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

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.

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 cluster Non-cluster

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 (  )= Li:  cluster states and neutron cluster states  2 (  )=0.52  2 ( n )=0  2 (  )=0.01  2 ( n )=0.26

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

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…

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

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 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.

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

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

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

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

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

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:

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

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

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

19 3. Application to 5 H and 5 He A. Adahchour and P.D., Nucl. Phys. A 813 (2008) Introduction 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)

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= He+p 3 H+n

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

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

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

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

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

26 3. Results for 5 H and 5 He(T=3/2) 5 He Weak coulomb effects: essentially threshold Energy curves

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) Th.[2]: P.D. and A. Kharbach, Phys. Rev. C 63 (2001) Th.[3]: K. Arai, Phys. Rev. C 68 (2003) Th.[4]: J. Broeckhove et al., J. Phys. G. 34 (2007) 1955 Exp.[1]: A.A. Korsheninnikov et al., PRL 87 (2001) Exp.[2] M.S. Golovkov et al., PRL 93 (2004)  broad state in 5 He:E x ~21.3 MeV  ~1 MeV

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?

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) Only 12 C spectroscopy (energies, B(E2), densities)

30 12 C energy curves 12 C Energy spectrum  phase shifts GCMexp 12 C microscopic

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= exp(-(r/2.13) 2 ) deep l independent Others:  phase shifts have a similar quality

32 ABD0ABBuck+sup x Buck+ projexp C spectrum, J=0 +  no satisfactory potential!! Ali-Bodmer potential (shallow) Buck potential (deep)

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) 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

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

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/ 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+

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 E cm ( 19 F) E cm ( 19 Ne) p+ 18 O p+ 18 O Fitted (NN int)

37 E cm ( 19 Ne) Near threshold ? 00 (5.34)  + 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) p+ 18 O p+ 18 O

38 Microscopic 18 F(p,  ) 15 O 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 19 F, unknown in 19 Ne) R matrix: allows to add resonances (3/2 + ) or to modify their properties (1/2 + ) n+ 18 F Theoryexp. 19 F, J=1/2+  modified 19 Ne spectrum J=1/  + 15 O p+ 18 F 19 Ne 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

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?

41 Two recent experiments J.-C. Dalouzy et al: Ganil + LLN, Ref: Phys. Rev. Lett. 102, (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

42 A.C. Murphy et al: Edinburgh + TRIUMF (radioactive 18 F beam): Phys. Rev. C79 (2009) 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)

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 C Impossible to reproduce 2  and 3  simultaneously (all models) 3  continuum: future microscopic studies possible (  +n+n in progress) 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