from
spectroscopy
Hunting for
instabilities
Extentions
Beyond
mean-field
Conclusion
Cooking recipe
ρ0, E/A
Compression modulus K∞ = 9ρ20
d2
dρ2
E
A(ρ)
∣
∣
∣
∣
ρ=ρ0
Giant breathing mode E0;T=0
(J.P. Blaizot)
K∞ = 220±20 MeV
Tensor
K. Bennaceur
Introduction
Skyrme EDF
Constraints from
nuclear matter
Constraints from
spectroscopy
Hunting for
instabilities
Extentions
Beyond
mean-field
Conclusion
Cooking recipe
ρ0, E/A K∞ = 220±20 MeV
Isoscalar effective mass
(
m∗
m
)−1
s
= 1+1
8
m
h2 [3t1 + t2(5+4x2)]ρ0
Giant mode E2;T=0
m∗
m = 0.8±0.1
Tensor
K. Bennaceur
Introduction
Skyrme EDF
Constraints from
nuclear matter
Constraints from
spectroscopy
Hunting for
instabilities
Extentions
Beyond
mean-field
Conclusion
Cooking recipe
ρ0, E/A K∞ = 220±20 MeV m∗
m = 0.8±0.1
150 200 250 300 350 400K∞ (MeV)
0.6
0.7
0.8
0.9
1.0m
*/mα = 1/6α = 1/3α = 1/2α = 1
T6
SkM*
RATP
SIII
α = 1/6
Tensor
K. Bennaceur
Introduction
Skyrme EDF
Constraints from
nuclear matter
Constraints from
spectroscopy
Hunting for
instabilities
Extentions
Beyond
mean-field
Conclusion
Cooking recipe
ρ0, E/A K∞ = 220±20 MeV m∗
m = 0.8±0.1 α = 1/6
Yp=0
ρ
Yp=0.5EA
symmetry energy: aI = 12
d2
dρ2EA (ρ)
∣
∣
∣
I=0
isovector effective mass:(
m∗
m
)−1
v= 1+ m
4h2 [t1(2+x1)+ t2(2+x2)]ρ= 1+κv
κv ≡TRK enhancement factor for the m1 sum rule
aI ≃ 32 MeV κv ∼ 0.4 to 0.5
Tensor
K. Bennaceur
Introduction
Skyrme EDF
Constraints from
nuclear matter
Constraints from
spectroscopy
Hunting for
instabilities
Extentions
Beyond
mean-field
Conclusion
Cooking recipe
ρ0 , EA
, K∞ , m∗
m−→ ti ,xi ,α
aI , neutron matter , κv −→ ti ,xi
(surface energy), Landau parameters −→ ti ,xi(x2 = -1)
spherical magic nuclei: −→ ti ,xi ,W016O, 40−48Ca, 56Ni, 90Zr, 100−132Sn, 208Pb
(binding energies, charge radii, s.p.e.)
Successful but:
• predictive power might be limited→ constraints from microscopic calculations
• Spectroscopic properties could be improved
Tensor
K. Bennaceur
Introduction
Skyrme EDF
Constraints from
nuclear matter
Constraints from
spectroscopy
Hunting for
instabilities
Extentions
Beyond
mean-field
Conclusion
Effective masses
Many people agree with the fact that ∆m∗ =m∗
n −m∗p
m
∣
∣
∣
∣
I=1
> 0
SLy interactions give ∆m∗ =m∗
n −m∗p
m
∣
∣
∣
∣
I=1
< 0
Is it bad for neutron rich nuclei ? Can it be corrected ?
Three “test” interactions:f− : ∆m∗ < 0 (like SLy5) , f0 : ∆m∗ = 0 , f+ : ∆m∗ > 0
Results
-4
-3
-2
-1
0
1
156Sn, n
f- f0 f+
2f7/2
3p3/2
1h9/2
3p1/2
2f5/2
1i13/2
-28
-26
-24
-22
-20
-18
-16
156Sn, p
f- f0 f+ 1f5/2
2p3/2
2p1/2
1g9/2
1g7/2
2d5/2
0
0.5
1
1.5
0 10 20 30 40 50 60 70 80 90 100
∆ κ [M
eV]
N-Z
Sn
0
0.5
1
1.5
∆ κ [M
eV]
Pbf-f0f+
∆(5), exp.
Not a tremendous effect, but ...
Tensor
K. Bennaceur
Introduction
Skyrme EDF
Constraints from
nuclear matter
Constraints from
spectroscopy
Hunting for
instabilities
Extentions
Beyond
mean-field
Conclusion
Isospin instabilities
Instabilites d'isospin
0 50 100 150 200 250 300 350
Itérations
10-10
10-8
10-6
10-4
10-2|
E /
E |
1 / 3+ 2 / 3
Force 4
( 1 = -38,4 )
( 1 = -15,7 )0 2 4 6 8
r [fm]
0.00
0.05
0.10
[fm
-3]
0 2 4 6 8
r [fm]
0.00
0.05
0.10
[fm
-3]
0 2 4 6 8
r [fm]
0.00
0.05
0.10
[fm
-3]
Taille nie C1 = 364t1[2x1 + 1 164t2[2x2 + 1Intera tions ee tives et theories de hamp moyen : de la matiere nu leaire aux noyaux
•
•
The suspect is C∆ρ1 ρ1∆ρ1 when C
∆ρ1 is large and positive
Missing term ?
Systematic way to predict that ?...
Tensor
K. Bennaceur
Introduction
Skyrme EDF
Constraints from
nuclear matter
Constraints from
spectroscopy
Hunting for
instabilities
Extentions
Beyond
mean-field
Conclusion
Spin-Isospin content
Same equation of state for symmetric nuclear matter and pureneutron matter but different spin-isopsin content and...
-25-20-15-10-5 0 5
10
0 0.1 0.2 0.3
E/A
S,T
=0,
0 [M
eV]
ρ [fm-3] 0 0.1 0.2 0.3 0.4
-30-25-20-15-10-5
E/A
S,T
=0,
1 [M
eV]
ρ [fm-3]
SLy5
-30-20-10
0 10 20
E/A
S,T
=1,
0 [M
eV]
-10
-5
0
5
E/A
S,T
=1,
1 [M
eV]
f-f0f+
Wrong sign inS = 0, T = 0
andS = 1, T = 1
channels
BHF calc., Baldo (2006)
Only 12 t2(
1+x2Pσ)
k′ ·δ k acts in these two channels...
Tensor
K. Bennaceur
Introduction
Skyrme EDF
Constraints from
nuclear matter
Constraints from
spectroscopy
Hunting for
instabilities
Extentions
Beyond
mean-field
Conclusion
Tensor interaction
The tensor force is an essential part of the NN interaction(3S-D1 phase shifts, deuteron quadrupole moment).
Early studies of its effects in a mean-field framework:
T.H.R. Skyrme suggested it from the beginning –considered standard s.o. too simple
F. Stancu, H. Flocard, D.M. Brink, PLB (1977) K. F. Liu et al., NPA (1991)
Recent attempts at adding a tensor term to mean-field/densityfunctional models:
Gogny: T. Otsuka, T. Matsuo and D. Abe, PRL (2005) RHF: W.H. Long, N. Van Giai and J. Meng, PLB (2006) Skyrme:
Perturbative studies: G. Colò, H. Sagawa, S. Fracassoand P. Bortignon, PLB (2007);D.M. Brink and F. Stancu, PRC 75, 064311 (2007)
Refit:B.A. Brown, T. Duguet, T. Otsuka, D. Abe and T.Suzuki, PRC 74, 061303 (2006);T. Lesinski, M. Bender, K.B., T. Duguet and J. Meyer,PRC 76, 014312 (2007).
Tensor
K. Bennaceur
Introduction
Skyrme EDF
Constraints from
nuclear matter
Constraints from
spectroscopy
Hunting for
instabilities
Extentions
Beyond
mean-field
Conclusion
Tensor terms in the nuclear density functional
Zero-range tensor force :
vt(r) = 12 te
[
3(σ 1 ·k′)(σ 2 ·k
′)− (σ 1 ·σ 2)k′2]
δ (r)+h.c.
+to
[
3(σ 1 ·k′)δ (r)(σ 2 ·k)− (σ 1 ·σ2)k′ · δ (r)k
]
Potential energy density in a spherical even-even nucleus:
HSkyrme = ∑
t=0,1
Cρt [ρ0]ρ2
t +C∆ρt ρt∆ρt +C τ
t ρtτt
+12 CJ
t J2t +C ∇J
t ρt∇ ·Jt
( C ∇J0 = −
3
4W0 , C ∇J
1 = −1
4W0 )
“Tensor terms”: CJt = AJ
t (central) +BJt (tensor)
Ht =
1
2
(
CJ0 J2
0 + CJ1 J2
1
)
= 12 α (J2
n +J2p) + β Jn ·Jp
Tensor
K. Bennaceur
Introduction
Skyrme EDF
Constraints from
nuclear matter
Constraints from
spectroscopy
Hunting for
instabilities
Extentions
Beyond
mean-field
Conclusion
Coupling constants for the tensor terms
EJ ∝ CJ0 J2 +CJ
1 (Jn −Jp)2 ∝1
2α ∑
q
J2q +β Jn ·Jp
. ...
. .
.
Region of “reasonable” parameters
(Stancu, Brink, Flocard ’77)
Existing forces
(non local contribution)
CJ0 = 1
2 (α +β )
CJ1 = 1
2 (α −β )
Tensor
K. Bennaceur
Introduction
Skyrme EDF
Constraints from
nuclear matter
Constraints from
spectroscopy
Hunting for
instabilities
Extentions
Beyond
mean-field
Conclusion
Parameterizations
-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 coupling(in spherical nuclei)∼ SLy4
36 parameter sets TIJ with
α = 60(J −2) MeVfm5
β = 60(I −2) MeVfm5
Tensor
K. Bennaceur
Introduction
Skyrme EDF
Constraints from
nuclear matter
Constraints from
spectroscopy
Hunting for
instabilities
Extentions
Beyond
mean-field
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 levels with ℓ ≤ 2(n ≥ 2) larger than empiricalvalues.
Also true for ν3p levelin 208Pb, constrained inthe fit
Splittings of ℓ ≥ 3 (n = 1)levels underestimated
Tensor
K. Bennaceur
Introduction
Skyrme EDF
Constraints from
nuclear matter
Constraints from
spectroscopy
Hunting for
instabilities
Extentions
Beyond
mean-field
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 levels with ℓ ≤ 2(n ≥ 2) larger than empiricalvalues.
Also true for ν3p levelin 208Pb, constrained inthe fit
Splittings of ℓ ≥ 3 (n = 1)levels underestimated
Tensor
K. Bennaceur
Introduction
Skyrme EDF
Constraints from
nuclear matter
Constraints from
spectroscopy
Hunting for
instabilities
Extentions
Beyond
mean-field
Conclusion
Spin-orbit doublet centroids: nodeless “intruder” states
Are s.o. splittings all there is to single-particle 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
Centroid energies too high in theory vs. experiment. Quantityrelated to the central potential
Tensor
K. Bennaceur
Introduction
Skyrme EDF
Constraints from
nuclear matter
Constraints from
spectroscopy
Hunting for
instabilities
Extentions
Beyond
mean-field
Conclusion
Spin-orbit doublet centroids: nodeless “intruder” states
Are s.o. splittings all there is to single-particle 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
Centroid energies too high in theory vs. experiment. Quantityrelated to the central potential
Tensor
K. Bennaceur
Introduction
Skyrme EDF
Constraints from
nuclear matter
Constraints from
spectroscopy
Hunting for
instabilities
Extentions
Beyond
mean-field
Conclusion
Spin-orbit doublet centroids: nodeless “intruder” states
Are s.o. splittings all there is to single-particle 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
Centroid energies too high in theory vs. experiment. Quantityrelated to the central potential
Tensor
K. Bennaceur
Introduction
Skyrme EDF
Constraints from
nuclear matter
Constraints from
spectroscopy
Hunting for
instabilities
Extentions
Beyond
mean-field
Conclusion
Spin-orbit doublet centroids: nodeless “intruder” states
Are s.o. splittings all there is to single-particle 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
Centroid energies too high in theory vs. experiment. Quantityrelated to the central potential
Tensor
K. Bennaceur
Introduction
Skyrme EDF
Constraints from
nuclear matter
Constraints from
spectroscopy
Hunting for
instabilities
Extentions
Beyond
mean-field
Conclusion
Spin-orbit doublet centroids: nodeless “intruder” states
Are s.o. splittings all there is to single-particle 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
Centroid energies too high in theory vs. experiment. Quantityrelated to the central potential
Tensor
K. Bennaceur
Introduction
Skyrme EDF
Constraints from
nuclear matter
Constraints from
spectroscopy
Hunting for
instabilities
Extentions
Beyond
mean-field
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
]
⇒ See the presentation by D. Davesne tomorrow
Tensor
K. Bennaceur
Introduction
Skyrme EDF
Constraints from
nuclear matter
Constraints from
spectroscopy
Hunting for
instabilities
Extentions
Beyond
mean-field
Conclusion
Isospin instabilities
0 0.16 0.32 0.48 0.64
0 1 2 3s.-is
os. ρ
c [fm
-3]
q [fm-1]
f-f0f+
SLy5
0 1 2 3 4 0 0.16 0.32 0.48 0.64
v.-is
os. ρ
c [fm
-3]
q [fm-1]
0 0.16 0.32 0.48 0.64 0.8
s.-is
ov. ρ
c [fm
-3]
0 0.16 0.32 0.48 0.64 0.8
v.-is
ov. ρ
c [fm
-3]
f− : ∆m∗ < 0 (like SLy5) , f0 : ∆m∗ = 0 , f+ : ∆m∗ > 0
Tensor
K. Bennaceur
Introduction
Skyrme EDF
Constraints from
nuclear matter
Constraints from
spectroscopy
Hunting for
instabilities
Extentions
Beyond
mean-field
Conclusion
Isospin instabilities
0
0.16
0.32
0.48
0.64
0.8
0.96
1.12
0 0.5 1 1.5 2 2.5 3 3.5 4
scal
ar-i
sov
ecto
r ρ c
[f
m-3
]
q [fm-1
]
SLY4SLY5
SIIISKM*
SKP
⇒ Useful tool to predict and avoid isospin instabilities
Tensor
K. Bennaceur
Introduction
Skyrme EDF
Constraints from
nuclear matter
Constraints from
spectroscopy
Hunting for
instabilities
Extentions
Beyond
mean-field
Conclusion
Spin instabilities
Spin instabilities only appear if one breaks time reversalsymmetry
HT = −BTT ∑
µνJT ,µνJT ,µν −
1
2BF
T ∑µµ
(
JT ,µµ)2
−1
2BF
T ∑µν
JT ,µνJT ,νµ
+ BTT sT ·TT +BF
T sT ·FT +1
2B∆s
T sT ·∆sT +B∇sT (∇ ·sT )2
Preliminary tests performed by P.H. Heenen and V. Hellemansshow that cranked mean-field calculations for rotational bandsdo not converge when using an interaction with tensor terms...
Linear response calculations by D. Davesne seem to prove thatmany (most ? (all ?) ) of the interactions we haveproduced are unstable...
The suspects are sT ·∆sT ⇒ Laplacian operator is dangerous ?
Tensor
K. Bennaceur
Introduction
Skyrme EDF
Constraints from
nuclear matter
Constraints from
spectroscopy
Hunting for
instabilities
Extentions
Beyond
mean-field
Conclusion
Extensions for the functional: D wave contribution
-
Tensor
K. Bennaceur
Introduction
Skyrme EDF
Constraints from
nuclear matter
Constraints from
spectroscopy
Hunting for
instabilities
Extentions
Beyond
mean-field
Conclusion
Extensions for the functional: D wave contribution
Page 4:
recentlyadded
Page 5:
∼ density dependent term
Not tested yet !
Who knows what we will discover on page 6 and beyond ?...
Tensor
K. Bennaceur
Introduction
Skyrme EDF
Constraints from
nuclear matter
Constraints from
spectroscopy
Hunting for
instabilities
Extentions
Beyond
mean-field
Conclusion
Extensions for the functional: D wave contribution
Effective D wave interaction
vD(r) = tD(
1+xD Pσ)
[
k′2 δ (r)k2 − ∑µ,ν
k′µ k′ν δ (r)kµ kν]
Energy density (time even densities)
HD =1
2tD
[
(
1+xD
2
)
(
∑µν
τµν ∇µ ∇ν ρ − τ∆ρ
)
−
(
1
2+xD
)
∑q
(
∑µν
τq,µν ∇µ ∇ν ρq − τq∆ρq
)]
Mean field : decouples −∇ ·h2
2m∗(r)∇ and
h2
2m∗(r)
ℓ(ℓ+1)
r2
vD =[
tD(
1+xDPσ)
k′2δ (r)k2 − t′D(
1+x ′DPσ)
∑µ,ν
k′µ k′ν δ (r)kµ kν]
⇒ higher order derivatives in the HF equation : good or bad news ?
Tensor
K. Bennaceur
Introduction
Skyrme EDF
Constraints from
nuclear matter
Constraints from
spectroscopy
Hunting for
instabilities
Extentions
Beyond
mean-field
Conclusion
Constraints from beyond mean-field calculations
Problems with → E [ρ] =∫
E [ρ]dr 6= 〈Ψ|T + V |Ψ〉
and → E [ρ] with ρα , α /∈ N
Poles
that can becorrected
and
branch cuts that can not
in the projectedenergy
(Fig. by M. Bender)
V = t0(
1+x0Pσ)
δ +t1
2
(
1+x1Pσ)(
k′2 δ +δ k2)
+ t2(
1+x2Pσ)
k′ ·δ k
+t3
6
(
1+x3Pσ)
ρ δ
+ iW0 σ ·[
k′×δ k]
⇒ K∞ &350 MeV
See arXiv:0809.2045, 0809.2041, 0809.2049
Tensor
K. Bennaceur
Introduction
Skyrme EDF
Constraints from
nuclear matter
Constraints from
spectroscopy
Hunting for
instabilities
Extentions
Beyond
mean-field
Conclusion
Conclusionxxxxxxxxxx Shopping list
Some constraints from microscopic calculations can hardly besatisfied with the current form of the SLy functional: ∆m∗,spin-isospin content
⇒ different treatment of C∆ρ1 ρ1∆ρ1 ?
⇒ t5(
1+x5Pσ)
k′ δ ρ ·k ?
Spin-orbit splittings are wrong and can not be corrected by atensor interaction
⇒ C ∇J0 6= 3C ∇J
1 ?⇒ density dependent spin-orbit coupling constants ?
Spin doublet centroids do not evolve correctly with ℓ
⇒ central part of the interaction ?⇒ Could be improved with a D wave ?
Lot of dangerous regions for the parameters
⇒ get rid of sT ·∆sT ?
Only integer powers of the density
⇒ How many ?⇒ What about coul-ex ?
Tensor
K. Bennaceur
Introduction
Skyrme EDF
Constraints from
nuclear matter
Constraints from
spectroscopy
Hunting for
instabilities
Extentions
Beyond
mean-field
Conclusion
People involved in the different parts of this work
M. Bender CENBGD. Davesne IPNLT. Duguet CEA/IRFUP.H. Heenen ULBW. Hellemans ULBD. Lacroix GANILT. Lesinski ORNLM. Martini IPNLJ. Meyer IPNL
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