Skyrme functional and halo systems - CEA-Irfu

Skyrme EDF
and halos
K. Bennaceur
Skyrme functional and halo systems
V.
Rotival1,2 ,
K.
Bennaceur3 ,
T.
Duguet2,4
1 DPTA/SPN - CEA/DAM Île-de-France - Bruyères-le-Châtel, France
2 NSCL, East Lansing, MI, USA
Dep. of Physics and Astronomy, Michigan State University, East Lansing, USA
3 Institut de Physique Nucléaire de Lyon, CNRS–IN2P3
Université Claude Bernard Lyon 1
4 CEA, Centre de Saclay, IRFU/SPhN, Gif-sur-Yvette, France
ESNT Workshop, Saclay, May 18–20, 2009
Introduction
Conditions for
halo formation
Quantitative
tools
Application to
the Skyrme EDF
Conclusion
Outline
Skyrme EDF
and halos
K. Bennaceur
Introduction
Conditions for
halo formation
■ Halo nuclei and mean field models
Quantitative
tools
■ Quantitative tools to characterize halo
Application to
the Skyrme EDF
Conclusion
■ Effective interaction in ph and pp channels
■ Systematic search among spherical nuclei
■ Conclusion
Overview on halo nuclei
Skyrme EDF
and halos
K. Bennaceur
Introduction
■ Unusual spatial extension
Conditions for
halo formation
Quantitative
tools
“An atomic nucleus is called a halo nucleus or is said to have a
nuclear halo if its radius is appreciably larger than that predicted
by the liquid drop model.”
⇒ How large is “appreciably larger” ?
■ Small one particle separation energy, role of low ℓ states
⇒ Couplings to continuum states
⇒ Sensitive to the spectroscopic details of the interaction
■ Coulomb effects
■ Importance of pairing correlations
Application to
the Skyrme EDF
Conclusion
Need for quantitative criteria
Skyrme EDF
and halos
K. Bennaceur
What about the Helm model ?
(S. Mizutori et al., PRC 61, 044326)
Introduction
Conditions for
halo formation
■ Radius of the realistic density Rr.m.s. =
H
■ Radius of the Helm density Rr.m.s.
=
sR
sR
R
R
ρ (r)r 4 dr
ρ (r)r 2 dr
ρH (r)r 4 dr
=
ρH (r)r 2 dr
R0 and σ extracted from the microscopic form factor
■ Geometric
r
rand Helm radii:
5
5 H
Rr.m.s. and RHelm =
R
Rgeom =
3
3 r.m.s.
(
∆Rskin = RHelm (n) − RHelm (p)
∆Rhalo = Rgeom (n) − RHelm (n)
Quantitative
tools
Application to
the Skyrme EDF
r
3
R02 + 5σ 2
5
Conclusion
Need for quantitative criteria
Skyrme EDF
and halos
K. Bennaceur
What about the Helm model ?
(S. Mizutori et al., PRC 61, 044326)
N
30
6.0
5.8
R [fm]
5.6
34
38
42
46
50
R geom (n)
R Helm (n)
R geom (p)
R Helm (p)
■ Radius of the realistic density Rr.m.s. =
sR
5.4
5.2
5.0 ■
H
Radius of the Helm density Rr.m.s.
=
4.8
4.6
54
Cr
sR
R
R
ρ (r)r 4 dr
ρ (r)r 2 dr
ρH (r)r 4 dr
=
ρH (r)r 2 dr
R0 and σ extracted from the microscopic form factor
58
62
66
70
74
78
R [fm]
A
■ Geometric
Helm radii:
r
rand
N
5 90
5 H
50
70
110
R
and
R
=
R
Rgeom
=
r.m.s.
Helm
R geom (n)
7.2
3
3 r.m.s.
7.0
6.8
6.6
6.4
6.2
6.0
5.8
5.6
100
R Helm (n)
R geom (p)
R Helm (p)
(
∆Rskin = RHelm (n) − RHelm (p)
∆Rhalo = Rgeom (n) − RHelm (n)
Sn
120
140
A
160
Introduction
Conditions for
halo formation
54
Quantitative
tools
Application to
the Skyrme EDF
r
3
R02 + 5σ 2
5
Conclusion
Need for quantitative criteria
Skyrme EDF
and halos
K. Bennaceur
What about the Helm model ?
(S. Mizutori et al., PRC 61, 044326)
N
30
6.0
R [fm]
5.6
38
42
46
50
10 0
54
R geom (n)
R Helm (n)
R geom (p)
R Helm (p)
10
■ Radius of the realistic density Rr.m.s. =
(r) [fm -3 ]
5.8
34
5.4
5.2
5.0 ■
H
Radius of the Helm density Rr.m.s.
=
4.8
4.6
54
Cr
Conditions for
halo formation
sR
-2
-4
10
s
R
R
10 -6
R
ρ (r)r 4 dr
ρ (r)r 2 dr
54
62
66
70
74
78
p
Helm
0.0
4.0
100
R Helm (n)
R geom (p)
R Helm (p)
(
(r) [fm -3 ]
R [fm]
A
■ Geometric
Helm radii:
r
rand
0
N
5 90
5 10H
50
70
110
R
and
R
=
R
Rgeom
=
r.m.s.
Helm
r.m.s.
R geom (n)
7.2
3
3 -2
7.0
6.8
6.6
6.4
6.2
6.0
5.8
5.6
Quantitative
tools
Application to
the Skyrme EDF
(r)r 4n dr
ρH Cr
=
n
dr
ρH (r)r 2pHelm
R0 and σ extracted from the microscopic
10 -8 form factor
58
r
3
R02 + 5σ 2
5
8.0
12.0
r [fm]
10
∆Rskin = RHelm (n) −
10 -4RHelm (p)
80
Cr
n
∆Rhalo = Rgeom (n) −
R
(n)
10 -6 Helm n
Helm
p
Sn
120
140
A
160
Introduction
10 -8
0.0
p
Helm
4.0
8.0
r [fm]
12.0
Conclusion
Core and halo from the one-body density in one slide
Skyrme EDF
and halos
(2 +1)|
_
(r)| 2 [fm -3 ]
K. Bennaceur
=
=
=
=
=
=
10 -2
10 -4
10 -6
0
1
2
3
4
5
Introduction
States with s.p.e. -100 keV in a
Conditions for
halo formation
spherical potential with radius 4 fm
Quantitative
tools
Application to
the Skyrme EDF
10 -8
Conclusion
10 -10
0
5
10
15
20
r [fm]
25
30
35
Core and halo from the one-body density in one slide
Skyrme EDF
and halos
(2 +1)|
_
(r)| 2 [fm -3 ]
K. Bennaceur
=
=
=
=
=
=
10 -2
10 -4
10 -6
0
1
2
3
4
5
Introduction
States with s.p.e. -100 keV in a
Conditions for
halo formation
spherical potential with radius 4 fm
Quantitative
tools
Application to
the Skyrme EDF
10 -8
Conclusion
10 -10
0
5
10
15
20
25
30
35
r [fm]
Halo
Core
Total
[fm -3 ]
10 0
State with smallest binding energy
dominates at large distance
10 -2
10 -4
State with the smallest binding energy:
10 -6
10
has nodes ⇒ smaller amplitude inside
than nodeless states
-8
10 -10
0
5
10
r [fm]
15
20
or has ℓ > 0 ⇒ smaller amplitude inside
than ℓ = 0 states
or the nucleus is hydrogen...
Core and halo from the one-body density in one slide
Skyrme EDF
and halos
(2 +1)|
_
(r)| 2 [fm -3 ]
K. Bennaceur
=
=
=
=
=
=
10 -2
10 -4
10 -6
0
1
2
3
4
5
Introduction
Conditions for
halo formation
Quantitative
tools
Application to
the Skyrme EDF
10 -8
Conclusion
10 -10
0
5
10
15
20
25
30
35
r [fm]
Halo
Core
Total
[fm -3 ]
10 0
10 -2
10 -4
10 -6
10 -8
10 -10
0
5
10
r [fm]
15
20
Core and halo from the one-body density in one slide
Skyrme EDF
and halos
(r)| 2 [fm -3 ]
K. Bennaceur
=
=
=
=
=
=
10 -2
10 -4
(2 +1)|
_
10 -6
Introduction
0
1
2
3
4
5
Conditions for
halo formation
Quantitative
tools
Application to
the Skyrme EDF
10 -8
Conclusion
10 -10
0
5
10
15
20
25
30
35
r [fm]
Halo
Core
Total
0.00
-0.28
-0.56
10 -2
0.00
10 -4
log 10’
[fm -3 ]
10 0
log 10’’
0.28
10 -6
-0.56
-1.12
-1.68
10 -8
10
Halo
Core
Total
10 0
-10
0
5
10
15
20
r [fm]
ρcore (r0 ) ∼
1
× ρhalo (r0 )
10
10 -2
10 -4
10 -6
10 -8
0
5
10
r [fm]
15
20
The halo region
Skyrme EDF
and halos
K. Bennaceur
r0 > rmax such that
∂ 2 log ρ (r) ∂ r2
r0
2 ∂ 2 log ρ (r) =
5
∂ r2
rmax
Error bars provided by
∂ 2 log ρ (r) ∂ 2 log ρ (r) ∂ 2 log ρ (r) 0.35 ×
6
6
0.5
×
∂ r2
∂ r2
∂ r2
rmax
r0
rmax
Introduction
Conditions for
halo formation
Quantitative
tools
Application to
the Skyrme EDF
Conclusion
The halo region
Skyrme EDF
and halos
K. Bennaceur
r0 > rmax such that
∂ 2 log ρ (r) ∂ r2
r0
2 ∂ 2 log ρ (r) =
5
∂ r2
rmax
Error bars provided by
∂ 2 log ρ (r) ∂ 2 log ρ (r) ∂ 2 log ρ (r) 0.35 ×
6
6
0.5
×
∂ r2
∂ r2
∂ r2
rmax
r0
rmax
Nhalo = 4π
Participating nucleons :
Z +∞
r0
ρ (r) r 2 dr
Nuclear extension :
δ Rhalo =
=
Rrms, tot
sR
+∞
R0+∞
0
ρ (r) r 4 dr
ρ (r) r 2 dr
−
Rrms, inner
sR
r0
ρ (r) r 4 dr
R0r0
−
2
0 ρ (r) r dr
Introduction
Conditions for
halo formation
Quantitative
tools
Application to
the Skyrme EDF
Conclusion
Benchmarks
Skyrme EDF
and halos
K. Bennaceur
■ Coupled-channels calculations (F. Nunes et al.) for
Introduction
(halo nucleus) 11 Be and (stable) 13 C: core + neutron
CC calculation provides with core, neutron and total densities
Conditions for
halo formation
Quantitative
tools
10 -2
10
-4
13
10 -6
11
11 Be
10 -8
10 -10
0
5
10
C
Be
Nhalo
0.000
0.000
Core
δ Rhalo
0.000
0.000
(fm)
Nhalo
7 10−4
0.270
Total
δ Rhalo
7 10−4
0.394
(fm)
15
r [fm]
■ Mean-field (SLy4) calculations for Cr isotopes
N
30
34
38
42
46
50
54
Cr
0.14
74 Cr
0.12
R halo [fm]
[fm -3 ]
10
Halo
Core
Total
0
0.10
76
0.08
80 Cr
0.04
0.02
0.00
Cr
78 Cr
0.06
54
58
62
66
A
70
74
78
Nhalo
0.000
0.057
0.194
0.472
δ Rhalo (fm)
0.000
0.018
0.055
0.134
Rr.m.s.
2.487
2.908
(fm)
Application to
the Skyrme EDF
Conclusion
Application: Skyrme EDF calculations for spherical nuclei
Skyrme EDF
and halos
K. Bennaceur
Introduction
Conditions for
halo formation
■ Which particle-hole interation ?
■ Which particle-particle interation ?
Quantitative
tools
Application to
the Skyrme EDF
Conclusion
■ Which nuclei ?
Skyrme EDF: particle-hole channel
Skyrme EDF
and halos
K. Bennaceur
Functional for spherical nuclei
Introduction
h̄ 2
∆ρ
ρ
E =
τ0 + ∑ Ct ρt2 + Ct ρr ∆ρt + Ctτ ρt τt + CtJ J2t + Ct∇J ρt ∇ · Jt
2m
t=0,1
Conditions for
halo formation
Quantitative
tools
Application to
the Skyrme EDF
SLy4
SIII
m∗ 1
1
ρsat
2
ρsat
3
ρsat
T6
SKa
T21-T26
ρsat
0.160
0.145
0.162
0.145
0.160
0.175
0.161
0.155
0.161
K∞
230
355
230
230
230
230
236
263
230
m ∗ /m
0.70
0.76
1.00
0.70
0.70
0.70
1.00
0.61
0.70
κv
0.25
0.53
0.25
0.25
0.25
0.25
0.00
0.94
0.25
Conclusion
E/A
-16.97
-15.85
-16.07
-15.69
-15.99
-16.22
-15.93
-15.99
-16.00
+ VT
Skyrme EDF: particle-particle channel
Skyrme EDF
and halos
K. Bennaceur
Functional (pp part) for spherical nuclei
"
′#
V0
ρ0 α
˜
E =
1−η
∑ ρ̃q2
4
ρsat
q=n,p
■ V0 overall strength,
Conditions for
halo formation
Quantitative
tools
Application to
the Skyrme EDF
Conclusion
■ η = 0 → volume pairing ...to... η = 1 → surface pairing,
■ α ′ < 1 stronger pairing at low density.
Overall strength : h∆in = 1.250 MeV in
Introduction
120 Sn
Zero range interaction ⇒ divergence of the energy
■ “Regularization” using a cutoff (60 MeV),
■ “Renormalization” (A. Bulgac recipe)
Strength of the pairing interaction
Skyrme EDF
and halos
K. Bennaceur
(Used on top of SLy4)
0
Introduction
Conditions for
halo formation
Quantitative
tools
V 0 [MeV]
-2000
Application to
the Skyrme EDF
Conclusion
-4000
-6000
0.0
REG-X
REN-X
0.5
Are all these interactions satisfying ?
1.0
0.5
... No.
η = 1/2 seems to be a good choice (Dobaczewski et al., nucl-th/0109073)
“REG”ularized interactions sometimes suspicious at low density...
Regularized surface pairing interactions
Skyrme EDF
and halos
Surface pairing interaction strong at low density, exotic nucleus, box
big enough, enough partial waves... then
=1.0
=1.0
=1.0
=1.0
=1.0
10 -2
=1.0
=0.8
=0.6
=0.4
=0.2
Quantitative
tools
Application to
the Skyrme EDF
Conclusion
10 -4
n
(r) [fm -3 ]
Introduction
Conditions for
halo formation
10 0
10 -6
10 -1
10 -3
n
(r) [fm -3 ]
K. Bennaceur
10 -5
80
0
Cr
5
10
15
20
25
30
35
40
r [fm]
... the halo hits the fan.
Renormalized surface pairing interactions
Skyrme EDF
and halos
K. Bennaceur
Introduction
Conditions for
halo formation
■ No suspicious results for nuclei
■ Correct gap equation in nuclear matter
Quantitative
tools
Application to
the Skyrme EDF
Conclusion
■ No cutoff... one parameter less.
Convergence of the results
Skyrme EDF
and halos
If calculations are converged for
80 Cr,
they will be for any nucleus...
K. Bennaceur
Introduction
Quantitative
tools
R halo
R halo
0.40
0.40
Application to
the Skyrme EDF
Conclusion
N halo
0.50
N halo
0.50
0.30
0.30
0.20
0.20
0.10
0.00
15
Conditions for
halo formation
0.14
0.12
0.10
0.08
0.06
0.04
0.02
0.00
0.14
0.12
0.10
0.08
0.06
0.04
0.02
0.00
80
25
35
45
R box [fm]
55
Cr
65
0.10
0.00
15
80
25
35
45
2J max
55
Cr
65
Convergence of the results
Skyrme EDF
and halos
If calculations are converged for
80 Cr,
they will be for any nucleus...
K. Bennaceur
Introduction
Conditions for
halo formation
0.14
0.12
0.10
0.08
0.06
0.04
0.02
0.00
Quantitative
tools
R halo
R halo
0.14
0.12
0.10
0.08
0.06
0.04
0.02
0.00
0.40
0.40
Conclusion
N halo
0.50
N halo
0.50
Application to
the Skyrme EDF
0.30
0.30
0.20
0.20
0.10
80
0.00
15
25
35
45
0.10
Cr
55
80
0.00
15
65
25
35
R box [fm]
55
Cr
65
2J max
0.4
J max =15/2
J max =31/2
J max =65/2
0.3
P nv 2
80
Cr
0.2
J max =15/2
J max =31/2
J max =65/2
0.3
80
P nuv
0.4
Cr
0.2
0.1
0.0
45
0.1
1
7
13
19
25
2j
31
37
43
0.0
1
7
13
19
25
2j
31
37
43
Results: impact ph functional
Skyrme EDF
and halos
K. Bennaceur
Introduction
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0.21
0.18
0.15
0.12
0.09
0.06
0.03
0.00
e i [MeV]
R halo
N halo
Conditions for
halo formation
Quantitative
tools
Application to
the Skyrme EDF
Conclusion
0
s1/2
d5/2
-3
g9/2
-6
-9
SLy4 T6
Ska
SIII
m *1
1
sat
2
sat
3
sat
80
T21
Cr
T22
T23
T24
T25
T26
Results: Large scale analysis Nhalo
Skyrme EDF
and halos
K. Bennaceur
500 “Spherical” nuclei ⇐ |β2 | < 0.1 with D1S
Introduction
Conditions for
halo formation
100
Quantitative
tools
0.70
80
Application to
the Skyrme EDF
Conclusion
0.56
60
Z
0.42
0.28
40
0.14
20
N halo
0
0
40
80
120
N
160
0.00
Results: Large scale analysis δ Rhalo
Skyrme EDF
and halos
K. Bennaceur
500 “Spherical” nuclei ⇐ |β2 | < 0.1 with D1S
Introduction
Conditions for
halo formation
100
Quantitative
tools
0.16
80
Application to
the Skyrme EDF
Conclusion
0.12
Z
60
0.08
40
0.04
20
R halo
0
0
40
80
120
N
160
0.00
Results: Large scale analysis
Skyrme EDF
and halos
K. Bennaceur
100
Introduction
100
0.70
80
0.16
Conditions for
halo formation
0.12
Quantitative
tools
0.08
Application to
the Skyrme EDF
80
0.56
60
60
Z
0.42
0.28
40
40
0.04
0.14
20
20
N halo
0
R halo
0.00
0.00
0
0
40
80
120
160
0
40
N
80
120
160
N
■ Several isotopic chains are predicted to display neutrons halos;
■ Halos are predicted to exist only at the very limit of stability;
■ The maximum value of Nhalo is about 0.7;
■ Nhalo decrease with mass between Z=20 and Z=100;
■ Nhalo has almost no influence on the nuclear extension of
massive nuclei.
Remember: I only talk about spherical even-even nuclei...
Conclusion
Conclusion: Best halo candidates
Skyrme EDF
and halos
K. Bennaceur
Introduction
N halo
0.6
Conditions for
halo formation
0.4
Quantitative
tools
0.2
0.0
0.18
R halo
Application to
the Skyrme EDF
Conclusion
0.12
0.06
0.00
0
eni [MeV]
s, p, or d
states
-3
-6
-9
84
Fe
80
Cr
88
Ni
86
Ni
140
Pd
136
Ru
78
Cr
Conclusion: Best halo candidates
Skyrme EDF
and halos
K. Bennaceur
Introduction
N halo
0.6
Conditions for
halo formation
0.4
0.2
Quantitative
tools
0.0
0.18
R halo
Application to
the Skyrme EDF
0.12
Conclusion
0.06
0.00
0
eni [MeV]
h
ac
e
r
of g...
t
ou e lon
ly
al efor
t
b
To
-3
-6
-9
84
Fe
80
Cr
88
Ni
86
Ni
140
Pd
136
Ru
78
s, p, or d
states
Cr
Work done in collaboration with
Skyrme EDF
and halos
K. Bennaceur
Introduction
T. Duguet
V. Rotival
CEA/IRFU - MSU/NSCL
CEA/DPTA - MSU/NSCL
+ thanks to F. Nunes, T. Lesinski and J. Mitroy.
Conditions for
halo formation
Quantitative
tools
Application to
the Skyrme EDF
Conclusion
• New analysis method of the halo phenomenon in finite many-fermion
systems: First applications to medium-mass atomic nuclei,
V. Rotival and T. Duguet,
Phys. Rev. C 79, 054308 (2009).
• Halo phenomenon in finite many-fermion systems: Atom-positron
complexes and large-scale study of atomic nuclei,
V. Rotival, K. B. and T. Duguet,
Phys. Rev. C 79, 054309 (2009).
Surface pairing in very exotic nuclei
Skyrme EDF
and halos
K. Bennaceur
0.0
0.0
-0.5
-0.5
[MeV]
0.5
150Sn
-1.0
-1.5
ULB
DFTS
RDFTS
DFTS+v2
-2.0
-2.5
N(r)
N(r)
[MeV]
Introduction
0.5
Conditions for
halo formation
Quantitative
tools
170Sn
-1.0
ULB
DFTS
RDFTS
DFTS+v2
-2.0
-2.5
-3.0
Application to
the Skyrme EDF
-1.5
Conclusion
-3.0
-3.5
-3.5
0
2
4
6
8
r [fm]
10
12
14
16
0
2
4
6
8
r [fm]
10
12
14
ULB :
cutoff at 5 MeV above the Fermi energy
DFTS :
cutoff at ∼ 60 MeV above the Fermi energy
16
Nulear Struture Near the Limits of Stability | Nulear density-funtional theory { INT-05-3