Cluster Models P. Descouvemont Physique Nucl

1
Cluster ModelsP. DescouvemontPhysique
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

2
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.

3
1. Evidence for clustering

• Large distance between the clusters? wave
  function important at large distancesExample
  a16O

3-
cluster
1-
a16O
4
Non-cluster
2
0
20Ne
Comparison of radiia1.4 fm, 16O2.7 fm For 20Ne
0 ltr2gt1/23.9 fm For 20Ne 1- ltr2gt1/25.6
fm
4
Evidence for clustering

• Large reduced width

Defines the reduced width g2 (Plpenetration
factor)
gW2Wigner limit3?2/2ma2
7Li a cluster states and neutron cluster
states
q2(a)0.01q2(n)0.26
q2(a)0.52q2(n)0
5
Evidence for clustering
Exotic cluster structure 6He6He in 12BeM.
Freer et al, Phys. Rev. Lett. 82 (1999) 1383
Rotational bandE(J)E0?2J(J1)/2mR2 With
Rdistance ? estimate CalculationP.D., D. Baye,
Phys. Lett. B505 (2001) 71Mixing of 6He6He and
a8He
Particular cluster structure halo nuclei
11Be10Ben 6Heannneutronsimplest
cluster
6
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)
• Greens Function Monte Carlo
• Etc

7
2. Cluster Models

• Several variants
• Non microscopic ? 2 clusters nucleus-nucleus
  interaction ? 3 clusters
• Microscopic ? 2 clusters nucleon-nucleon
  interaction
• ? 3 clusters

y
x
8
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

9
2-body models
Non-microscopic 2 particles without structure
potential model

• Microscopic (cluster approx.)RGM, GCM

r
r
ex aa, p16O, etc.
F1 ,F2internal wave functions Solved by the
GCM ex 12Ca, 18Fp, etc.
10
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
? rquantal relative coordinate Rngenerator
coordinate (variational calculation)
11
The Generator Coordinate Method (GCM) for 2
clusters
2. Microscopic
RGM notation
GCM expansion
Slater Determinants
GCM notation
? the basis functions are projected Slater
determinants (b1b2b) ? variational calculation
needs matrix elements
? matrix elements between Slater determinants
(projection numerical)? can be extended to
3-clusters
12
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 gt a Coulomb only
  antisymetrization negligible

13
The R-matrix method phase-shift calculation

• 2 body calculations (spins zero)

Internal wave function combination of Slater
determinants
External wave function Coulomb (Ulcollision
matrix)
Bloch-Schrödinger equation

• With L Bloch operator
• restore the hermiticity of H over the internal
  region)
• ensures

14
The R-matrix method phase-shift calculation
Solution of the Bloch-Schrödinger equation
R-matrix equations
? N1 unknown quatities (Ul, fl(Rn)), N1
equations ? ltgtImatrix element over the internal
region ? stability with the channel radius a is
a strong test
15
3-body models Hyperspherical coordinates
Jacobi coordinates x1, y1 3 sets (xi, yi), i1,2,3
y1

x1
Hyperspherical coordinates
6 coordinates
Hamiltonian
16
Kinetic energy

• With K2(W) angular operator (equivalent to L2
  in two-body systems)
• Eigenfunctions
• hyperspherical harmonics
• Eigenvalues K(K4)
• With lx, ly angular momenta associated with
  x, y
• K hypermoment
• FK(a) Jacobi polynomial
• Spin
  with S total spin

17
Schrödinger equation
YJMp is expanded over the hyperspherical harmonics
To be determined
Known functionshyperspherical harmonics

• glx,ly,L,S
• Set of equations for
• Truncation at K Kmax
• Can be extended to 4-body, 5-body, etc

18
Three-body Models
Non microscopic
Microscopic
y1
x1
Hamiltonian
Hamiltonian
Vijnucleus-nucleus interactionProblems with
forbidden states Ex 6Heann 12Caaa 14Be1
2Benn
Vijnucleon-nucleon interaction Ex
6Heann 5Htnn
Projection 7-dim integrals
19
3. Application to 5H and 5HeA. Adahchour and
P.D., Nucl. Phys. A 813 (2008) 252

• 3.1Introduction
• 5H unbound, with N/Z4 very large value
• Expected 3-body structure 3Hnn
• 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(T3/2) analog
  states (suggested by Ter-Akopian et al., EPJ
  A25 (2005) 315)

20
Application to 5H and 5He
3.2 Conditions of the calculation microscopic
3-cluster
NN interaction Minnesota HH0uV (uadmixture
parameter in the Minnesota interaction u1) From
3Hep u1.12
21
Application to 5H and 5He
Cluster structure
n
x
5H3Hnn Tz3/2,T3/2
y
5He3Henn coupled with 3Hnp Tz1/2,
T1/2,3/2
Main difficulty unbound states ? need for
specific methods ACCC
22
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 HH0lV (l1 is the physical value,
  E(l1)gt0? unbound state)
• Determine l0 such as E(l0)0
• For l gt l0 E(l)lt0 ? bound-state calculation
• l gt l0 x real, k imaginary, E real lt0
• l lt l0 x imaginary, k complex, Ek2ER-iG/2 ?
  the width can be computed

Padé approximant

• Choose MN1 l values l gt l0 ? determine ci,dj
• Use l1 ? k complex
• ? Main problem stability!

23
Application to 5H and 5He(T3/2)
5H,5He
T3/2 state??3-body decay an and td forb.
3Henn
3Hnp
5H
3Hnn
5He
T1/2 states an structure
4Hen
24
Application to 5H and 5He(T3/2)
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
Different J values
? fast convergence
? 1/2 expected to be g.s.
25
3. Results for 5H and 5He(T3/2)
Application of the ACCC method ? search for
resonance energies and widths ? test of the
stability with N (Padé approximant)
Er 2 MeVG 0.6 MeV ? theoretical
uncertainties
26
3. Results for 5H and 5He(T3/2)
5He
Energy curves
Weak coulomb effectsessentially threshold
27
3. Results for 5H and 5He(T3/2)
Th.1 N.B. Shulgina 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 Ex21.3 MeV G1 MeV
28
4. Application to 12C
poorly known

• Main issues
• Simultaneous description of a-a scattering and of
  12C?
• Bose-Einstein condensate?
• Astrophysics (Triple-a process, Hoyle state
  others?)

aaa
Well known

• Two approaches
• Microscopic theory
• Non microscopic theory ? 3a continuum states?

29
4. Application to 12C

• Microscopic models
• RGM M. Kamimura (Nucl. Phys.A 351 (1981) 456)
  form factors of 12C
• GCM E. Uegaki et al., PTP62 (1979) 1621
  triangle structure of 12C P.D., D.Baye, Phys.
  Rev. C36 (1987) 54 8Bea model 8Be(a,g)12C S
  factor 2 resonance (with the 02 state as
  bandhead)
• GCM hyperspherical formalism aaaM. Theeten
  et al., Phys. Rev. C 76 (2007) 054003 Only 12C
  spectroscopy (energies, B(E2), densities)

30
a-a phase shifts
12C microscopic
12C Energy spectrum
12C energy curves
GCM
exp
31
Application to 12C
B. Non-Microscopic model

• aa 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)

• Buck potential (Nucl. Phys. A275 (1977) 246)
• V-122.6 exp(-(r/2.13)2)
• deep
• l independent
• Others a-a phase shifts have a similar quality

32
12C spectrum, J0
0
-2
-4
Ali-Bodmer potential(shallow)
Buck potential (deep)
-6
ABD0
AB
Bucksup
Bucksup x 1.088
Buck proj
exp
? no satisfactory potential!!
33
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 aa and aaa
• 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 ann

34
5. Application to 18F(p,a)15O Ref. M. Dufour
and P.D., Nucl. Phys. A785 (2007) 381

• Very important for novae
• Many experimental works
• Direct (18F beam)
• Indirect (spectroscopy of 19Ne)
• 2 recent experiments

• Microscopic cluster calculation (19-nucleon
  system)
• High level density ? limit of applicability
• Questions to address
• Spectroscopy of 19F and 19Ne (essentially
  J1/2,3/2 s waves)
• 18F(p,a)15O S-factor
• How to improve the current status on 18F(p,a)15O?

35
Application to 18F(p,a)15O

• NN interaction modified Volkov (reproduces the Q
  value) spin-orbit
• Multichannel p18F a15O n18Ne
• Shell model space sd shell for 18F, 18Ne, p
  shell for 15O ? 18F J1 (x7), 0 (x3), 2
  (x8), 3 (x6), 4 (x3), 5 (x1) 15O J1/2-,
  3/2- 18Ne J0 (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 ? J1/2 and 3/2

36
J3/2
19
E
(
Ne)
19
E
(
F)
cm
cm
Experiment
Theory
1
5
18
n
F
18
n
F
0
4
18
p
O
3
-1
-2
2
18
Fitted (NN int)
p
O
7.24
7.26
-3
1
7.08
6.53
18
18
p
F
p
F
6.44
0
-4
6.50
6.42
5.50
-5
-1
15
15
-6
-2
a

N
a

N
4.03
15
a

O
3.91
-7
-3
15
a

O
-4
-8
1.54
1.55
-9
-5
19
19
19
19
Ne
F
F
Ne
37
J1/2 (no parameter)
Ecm (19Ne)
19
E
(
F)
cm
Experiment
Theory
1
5
18
18
n
F
n
F
0
4
18
p
O
-1
3
8.65
Near threshold
-2
2
8.14
?
18
p
O
7.36
-3
1
18
18
p
F
p
F
6.26
-4
0
5.94
(5.34)
5.35
-5
-1
-6
-2
15
15
a

N
a

N
15
15
a

O
a

O
-7
-3
-8
-4
-9
-5
-10
-6
0
0
-11
19
19
19
19
-7
Ne
F
F
Ne
38

• Microscopic 18F(p,a)15O S factor

1/2 s wave ? important down to low energies?
(constructive) interference with the subthreshold
state
39

• 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)

E
(MeV)
cm
2
2
7.90
known in 19Funknown in 19Ne
1
1
18
p
F
?modified 19Ne spectrum
0
18
n
F
6.00
0
-1
5.35
-1
-2
Theory
exp.
15
a

O
-3

J1/2
19F, J1/2
-4
-5
-6
0
19
-7
Ne
? prediction of two 1/2 states E-0.41 MeV,
G231 keV E 1.49 MeV, G296 keV, Gp/G0.53
40

• 18F(p,a)15O S factor

3/2 resonancesinterferences?

• Consistent with experiment
• Uncertainties due to 3/2 strongly reduced near
  0.2 MeV (1/2 dominant)

41

• Two recent experiments

J.-C. Dalouzy et al Ganil LLN, Ref Phys. Rev.
Lett. 102, 162503 (2009) 19Nep? 19Nep ? 18Fpp
18Fp
19Ne
? evidence for a broad 1/2 peak (E) near
Ecm1.45 MeV, G292?107 keV Cluster calculation
Ecm1.49 MeV, G296 keV
42

• 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

? no evidence for a 1/2 resonance (E too low?)
43
6. Conclusions

• Cluster models
• Different variants microscopic semi-microscopi
  c non microscopic
• Continuum accessible (R-matrix)
• 5H, 5He(T3/2)
• 5H resaonable agreement with other works
• 5He (T3/2) analog state of 5H above 3Hnp
  threshold ? Ex21.3 MeV, G1 MeV
• 12C
• Impossible to reproduce 2a and 3a simultaneously
  (all models)
• 3a continuum future microscopic studies possible
  (ann in progress)
• 18F(p,a)15O
• The GCM predicts a 1/2 resonance (s wave) near
  the 18Fp threshold
• Observed in an indirect experiment
• Not observed in a direct experiment