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Cluster Models P. Descouvemont Physique Nucléaire Théorique et Physique Mathématique, CP229, Université Libre de Bruxelles, B1050 Bruxelles - Belgium. Evidences for clustering Cluster models: non-microscopic (nucleus-nucleus interaction) microscopic (NN interaction) continuum states

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## Evidences for clustering

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**Cluster ModelsP. DescouvemontPhysique Nucléaire Théorique**et Physique Mathématique, CP229,Université Libre de Bruxelles, B1050 Bruxelles - Belgium Evidences for clustering Cluster models: non-microscopic (nucleus-nucleus interaction) microscopic (NN interaction) continuum states Application 1 : 5H and 5He (microscopic 3 body) Application 2 : triple a process (non-microscopic) Application 3: 18F(p,a)15O (reaction, microscopic 2 body) Conclusions**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.**1. Evidence for clustering**Large distance between the clusters wave function important at large distancesExample :a+16O 3- cluster 1- a+16O 4+ Non-cluster 2+ 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**Evidence for clustering**8Be: a cluster states q2(a)= 0.40 q2(a)=0.28 Large reduced width Defines the reduced width g2 (Pl=penetration factor) gW2=Wigner limit=32/2ma2 7Li: a cluster states and neutron cluster states q2(a)=0.01q2(n)=0.26 q2(a)=0.52q2(n)=0**Evidence for clustering**Exotic cluster structure: 6He+6He in 12BeM. 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) 71Mixing of 6He+6He and a+8He Particular cluster structure: halo nuclei: 11Be=10Be+n6He=a+n+nneutron=simplest cluster**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…**2. Cluster Models**x r y r • Several variants • Non microscopic 2 clusters nucleus-nucleus interaction 3 clusters • Microscopic: 2 clustersnucleon-nucleon interaction • 3 clusters y x**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**2-body models**Microscopic (+cluster approx.)RGM, GCM Non-microscopic:2 particles without structure= potential model r r ex: a+a, p+16O, etc. F1 ,F2=internal wave functions Solved by the GCM ex: 12C+a, 18F+p, etc.**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)**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 (b1=b2=b) variational calculation needs matrix elements matrix elements between Slater determinants (projection numerical) can be extended to 3-clusters**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**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**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**3-body models: Hyperspherical coordinates**Jacobi coordinates x1, y1 3 sets (xi, yi), i=1,2,3 y1 x1 Hyperspherical coordinates: 6 coordinates Hamiltonian:**ly**lx Schrödinger equation YJMp is expanded over the hyperspherical harmonics To be determined Known functionshyperspherical harmonics • g=lx,ly,L,S • Set of equations for • Truncation at K = Kmax • Can be extended to 4-body, 5-body, etc…**Three-body Models**R r Non microscopic Microscopic y1 x1 Hamiltonian Hamiltonian: Vij=nucleus-nucleus interactionProblems with forbidden states Ex: 6He=a+n+n12C=a+a+a14Be=12Be+n+n Vij=nucleon-nucleon interaction Ex: 6He=a+n+n5H=t+n+n Projection: 7-dim integrals**3. Application to 5H and 5HeA. 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)**Application to 5H and 5He**3He+p 3H+n 3.2 Conditions of the calculation: microscopic3-cluster NN interaction: Minnesota H=H0+u*V (u=admixture parameter in the Minnesota interaction: u~1) From 3He+p: u=1.12**Application to 5H and 5He**n 3H n n n 3He p 3H Cluster structure: n x 5H=3H+n+n Tz=3/2,T=3/2 y 5He=3He+n+n coupled with 3H+n+p Tz=1/2, T=1/2,3/2 Main difficulty: unbound states need for specific methods: ACCC**Application to 5H and 5He**E(l) 1 l0 l • 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 • 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 Padé approximant • Choose M+N+1 l values l > l0 determine ci,dj • Use l=1 k complex • Main problem: stability!**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 5H 3H+n+p 5He T=1/2 states: a+n structure 4He+n**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 Different J values fast convergence 1/2+ expected to be g.s.**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 MeVG ~ 0.6 MeV “theoretical” uncertainties**3. Results for 5H and 5He(T=3/2)**5He Energy curves Weak coulomb effects:essentially threshold**3. Results for 5H and 5He(T=3/2)**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 MeVG~1 MeV**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?) a+a+a Well known • Two approaches • Microscopic theory • Non microscopic theory 3a continuum states?**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]: 8Be+a model 8Be(a,g)12C S factor 2+ resonance (with the 02 state as bandhead) • GCM + hyperspherical formalisma+a+aM. Theeten et al., Phys. Rev. C 76 (2007) 054003 Only 12C spectroscopy (energies, B(E2), densities)**4+**6 4 1- 1- 3- 2 0+ 3- 0+ 0 -2 2+ 4+ -4 -6 0+ -8 2+ -10 0+ a-a phase shifts 12C microscopic 12C Energy spectrum 12C energy curves GCM exp**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) • 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**12C spectrum, J=0+**0 -2 -4 Ali-Bodmer potential(shallow) Buck potential (deep) -6 exp ABD0 AB Buck+sup Buck+sup x 1.088 Buck+ proj no satisfactory potential!!**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**5. Application to 18F(p,a)15ORef.: M. Dufour and P.D., Nucl.**Phys. A785 (2007) 381 18F+n 18F+p 19Ne 19F • 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 J=1/2+,3/2+: s waves) • 18F(p,a)15O S-factor • How to improve the current status on 18F(p,a)15O?**Application to 18F(p,a)15O**• NN interaction: modified Volkov (reproduces the Q value) + spin-orbit • Multichannel: p+18Fa+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-, 3/2-18Ne: 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+**J=3/2+**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 18 6.53 18 p+ F p+ F 0 6.44 -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 Ne F F**J=1/2+ (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 1 -3 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 -4 -8 -5 -9 -10 -6 0 0 -11 19 19 19 19 -7 F Ne Ne F**Microscopic 18F(p,a)15O S factor**1/2+= s wave important down to low energies (constructive) interference with the subthreshold state**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 + J=1/2 19F, J=1/2+ -4 -5 -6 0 19 -7 Ne 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? • Consistent with experiment • Uncertainties due to 3/2+ strongly reduced near 0.2 MeV (1/2+ dominant)**J.-C. Dalouzy et al: Ganil + LLN, Ref: Phys. Rev. Lett. 102,**162503 (2009) 19Ne+p 19Ne*+p 18F+p+p Two recent experiments 18F+p 19Ne evidence for a broad 1/2+ peak (E) near Ecm=1.45 MeV, G=292107 keV Cluster calculation Ecm=1.49 MeV, G=296 keV**18F(p,p)18F**18F(p,a)15O ds/dW (mb/sr) Ecm (MeV) Ecm (MeV) • 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?)**6. Conclusions**• Cluster models • Different variants: microscopic semi-microscopic non microscopic • Continuum accessible (R-matrix) • 5H, 5He(T=3/2) • 5H: 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 • 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

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