Single-particle and cluster excitations

Single-particle and cluster excitations TALENT Module Nb. 6 A. M. Moro Universidad de Sevilla 9/32

Inelastic scattering: cluster model Some nuclei permit a description in terms of two or more clusters: d=p+n, 6 Li=α+d, 7 Li=α+ 3 H. Projectile-target interaction: Example: 7 Li=α+t V(R, r)=u 1 (r 1 )+U 2 (r 2 ) r α = R Internal states: m t m α r; r t = R+ r m α + m t m α + m t [T r + V α t (r) ε n ]φ n (r)=0 Transition potentials: V n,n (R)= drφ n(r) [U 1 (r 1 )+U 2 (r 2 )]φ n (r) 7 Li α r t R r α r t T TALENT Module Nb. 6 A. M. Moro Universidad de Sevilla 10/32

Coupling potentials in the angular momentum basis Projectile states: φ m nlj (r)= u nlj(r) [Y l (ˆr) χ s ] jm r So, we need to evaluate: (from J.A.Tostevin) (l f s)j f m f V(R,ξ) (l i s)j i m i = φ m f n f l f j f V(R,ξ) φ m i n i l i j i TALENT Module Nb. 6 A. M. Moro Universidad de Sevilla 11/32

Inelastic scattering: cluster model Example: 7 Li(α+t)+ 208 Pb at 68 MeV (Phys. Lett. 139B (1984) 150): CC calculation with 2 channels (3/2, 1/2 ) TALENT Module Nb. 6 A. M. Moro Universidad de Sevilla 13/32

Application of the CC method to weakly-bound systems Example: Three-body calculation (p+n+ 58 Ni) with Watanabe potential: V dt (R)= drφ gs (r) ( V pt (r pt )+V nt (r nt ) ) φ gs (r) 10 1 d + 58 Ni at E=80 MeV (dσ/dω)/(dσ R /dω) 10 0 10-1 10-2 Stephenson et al (1982) No continuum (Watanabe potential) 10-3 0 30 60 90 120 θ c.m. 2 H n r R p T TALENT Module Nb. 6 A. M. Moro Universidad de Sevilla 14/32

Application of the CC method to weakly-bound systems Example: Three-body calculation (p+n+ 58 Ni) with Watanabe potential: V dt (R)= drφ gs (r) ( V pt (r pt )+V nt (r nt ) ) φ gs (r) 10 1 d + 58 Ni at E=80 MeV (dσ/dω)/(dσ R /dω) 10 0 10-1 10-2 Stephenson et al (1982) No continuum (Watanabe potential) 10-3 0 30 60 90 120 θ c.m. Three-body calculations omitting breakup channels fail to describe the experimental data. TALENT Module Nb. 6 A. M. Moro Universidad de Sevilla 14/32 2 H n r R p T

Inelastic scattering of weakly bound nuclei Single-particle (or cluster) excitations become dominant. Excitation to continuum states important. Exotic nuclei are weakly bound coupling to continuum states becomes an important reaction channel TALENT Module Nb. 6 A. M. Moro Universidad de Sevilla 15/32

Normal versus halo nuclei How does the halo structure affect the elastic scattering? 4 He+ 208 Pb @ E=22 MeV 6 He+ 208 Pb @ E=22 MeV 1 1 0.8 0.8 σ/σ R 0.6 σ/σ R 0.6 V 0 =5.89 MeV r 0 =1.33 fm, a=1.15 fm 0.4 0.4 W 0 =9.84 MeV r i =1.33 fm a i =1.7 fm 0.2 V 0 =96.4 MeV r 0 =1.376 fm a 0 =0.63 fm W 0 =32 MeV r i =1.216 fm a i =0.42 fm 0 0 30 60 90 120 150 180 θ c.m. (deg) 0.2 0 0 30 60 90 120 150 180 θ c.m. (deg) 4 He+ 208 Pb shows typical Fresnel pattern strong absorption 6 He+ 208 Pb shows a prominent reduction in the elastic cross section due to the flux going to other channels (mainly break-up) 6 208 TALENT Module Nb. 6 A. M. Moro Universidad de Sevilla 16/32

Inclusion of the continuum in CC calculations: continuum discretization Quantum Hamiltonian Bound states - Discrete - Finite - Normalizable Unbound states - Continuous - Infinite - Non-normalizable Continuum discretization: represent the continuum by a finite set of square-integrable states True continuum Non normalizable Continuous Discretized continuum Normalizable Discrete TALENT Module Nb. 6 A. M. Moro Universidad de Sevilla 17/32

Bound versus scattering states ε V(r) [MeV] 0-50 V(r) φ ε (r) φ gs (r) 8 B Continuum state: φ m k,lj (r)= u k,lj(r) [Y l (ˆr) χ s ] jm r -100 TALENT Module Nb. 6 A. M. Moro Universidad de Sevilla 18/32

Bound versus scattering states ε V(r) [MeV] 0-50 V(r) φ ε (r) φ gs (r) 8 B Continuum state: φ m k,lj (r)= u k,lj(r) [Y l (ˆr) χ s ] jm r -100 Unbound states are not suitable for CC calculations: Continuous (infinite) distribution in energy. Non-normalizable: u k,lsj (r) u k,lsj(r) δ(k k ) SOLUTION continuum discretization TALENT Module Nb. 6 A. M. Moro Universidad de Sevilla 18/32

CDCC formalism: construction of the bin wavefunctions Bin wavefunction: u lsj,n (r)= 2 πn k2 k 1 w(k)u k,lsj (r)dk k: linear momentum u k,lsj (r): scattering states (radial part) w(k): weight function 1 0.5 u(r) 0-0.5 Continuum WF at ε x =2 MeV Bin WF: ε x =2 MeV, Γ=1 MeV -1 0 20 40 60 80 100 120 r (fm) TALENT Module Nb. 6 A. M. Moro Universidad de Sevilla 19/32

Continuum discretization for deuteron scattering φ l,2 φ l,2 φ l,1 n=3 n=2 n=1 l=0 l=1 l=2 s waves p waves Breakup threshold d waves ε max ε=0 ε = 2.22 MeV ground state Select a number of partial waves (l=0,...,l max ). For eachl, set a maximum excitation energyε max. Divide the intervalε=0 ε max in a set of sub-intervals (bins). For each bin, calculate a representative wavefunction. TALENT Module Nb. 6 A. M. Moro Universidad de Sevilla 20/32

CDCC equations for deuteron scattering Hamiltonian: H= T R +h r (r)+v pt (r pt )+V nt (r nt ) Model wavefunction: N Ψ(R, r)=φ gs (r)χ 0 (R)+ φ n (r)χ n (R) n>0 Coupled equations: [H E]Ψ(R, r)=0 [ E εn T R V n,n (R) ] χ n (R)= V n,n (R)χ n (R) 2 H n n n r R p T Transition potentials: V n;n (R)= drφ n (r) [ V pt (R+ r 2 )+V nt(r r 2 ) ]φ n (r) TALENT Module Nb. 6 A. M. Moro Universidad de Sevilla 21/32

What observables can we study with CDCC Elastic scattering Breakup angular distribution, as a function of excitation energy: Breakup energy distribution, as a function of c.m. angle: TALENT Module Nb. 6 A. M. Moro Universidad de Sevilla 22/32

What observables can we study with CDCC Elastic scattering Breakup angular distribution, as a function of excitation energy: Breakup energy distribution, as a function of c.m. angle: From the S-matrices, more complicated breakup observables can be obtained, such as angular/energy distribution of one of the fragments TALENT Module Nb. 6 A. M. Moro Universidad de Sevilla 22/32

Application of the CDCC formalism: d+ 58 Ni 10 1 (dσ/dω)/(dσ R /dω) 10 0 10-1 10-2 Exp. (80.0 MeV) Exp. (79.0 MeV) CDCC: No continuum CDCC 10-3 0 30 60 90 120 θ c.m. l=0, 2 continuum p+ 58 Ni and n+ 58 Ni from Koning-Delaroche OMP. V pn (r)= 72.15 exp[ (r/1.484) 2 ] Coupling to breakup channels has a important effect on the reaction dynamics TALENT Module Nb. 6 A. M. Moro Universidad de Sevilla 23/32

Application of the CDCC method: 6 Li and 6 He scattering The CDCC has been also applied to nuclei with a cluster structure: 6 Li=α+d 11 Be= 10 Be+n 10 0 6 Li+ 40 Ca @ Elab =156 MeV No continuum 10-1 10 0 σ/σ R 10-2 10-3 σ/σ R Including continuum 10-4 Experiment No continuum CDCC 10-5 0 20 40 60 80 θ c.m. (deg) 10 1 11 12 Be + C @ 49 MeV/A 0 5 10 15 20 θ (degrees) c.m. TALENT Module Nb. 6 A. M. Moro Universidad de Sevilla 24/32

Application of the CDCC method: 6 Li and 6 He scattering The CDCC has been also applied to nuclei with a cluster structure: 6 Li=α+d 11 Be= 10 Be+n 10 0 10-1 6 Li+ 40 Ca @ Elab =156 MeV 10 0 No continuum σ/σ R 10-2 10-3 σ/σ R Including continuum 10-4 Experiment No continuum CDCC 10-5 0 20 40 60 80 θ c.m. (deg) 10 1 11 12 Be + C @ 49 MeV/A 0 5 10 15 20 θ (degrees) c.m. In Fraunhofer scattering the presence of the continuum produces a reduction of the elastic cross section TALENT Module Nb. 6 A. M. Moro Universidad de Sevilla 24/32