Transcript irfu.cea.fr

Cluster Models
P. Descouvemont
Physique Nucléaire Théorique et Physique Mathématique, CP229,
Université Libre de Bruxelles, B1050 Bruxelles - Belgium
1. Evidences for clustering
2. Cluster models: non-microscopic (nucleus-nucleus interaction)
microscopic (NN interaction)
continuum states
3. Application 1 : 5H and 5He (microscopic 3 body)
4. Application 2 : triple a process (non-microscopic)
5. Application 3: 18F(p,a)15O (reaction, microscopic 2 body)
6. Conclusions
1
Introduction
• Clustering: well known effect in light nuclei
• Nucleons are grouped in “clusters”
Best candidate: a particle (high binding energy, almost elementary particle)
 Ikeda diagram: cluster states near a threshold (8Be, 20Ne, etc
• Halo nuclei: special case of cluster states
• Beyond the nucleon level: hypernuclei
quarks
etc.
2
1. Evidence for clustering
Large distance between the clusters wave function important at large distances
Example :a+16O
3a+16O
cluster
4+
2+
Non-cluster
wave function
1-
a+16O wave function
1-
+
0
0
2
4
r (fm)
6
8
10
0+
20Ne
Comparison of radii:a~1.4 fm, 16O~2.7 fm
For 20Ne 0+: <r2>1/2=3.9 fm
For 20Ne 1-: <r2>1/2=5.6 fm
3
Evidence for clustering
Large reduced width
Defines the reduced width g2 (Pl=penetration factor)
gW2=Wigner limit=32/2ma2
8Be:
a cluster states
7Li:
a cluster states and
neutron cluster states
q2(a)=0.01
q2(n)=0.26
q2(a)=0.52
q2(n)=0
q2(a)= 0.40
q2(a)=0.28
4
Evidence for clustering
Exotic cluster structure: 6He+6He in 12Be
M. Freer et al, Phys. Rev. Lett. 82 (1999) 1383
Rotational band:
E(J)=E0+2J(J+1)/2mR2
With R=distance  estimate
Calculation:
P.D., D. Baye, Phys. Lett. B505 (2001) 71
Mixing of 6He+6He and a+8He
Particular cluster structure: halo nuclei:
11Be=10Be+n
6He=a+n+n
neutron=simplest cluster
5
Cluster models vs ab initio models
cluster models: assume a cluster structure
 effective nucleon-nucleon interaction
 direct access to continuum states
• microscopic (full antisymmetrization, depend on all nucleons)
• non microscopic (nucleus-nucleus interaction)
• semi-microscopic (approximate treatment of antisymmetrization)
ab initio models:
more general
try to determine a cluster structure
realistic nucleon-nucleon interaction
• Antisymmetrized Molecular Dynamics (AMD)
• Fermionic Molecular Dynamics (FMD)
• No Core Shell Model (NCSM)
• Green’s Function Monte Carlo
• Etc…
6
2. Cluster Models
Several variants
• Non microscopic  2 clusters
nucleus-nucleus interaction
 3 clusters
• Microscopic:
 2 clusters
nucleon-nucleon interaction
 3 clusters
r
y
x
r
x
y
7
Cluster Models
2-cluster models
• General description
• Microscopic approach: The generator coordinate method (GCM)
• Continuum states: the R-matrix method
3-cluster models
• Hyperspherical coordinates
• General description
8
2-body models
Non-microscopic:
2 particles without structure
= potential model
Microscopic (+cluster approx.)
RGM, GCM
r
r
A
A
i 1
i j
H  Tr  V (r )
H   Ti   Vij
 lm  g l (r )Yl m ()
 lm  A 1 2 g l (r )Yl m ()
ex: aa, p+16O, etc.
1 ,2=internal wave functions
Solved by the GCM
ex: 12C+a, 18F+p, etc.
9
The Generator Coordinate Method (GCM) for 2 clusters
The wave functions are expanded on a gaussian basis
1. potential model (non microscopic)
Schrödinger equation:
Expansion:
 r=quantal relative coordinate
Rn=generator coordinate (variational calculation)
10
The Generator Coordinate Method (GCM) for 2 clusters
2. Microscopic
A
A
i 1
i j
H   Ti   Vij
 lm  A 1 2 g l (r )Yl m ()
RGM notation
GCM expansion
Slater Determinants
GCM notation
 the basis functions are projected Slater determinants (b1=b2=b)
 variational calculation needs matrix elements
 matrix elements between Slater determinants (projection numerical)
 can be extended to 3-clusters
11
Continuum states
• Necessary for reactions
• Exotic nuclei: low Q value  continuum important
• Simple for 2 clusters, difficult for 3 clusters
• Various methods:
• Exact: calculation of the phase shift
• Approximations: Complex scaling, Analytic continuation (ACCC), box, etc.
(in general, only resonances)
• Use of the R-matrix method: the space is divided into 2 regions (radius a)
•Internal: r ≤ a
: Nuclear + coulomb interactions
: antisymetrization important
•External: r > a
: Coulomb only
: antisymetrization negligible
12
The R-matrix method: phase-shift calculation
•2 body calculations (spins zero)
Internal wave function:
combination of Slater determinants
External wave function:
Coulomb (Ul=collision matrix)
Bloch-Schrödinger equation:
With L = Bloch operator
• restore the hermiticity of H over the internal region)
• ensures
13
The R-matrix method: phase-shift calculation
Solution of the Bloch-Schrödinger equation:
R-matrix equations
 N+1 unknown quatities (Ul, fl(Rn)), N+1 equations
 <>I=matrix element over the internal region
 stability with the channel radius a is a strong test
14
3-body models: Hyperspherical coordinates
y1
Jacobi coordinates x1, y1
x1
3 sets (xi, yi), i=1,2,3
Hyperspherical coordinates:
6 coordinates
Hamiltonian:
15
Schrödinger equation
JMp is expanded over the hyperspherical harmonics
To be determined
Known functions
hyperspherical harmonics
• g=lx,ly,L,S
• Set of equations for
• Truncation at K = Kmax
ly
lx
• Can be extended to 4-body, 5-body, etc…
17
Three-body Models
Non microscopic
y1
Microscopic
R
x1
r
Hamiltonian
Vij=nucleus-nucleus interaction
Problems with forbidden states
Ex:
6He=a+n+n
12C=aaa
14Be=12Be+n+n
A
A
i 1
i j
Hamiltonian: H  T  V
 i  ij
Vij=nucleon-nucleon interaction
Ex:
6He=a+n+n
5H=t+n+n
Projection: 7-dim integrals
18
3. Application to 5H and 5He
A. Adahchour and P.D., Nucl. Phys. A 813 (2008) 252
3.1Introduction
• 5H unbound, with N/Z=4: very large value
• Expected 3-body structure: 3H+n+n
• Many works: experiment
theory
• Difficult for theory and experiment (unbound AND 3-body structure)
 still large uncertainties on
• ground state (Energy, width)
• level scheme?
• Isospin symmetry 
expected 5He(T=3/2) analog states
(suggested by Ter-Akopian et al., EPJ A25 (2005) 315)
19
Application to 5H and 5He
3.2 Conditions of the calculation: microscopic 3-cluster
NN interaction: Minnesota
H=H0+u*V (u=admixture parameter
in the Minnesota interaction: u~1)
3He+p
From 3He+p: u=1.12
3H+n
20
Application to 5H and 5He
Cluster structure:
n
x
5H=3H+n+n
Tz=3/2,T=3/2
y
3H
n
n
5He=3He+n+n
coupled with 3H+n+p
Tz=1/2, T=1/2,3/2
3He
n
n

3H
p
Main difficulty: unbound states  need for specific methods: ACCC
21
Application to 5H and 5He
3.3 Analytic Continuation in the Coupling Constant (ACCC)
[V.I. Kukulin et al., J. Phys. A 10 (1977) 33]
•
Write H as H=H0+lV (l=1 is the physical value, E(l=1)>0 unbound state)
•
Determine l0 such as E(l0)=0
•
For l > l0 : E(l)<0  bound-state calculation
Padé approximant
•
l > l0 : x real, k imaginary, E real <0
•
l < l0: x imaginary, k complex, E=k2=ER-iG/2  the width can be computed
E(l)
1
l0
l
• Choose M+N+1 l values l > l0
 determine ci,dj
• Use l=1  k complex
 Main problem: stability!
22
Application to 5H and 5He(T=3/2)
5H,5He
T=3/2 state??
3-body decay: a+n and t+d forb.
3He+n+n
3H+n+n
3H+n+p
5H
5He
T=1/2 states: a+n structure
4He+n
23
Application to 5H and 5He(T=3/2)
Microscopic wave function:
ci(r) expanded in gaussians centred at R = Generator Coordinate Method
Energy curves E(R): eigenvalue for a fixed R value
5H
Convergence with Kmax
10
Different J values
15
5
H, J=1/2+
3/2
E(R) (MeV)
E(R) (MeV)
8
6
4
4
5
H
10
5/2+
1/2-
4
6
5
8
2
12
2
+
1/2+
0
0
0
2
4
6
R (fm)
 fast convergence
8
10
12
0
2
8
10
12
R (fm)
 1/2+ expected to be g.s.
24
3. Results for 5H and 5He(T=3/2)
Application of the ACCC method
 search for resonance energies and widths
 test of the stability with N (Padé approximant)
Er ~ 2 MeV
G ~ 0.6 MeV
 “theoretical” uncertainties
25
3. Results for 5H and 5He(T=3/2)
5He
Energy curves
15
3/2+
10
1/2
5/2
+
Weak coulomb effects:
essentially threshold
+
E(R) (MeV)
5
T=3/2
0
0
2
-5
5/2
+
4
6
8
10
12
3/2+
-10
-15
1/2+
T=1/2
-20
R (fm)
26
3. Results for 5H and 5He(T=3/2)
5
H
J=1/2+
J=3/2+
E=
G
E=
G
present
1.9 ± 0.1
0.6 ± 0.2
4±1
3±1
E=
G
2.2 ± 0.2
1.0 ± 0.2
Th.[1]
Th.[2]
2.5-3.0 2.8-3.0
3-4
1-2
6.4-6.9
8
Th.[3]
1.59
2.48
3
4.8
Th.[4] Exp.[1] Exp.[2]
1.39 1.7 ± 0.3
~2
1.6 1.9 ± 0.4
2.11
2.87
5
He
J=1/2+
Th.[1]: N.B. Shul’gina et al., Phys. Rev. C 62 (2000) 014312
Th.[2]: P.D. and A. Kharbach, Phys. Rev. C 63 (2001) 027001
Th.[3]: K. Arai, Phys. Rev. C 68 (2003) 03403
Th.[4]: J. Broeckhove et al., J. Phys. G. 34 (2007) 1955
Exp.[1]: A.A. Korsheninnikov et al., PRL 87 (2001) 092501
Exp.[2] M.S. Golovkov et al., PRL 93 (2004) 262501
 broad state in 5He:
Ex~21.3 MeV
G~1 MeV
27
4. Application to 12C
poorly known
aaa
Main issues:
• Simultaneous description of a-a
scattering and of 12C?
• Bose-Einstein condensate?
Well known
• Astrophysics (Triple-a process, Hoyle
state + others?)
Two approaches
• Microscopic theory
• Non microscopic theory
 3a continuum states?
28
4. Application to 12C
a. Microscopic models
1) RGM: M. Kamimura (Nucl. Phys.A 351 (1981) 456) :
form factors of 12C
2) GCM:
E. Uegaki et al., PTP62 (1979) 1621: triangle structure of 12C
P.D., D.Baye, [Phys. Rev. C36 (1987) 54]: 8Be+a model
8Be(a,g)12C S factor
2+ resonance (with the 02 state as bandhead)
3) GCM + hyperspherical formalism aaa
M. Theeten et al., Phys. Rev. C 76 (2007) 054003
Only 12C spectroscopy (energies, B(E2), densities)
29
a-a phase shifts
12C
microscopic
12C
Energy spectrum
4+
6
4
1-
1-
3-
2
0+
3-
0+
0
-2
12C
2+
4+
energy curves
-4
-50
-6
-55
-
1
E (MeV)
-60
0+
-8
3-
2+
-65
-10
+
-70
4
-75
0+
+
2
0+
-80
GCM
exp
-85
0
5
10
R (fm)
15
30
Application to 12C
B. Non-Microscopic model
 aa scattering well described by different potentials
– deep potentials (Buck potential)
– shallow potentials (Ali-Bodmer potentials)
 we may expect a good description of the 3a system
 Removal of a-a forbidden states:
projection method (V. Kukulin)
supersymmetric transformation (D. Baye)
200
l=0
150
Buck potential (Nucl. Phys. A275 (1977) 246)
• V=-122.6 exp(-(r/2.13)2)
• deep
• l independent
l=2
delta (degres)
l=4
100
50
0
0
5
10
-50
Ecm (MeV)
15
Others: a-a phase shifts have a similar quality
31
12C
spectrum, J=0+
0
-2
-4
Ali-Bodmer potential
(shallow)
Buck potential (deep)
-6
exp
ABD0
AB
Buck+sup Buck+sup Buck+ proj
x 1.088
 no satisfactory potential!!
32
Application to 12C
Calculation of 3a phase shifts:
• Need for appropriate a-a potentials (3a potentials?)
• Derivation of a-a potentials
– from RGM kernels (non local)
• M. Theeten et al., PRC 76 (2007) 054003
• Y. Suzuki et al., Phys. Lett. B659 (2008) 160
– Fish-bone model: reproduces aa and aaa
• Z. Papp and S. Moszkowski, Mod. Phys. Lett. 22 (2008) 2201
Non local potentials  difficult for 3-body continuum states
• Microscopic approach to 3-body continuum states?
In progress for a+n+n
33
5. Application to 18F(p,a)15O
Ref.: M. Dufour and P.D., Nucl. Phys. A785 (2007) 381
Very important for novae
18F+n
Many experimental works:
• Direct (18F beam)
18F+p
• Indirect (spectroscopy of 19Ne)
• 2 recent experiments
19Ne
19F
Microscopic cluster calculation (19-nucleon system)
 High level density  limit of applicability
 Questions to address:
• Spectroscopy of 19F and 19Ne (essentially J=1/2+,3/2+: s waves)
• 18F(p,a)15O S-factor
• How to improve the current status on 18F(p,a)15O?

34
Application to 18F(p,a)15O
• NN interaction: modified Volkov (reproduces the Q value) + spin-orbit
• Multichannel: p+18F
a+15O
n+18Ne
• Shell model space: sd shell for 18F, 18Ne, p shell for 15O
18F: J=1+ (x7), 0+ (x3), 2+ (x8), 3+ (x6), 4+ (x3), 5+ (x1)

15O : J=1/2-, 3/218Ne: J=0+ (x3), 1+ (x2), 2+ (x5), 3+ (x2), 4+ (x2)
 many configurations
• Spectroscopy of 19Ne and continuum states (R-matrix theory)
• At low energies (below the Coulomb barrier), s waves are dominant
 J=1/2+ and 3/2+
35
J=3/2+
19
E cm ( Ne)
19
E cm ( F)
5
1
Experiment
Theory
n+
18
p+
18
4
F
18
n+
F
0
O
3
-1
2
1
0
Fitted
(NN int)
p+
18
18
p+
-2
O
7.24
7.26
6.53
6.50
F
-3
7.08
6.44
p+
18
F
5.50
-1
a+
-2
a+
15
15
N
O
a+
15
-4
6.42
-5
N
4.03
-6
3.91
-3
a+
15
O
-4
-7
-8
1.55
1.54
-9
-5
19
Ne
19
F
19
F
19
Ne
36
J=1/2+ (no parameter)
Ecm (19Ne)
19
Experiment
Theory
5
n+
18
p+
18
F
n+
18
0
O
-1
3
8.65
Near threshold
8.14
7.36
1
0
1
F
4
2
E cm ( F)
p+
18
F
6.26
5.94
(5.34)
-1
-2
a+
a+
15
15
N
a+
15
p+
18
-2
O
?
p+
F
5.35
-4
-5
-6
N
a+
O
18
-3
15
O
-3
-7
-4
-8
-5
-9
-6
-7
0
19
Ne
19
F
19
F
0
19
Ne
-10
-11
37
Microscopic 18F(p,a)15O S factor
105
18
S (MeV-b)
10
F(p,a)15O
4
Oak Ridge
Louvain-la-Neuve
103
total
10
2
10
1
1/2
+
3/2+
100
0
0.5
1
1.5
Ecm (MeV)
1/2+= s wave
 important down to low energies
 (constructive) interference with the subthreshold state
38
Drawbacks of the model:
• Some 3/2+ resonances missing
• 1/2+ properties not exact (in 19F, unknown in 19Ne)
• R matrix: allows to add resonances (3/2+) or to modify their properties (1/2+)
Ecm (MeV)
2
2
7.90
known in 19F
unknown in 19Ne
1
1
18
p+ F
0
n+
18

F
-1
Theory
19F,
exp.
J=1/2+
19Ne
modified
spectrum
0
-1
-2
6.00
5.35
15
a+ O
-3
J=1/2+
-4
-5
-6
-7
0
19
Ne
39
 prediction of two 1/2+ states: E=-0.41 MeV, G=231 keV
E= 1.49 MeV, G=296 keV, Gp/G=0.53
18F(p,a)15O
S factor
3/2+ resonances:
interferences?
S (MeV-b)
105
10
4
10
3
10
2
F(p,a)15O
18
total
+/+
+/-
1/2+
101
0
1
0.5
1.5
Ecm (MeV)
• Consistent with experiment
• Uncertainties due to 3/2+ strongly reduced near 0.2 MeV (1/2+ dominant)
40
Two recent experiments
J.-C. Dalouzy et al: Ganil + LLN, Ref: Phys. Rev. Lett. 102, 162503 (2009)
19Ne+p 19Ne*+p
 18F+p+p
18F+p
19Ne
 evidence for a broad 1/2+ peak (E) near Ecm=1.45 MeV, G=292107 keV
Cluster calculation
Ecm=1.49 MeV, G=296 keV
41
A.C. Murphy et al:
• Edinburgh + TRIUMF (radioactive 18F beam): Phys. Rev. C79 (2009) 058801
• Simultaneous measurement of 18F(p,p)18F and 18F(p,a)15O cross sections
• R-matrix analysis  many resonances
ds/d (mb/sr)
400
60
18F(p,p)18F
300
18F(p,a)15O
50
40
200
30
20
100
10
0
0
0.50
0.75
1.00
1.25
1.50
1.75
0.50
Ecm (MeV)
 no evidence for a 1/2+ resonance (E too low?)
0.75
1.00
1.25
1.50
1.75
Ecm (MeV)
42
6. Conclusions
1. Cluster models
• Different variants: microscopic
semi-microscopic
non microscopic
• Continuum accessible (R-matrix)
2.
5H, 5He(T=3/2)
•
•
3.
resaonable agreement with other works
5He (T=3/2): analog state of 5H above 3H+n+p threshold
 Ex~21.3 MeV, G~1 MeV
12C
•
•
4.
5H:
Impossible to reproduce 2a and 3a simultaneously (all models)
3a continuum: future microscopic studies possible (a+n+n in progress)
18F(p,a)15O
•
•
•
The GCM predicts a 1/2+ resonance (s wave) near the 18F+p threshold
Observed in an indirect experiment
Not observed in a direct experiment
43