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1 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 : 5 H and 5 He (microscopic 3 body) 4.Application 2 : triple process (non-microscopic) 5.Application 3: 18 F(p, ) 15 O (reaction, microscopic 2 body) 6.Conclusions
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2 Introduction Clustering: well known effect in light nuclei Nucleons are grouped in “clusters” Best candidate: particle (high binding energy, almost elementary particle) Ikeda diagram: cluster states near threshold ( 8 Be, 20 Ne, etc Halo nuclei: special case of cluster states Beyond the nucleon level: hypernuclei quarks etc.
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3 1. Evidence for clustering Large distance between the clusters wave function important at large distances Example : + 16 O Comparison of radii: ~1.4 fm, 16 O~2.7 fm For 20 Ne 0 + : 1/2 =3.9 fmFor 20 Ne 1 - : 1/2 =5.6 fm 20 Ne + 16 O 0+0+ 4+4+ 1-1- 2+2+ 3-3- cluster Non-cluster
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4 Evidence for clustering Large reduced width Defines the reduced width 2 (P l =penetration factor) W 2 =Wigner limit=3 2 /2ma 2 8 Be: cluster states 2 ( )= 0.40 2 ( )=0.28 7 Li: cluster states and neutron cluster states 2 ( )=0.52 2 ( n )=0 2 ( )=0.01 2 ( n )=0.26
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5 Evidence for clustering Exotic cluster structure: 6 He+ 6 He in 12 Be M. Freer et al, Phys. Rev. Lett. 82 (1999) 1383 Particular cluster structure: halo nuclei: 11 Be= 10 Be+n 6 He= +n+n neutron=simplest cluster Rotational band: E(J)=E 0 + 2 J(J+1)/2mR 2 With R=distance estimate Calculation: P.D., D. Baye, Phys. Lett. B505 (2001) 71 Mixing of 6 He+ 6 He and + 8 He
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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) Green’s Function Monte Carlo Etc…
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7 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 y x r
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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
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9 2-body models Microscopic (+cluster approx.) RGM, GCM r Non-microscopic: 2 particles without structure = potential model r ex: , p+ 16 O, etc. 1, 2 =internal wave functions Solved by the GCM ex: 12 C+ , 18 F+p, etc.
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10 The wave functions are expanded on a gaussian basis 1. potential model (non microscopic) Schrödinger equation: Expansion: r=quantal relative coordinate R n =generator coordinate (variational calculation) The Generator Coordinate Method (GCM) for 2 clusters
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11 2. Microscopic Slater Determinants matrix elements between Slater determinants (projection numerical) can be extended to 3-clusters The Generator Coordinate Method (GCM) for 2 clusters the basis functions are projected Slater determinants (b 1 =b 2 =b) variational calculation needs matrix elements GCM expansion RGM notation GCM notation
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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 > a: Coulomb only : antisymetrization negligible
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13 The R-matrix method: phase-shift calculation 2 body calculations (spins zero) Internal wave function: combination of Slater determinants External wave function: Coulomb (U l =collision matrix) Bloch-Schrödinger equation: With L = Bloch operator restore the hermiticity of H over the internal region) ensures
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14 The R-matrix method: phase-shift calculation Solution of the Bloch-Schrödinger equation: R-matrix equations N+1 unknown quatities (U l, f l (R n )), N+1 equations <> I =matrix element over the internal region stability with the channel radius a is a strong test
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15 3-body models: Hyperspherical coordinates y1y1 x1x1 Jacobi coordinates x 1, y 1 3 sets (x i, y i ), i=1,2,3 Hyperspherical coordinates: 6 coordinates Hamiltonian:
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16 Kinetic energy With K 2 ( ) = angular operator (equivalent to L 2 in two-body systems) Eigenfunctions: =hyperspherical harmonics Eigenvalues : K(K+4) With l x, l y = angular momenta associated with x, y K = hypermoment K ( ) = Jacobi polynomial Spin: with S = total spin
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17 Schrödinger equation JM is expanded over the hyperspherical harmonics To be determinedKnown functions hyperspherical harmonics =l x,l y,L,S Set of equations for Truncation at K = K max Can be extended to 4-body, 5-body, etc… lyly lxlx
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18 Hamiltonian Three-body Models y1y1 x1x1 Non microscopic Microscopic Hamiltonian: r R V ij =nucleus-nucleus interaction Problems with forbidden states Ex: 6 He= +n+n 12 C= 14 Be= 12 Be+n+n V ij =nucleon-nucleon interaction Ex: 6 He= +n+n 5 H=t+n+n Projection: 7-dim integrals
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19 3. Application to 5 H and 5 He A. Adahchour and P.D., Nucl. Phys. A 813 (2008) 252 3.1Introduction 5 H unbound, with N/Z=4: very large value Expected 3-body structure: 3 H+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 5 He(T=3/2) analog states (suggested by Ter-Akopian et al., EPJ A25 (2005) 315)
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20 Application to 5 H and 5 He 3.2 Conditions of the calculation: microscopic 3-cluster NN interaction: Minnesota H=H 0 +u*V (u=admixture parameter in the Minnesota interaction: u~1) From 3 He+p: u=1.12 3 He+p 3 H+n
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21 Application to 5 H and 5 He 5 H= 3 H+n+n T z =3/2,T=3/2 3H3H n n 5 He= 3 He+n+n coupled with 3 H+n+p T z =1/2, T=1/2,3/2 3 He n n 3H3H p n Cluster structure: Main difficulty: unbound states need for specific methods: ACCC x y
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22 Application to 5 H and 5 He 3.3 Analytic Continuation in the Coupling Constant (ACCC) [V.I. Kukulin et al., J. Phys. A 10 (1977) 33] Write H as H=H 0 + V ( =1 is the physical value, E( =1)>0 unbound state) Determine 0 such as E( 0 )=0 For > 0 : E( )<0 bound-state calculation > 0 : x real, k imaginary, E real <0 < 0: x imaginary, k complex, E=k 2 =E R -i /2 the width can be computed Choose M+N+1 values > 0 determine c i,d j Use =1 k complex Main problem: stability! 1 Padé approximant
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23 Application to 5 H and 5 He(T=3/2) 3 H+n+p 3 He+n+n T=3/2 state?? 3-body decay: +n and t+d forb. 5 H, 5 He 5H5H 5 He T=1/2 states: +n structure 3 H+n+n 4 He+n
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24 Application to 5 H and 5 He(T=3/2) 5H5H i ( ) expanded in gaussians centred at R = Generator Coordinate Method Energy curves E(R): eigenvalue for a fixed R value Convergence with K max Different J values Microscopic wave function: 1/2 + expected to be g.s. fast convergence
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25 3. Results for 5 H and 5 He(T=3/2) Application of the ACCC method search for resonance energies and widths test of the stability with N (Padé approximant) E r ~ 2 MeV ~ 0.6 MeV “theoretical” uncertainties
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26 3. Results for 5 H and 5 He(T=3/2) 5 He Weak coulomb effects: essentially threshold Energy curves
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27 3. Results for 5 H and 5 He(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 5 He:E x ~21.3 MeV ~1 MeV
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28 4. Application to 12 C Well known poorly known Main issues: Simultaneous description of scattering and of 12 C? Bose-Einstein condensate? Astrophysics (Triple- process, Hoyle state + others?) Two approaches Microscopic theory Non microscopic theory 3 continuum states?
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29 4. Application to 12 C a.Microscopic models 1)RGM: M. Kamimura (Nucl. Phys.A 351 (1981) 456) : form factors of 12 C 2)GCM: E. Uegaki et al., PTP62 (1979) 1621: triangle structure of 12 C P.D., D.Baye, [Phys. Rev. C36 (1987) 54]: 8 Be+ model 8 Be( ) 12 C S factor 2 + resonance (with the 0 2 state as bandhead) 3)GCM + hyperspherical formalism M. Theeten et al., Phys. Rev. C 76 (2007) 054003 Only 12 C spectroscopy (energies, B(E2), densities)
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30 12 C energy curves 12 C Energy spectrum 0+ 1- 2+ 3- 4+ 0+ 0 2 4 6 -8 -6 -4 -2 -10 phase shifts GCMexp 12 C microscopic
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31 Application to 12 C B. Non-Microscopic model scattering well described by different potentials –deep potentials (Buck potential) –shallow potentials (Ali-Bodmer potentials) we may expect a good description of the 3 system Removal of 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: phase shifts have a similar quality
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32 ABD0ABBuck+sup x 1.088 Buck+ projexp -6 -4 -2 0 12 C spectrum, J=0 + no satisfactory potential!! Ali-Bodmer potential (shallow) Buck potential (deep)
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33 Calculation of 3 phase shifts: Need for appropriate potentials (3 potentials?) Derivation of 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 and 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 +n+n Application to 12 C
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34 5. Application to 18 F(p, ) 15 O Ref.: M. Dufour and P.D., Nucl. Phys. A785 (2007) 381 Microscopic cluster calculation (19-nucleon system) High level density limit of applicability Questions to address: Spectroscopy of 19 F and 19 Ne (essentially J=1/2 +,3/2 + : s waves) 18 F(p, ) 15 O S-factor How to improve the current status on 18 F(p, ) 15 O? 19 F 19 Ne 18 F+n 18 F+p Very important for novae Many experimental works: Direct ( 18 F beam) Indirect (spectroscopy of 19 Ne) 2 recent experiments
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35 Application to 18 F(p, ) 15 O NN interaction: modified Volkov (reproduces the Q value) + spin-orbit Multichannel:p+ 18 F + 15 O n+ 18 Ne Shell model space: sd shell for 18 F, 18 Ne, p shell for 15 O 18 F: J=1 + (x7), 0 + (x3), 2 + (x8), 3 + (x6), 4 + (x3), 5 + (x1) 15 O : J=1/2 -, 3/2 - 18 Ne: J=0 + (x3), 1 + (x2), 2 + (x5), 3 + (x2), 4 + (x2) many configurations Spectroscopy of 19 Ne and continuum states (R-matrix theory) At low energies (below the Coulomb barrier), s waves are dominant J=1/2+ and 3/2+
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36 1.55 3.91 5.50 6.44 6.50 6.42 6.53 7.08 7.26 7.24 19 Ne 19 F F Ne Theory Experiment J=3/2 + + 15 O + O + N + N p+ 18 F n+ 18 F n+ 18 F p+ 18 F 4.03 1.54 -9 -8 -7 -6 -5 -4 -3 -2 0 1 E cm ( 19 F) 2 3 4 5 -5 -4 -3 -2 0 1 E cm ( 19 Ne) p+ 18 O p+ 18 O Fitted (NN int)
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37 E cm ( 19 Ne) Near threshold ? 00 (5.34) 5.35 5.94 6.26 7.36 8.14 8.65 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 0 1 + 15 O + O + N + N p+ 18 F p+ 18 F n+ 18 Fn+ 18 F Theory Experiment 19 Ne 19 F F Ne J=1/2 + ( no parameter) E cm ( 19 F) 2 3 4 5 -7 -6 -5 -4 -3 -2 0 1 p+ 18 O p+ 18 O
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38 Microscopic 18 F(p, ) 15 O S factor 1/2 + = s wave important down to low energies (constructive) interference with the subthreshold state
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39 Drawbacks of the model: Some 3/2 + resonances missing 1/2 + properties not exact (in 19 F, unknown in 19 Ne) R matrix: allows to add resonances (3/2 + ) or to modify their properties (1/2 + ) n+ 18 F 2 0 1 Theoryexp. 19 F, J=1/2+ modified 19 Ne spectrum J=1/2 + 0 5.35 + 15 O p+ 18 F 19 Ne 2 -7 -6 -5 -4 -3 -2 0 1 7.90 6.00 E cm (MeV) known in 19 F unknown in 19 Ne prediction of two 1/2+ states:E=-0.41 MeV, =231 keV E= 1.49 MeV, =296 keV, p / =0.53
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40 18 F(p, ) 15 O S factor Consistent with experiment Uncertainties due to 3/2 + strongly reduced near 0.2 MeV (1/2 + dominant) 3/2+ resonances: interferences?
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41 Two recent experiments J.-C. Dalouzy et al: Ganil + LLN, Ref: Phys. Rev. Lett. 102, 162503 (2009) 19 Ne+p 19 Ne*+p 18 F+p+p evidence for a broad 1/2+ peak (E) near E cm =1.45 MeV, =292 107 keV Cluster calculation E cm =1.49 MeV, =296 keV 18 F+p 19 Ne
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42 A.C. Murphy et al: Edinburgh + TRIUMF (radioactive 18 F beam): Phys. Rev. C79 (2009) 058801 Simultaneous measurement of 18 F(p,p) 18 F and 18 F(p, ) 15 O cross sections R-matrix analysis many resonances no evidence for a 1/2 + resonance (E too low?) 18 F(p,p) 18 F 18 F(p, ) 15 O E cm (MeV) d /d (mb/sr)
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43 6. Conclusions 1.Cluster models Different variants:microscopic semi-microscopic non microscopic Continuum accessible (R-matrix) 2. 5 H, 5 He(T=3/2) 5 H: resaonable agreement with other works 5 He (T=3/2): analog state of 5 H above 3 H+n+p threshold E x ~21.3 MeV, ~1 MeV 3. 12 C Impossible to reproduce 2 and 3 simultaneously (all models) 3 continuum: future microscopic studies possible ( +n+n in progress) 4. 18 F(p, ) 15 O The GCM predicts a 1/2 + resonance (s wave) near the 18 F+p threshold Observed in an indirect experiment Not observed in a direct experiment
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