Effect of resonances on elastic scattering Continuum Discretized Coupled-Channel Calculations (CDCC) for the systems 6Li with 28Si, 58Ni, 144Sm and 208Pb. Effect of resonances on elastic scattering Arturo Gómez Camacho Instituto Nacional de Investigaciones Nucleares, México

Contents 1 ).- Nuclear reaction mechanisms with weakly bound projectiles 2 ).- The case of 6Li and structure of resonant states 3 ).- CDCC 4). Apply CDCC to calculate elastic scattering angular distributions for 6Li with a variety of targets using systematic fragment-target and fragment-fragment potentials. Effect of resonant states 30 minutes 5.-) Conclusions

Weakly bound projectiles α α 6Li 9Be α d n Reactions involving weakly bound projectiles, important in nuclear astrophysical processes. 9Be α + α + n (Sthre = 1.67 MeV) 6Li α + d (Sthre = 1.47 MeV) 7Li α + t (Sthre = 2.46 MeV) 8B 7Be + p (Sthre = 0.137 MeV) 6He 2 n + α (S2n = 0.98 MeV) Breakup separation energies. T1/2 9Be=10-16 sec., 7Be and 8B are radioactive, 9Be and 6Li are stable,10,11Be unstable Short-lived, dificult to produce, easily broken, important coupling to the continuum

6Li structure and resonant states Ethre=1.47 MeV α d Breakup continuum states 6Li target α d Resonant states 6Li l = 2, J = 3+ , 2+ , 1+ l – 1 < j < l + 1, π = (-1) l Eres (MeV) Γres (MeV) 3+ 0.716 0.024 2+ 2.84 1.7 1+ 4.18 1.5

6Li internal states α d d α α internal wave function χI, μ d internal wave function χσ, s=1 d r, k α

CDCC. Discretization of continuum space 6Li Continuum excitation energy εmax ε0 = 0 εmax 4.18 MeV Δε (step) BINS ε0 = 0 Δε 2.84 MeV l = 0, 1, 2, 3, 4…. 0.71 MeV ε0 = 0 (approximation)Take limiting values for l and k (restrict configuration space) CONVERGENCE l, step, k. BINS boxes, wave functions up to llmax l=0 l=1 l=1 l=1 l=2 l=2 l=2 l=3,. j =1 j=2 j=1 j=0 j=3 j=2 j=1 j=4…. aaaaaaa 1.47 MeV Convergence tests: ε max , l max , Δε (step), relative angular momentum L (projetile-target system) , potential multipoles

Ground and scattering state bin wave functions Ground state α-d scattering state bin wave function Relative motion wave functions Convergence tests Internal wave function of the target

Normalized radial scattering bin wave functions Convergence tests The radial wave functions ul ( r, k ) = φl ( r, k ) / r are not square integrable, Introduce weight functions Wl ( k ) and define new square integrable wave functions ϕl ( r ) CCCC AAAA

Radial coupled-channel equations Coupled Equations elastic channel Uββ diagonal β=0 elastic channel elastic-continumm, continuum-continuum couplings          

The interactions V ( r ) = Vnuclear ( r ) + Vcoul ( r ) 6Li d+α projectile target 6Li d+α dd d-α A. Dìaz Torres et al., Phys. Rev. C 68, 044607 (22003) α-target double folding density dependent SPP, L.C. Chamon, et al., Phys. Rev. Lett. 79, (1997) d-target systematic potentials, H. An, C. Cai, Phys. Rev. C 73, 054601 (20016)

Effect of resonances BINS 6Li up to llmax εmax 4.18 MeV 2.84 MeV extract εmax ε0 = 0 4.18 MeV BINS 2.84 MeV 0.71 MeV ε0 = 0 (approximation)Take limiting values for l and k (restrict configuration space) CONVERGENCE l, step, k. BINS boxes, wave functions up to llmax l=0 l=1 l=1 l=1 l=2 l=2 l=2 l=3…. j =1 j=2 j=1 j=0 j=3 j=2 j=1 j=4…. aaaaaaa 1.47 MeV

. . . .Elastic channel couplings elastic scattering all couplings - - - - extract 3+, 2+ , 1+ . . . .Elastic channel couplings

Effect of resonances on elastic scattering ε0 = 0 4.18 MeV BINS 2.84 MeV 0.71 MeV

Solid all couplings Dashed extract 3+, 2+, 1+ Dotted elastic channel

Conclusions Basic features of CDCC has been presented and applied to reactions of 6Li with several targets. By modifying the continuum space, the effect of the resonances of 6Li on elastic scattering can be studied. It ha been found that the effect of the resonances 3+, 2+, 1+ of 6Li on elastic scattering angular distributions is in all cases small, decreases as the collision energy increases but at low energies is necessary to include them to fit the data Resonance 3+,2+ and 1+ continuum space is sufficiently close to the data at low energies except for 208Pb 19

Grazie Tante

Elastic scattering angular distributions

Elastic scattering of 6Li+28Si. Effect of resonances εmax = 6.8 Mev Lmax = 200 lmax=4

Elastic scattering of 6Li+58Ni. Effect of resonances

Elastic scattering of 6Li+144Sm. Effect of resonances

Elastic scattering of 6Li+208Pb. effect of resonances

CDCC formalism. Internal and relative wave functions target ϕ α T H ψtotal = E ψtotal ; H=T + V proyectile c.m ϕ α p Total wave function o α reaction channel internal wave function relative motion wave function internal states of proyectile and target Ψα ( ξ α ) = ϕ α p X ϕ α T

Radial Coupled-Channel Equations where relative motion total nternal wave function Multiplying by ψ*α’ ( ξ α’ ) CCC CC ψ*α’ ( ξ α’ )[ ddddDddd Ddddd dddddd kinetic energy of relative motion Integrating over internal state coordinates ξ and using,

Radial Coupled-Channel Equations (diagonal α= α´ ) ddddDddd Ddddd dddddd dddd then ddddDddd Ddddd dddddd (non-diagonal α ≠ α´ ) CCC CC Diagonal and non-diagonal interaction matrix elements

Total wave function, Aaa Aaaa aaaa Ddddd Dddddd dddddd dd angular momentum coupling Relative motion wave function satisfies dd

The interaction V ( r ) = Vnuclear ( r ) + Vcoul ( r ) projectile target dd

α+target interaction. São Paulo Potential V (r,E) = VCoul (r) – UN (r,E) Nuclear Potential UN ( r, E ) = [ NR ( E ) + i Ni ( E ) ] F( r, E ) F( r,E ) = VF (r) e – 4 v² / c² Relative velocity v² ( r, E ) = ( 2/μ ) [ E – VCoul (r) – UN (r,E) ] Folding potential VF (r) = ∫ ρ ( r₁ ) ρ( r₂ ) V₀ δ( r - r₁ + r₂ ) dr1 dr2 ρ = ρ0 / ( 1 + e ( r – R ) / a ) ; R = 1.31 A1/3 - 0.84 fm ; a = 0.56 fm L.C. Chamon, et al., Phys. Rev.C 66 (2002) 014610

d+target interaction. The nuclear potential of deuteron+target is taken from an extensive experimental analysis for any nuclear systems, Volume and surface Woods_Saxon shapes V( r, E ) = V0 ( E ) f ( r ), WV ( r, E ) = W0,V ( E ) f ( r ), WS ( r, E ) = 4 aS WS,0 (E) df ( r ) / dr , Here you draw the potentials f ( r ) = [ 1 + exp ( r – Ri ) / ai ] -1 R = ri (A d 1/3 + A T 1/3) Vo, WV and Ws are the strengths of the potentials, ai is the diffuseness ri the reduced radius. All of these parameters are determined from the experimental data Haixa An and Chongai Can, Phys, Rev. C 73 (2006) 054605

d+α interaction. Ψ ( 6 Li ) is determined from the potentials Woods-Saxon shape, Ground state V0 = 78.46 MeV, r0=1.15 fm, a0 = 0.7 fm Resonant states V0 = 80.0 MeV, r0 = 1.15 fm a0 = 0.7 fm VSO = 2.5 MeV rSO = 1.15 fm aSO = 0.7 fm Resonant states 6Li l = 2, J = 3+ , 2+ , 1+ l – 1 < j < l + 1, π = (-1) l Eres (MeV) Γres (MeV) 3+ 0.716 0.024 2+ 2.84 1.7 1+ 4.18 1.5 A. Diaz Torres, Phys. Rev. C 68 (2003) 044607

Elastic scattering of 6Li+28Si and effect of resonance continuum εmax = 6.8 Mev Lmax = 100

Elastic scattering of 6Li+144Sm and effect of resonance continuum εmax = 6.8 Mev Lmax = 120

Elastic scattering of 6Li+58Ni and effect of resonance continuum εmax = 6.8 Mev Lmax = 120

Summary Basic features CDCC has been presented and applied to reactions of 6Li with several targets. With the use of phenomenological fragment-fragment and fragment-target potentials, good agreement to elastic scattering angular distributions is obtained. By modifying the continuum space of the fragments, the effect of the resonances of 6Li on elastic scattering can be studied. Breakup cross section angular distributions are equally obtained. Need data. 37

NUCLEAR REACTION MECHANISMS WEAKLY BOUND PROJECTILES σR = σ elastic + σ inelastic + σ fusion+ σ breakup + σ transfer + σ fission DIRECT AND SEQUENTIAL COMPLETE FUSION NON-CAPTURE BREAKUP P T INCOMPLETE FUSION NUCLEON TRANSFER σF = σ SCF + σ DCF + σ ICF

Radial coupled-channel equations, Effect of breakup couplings on 8B+58Ni elastic scattering CDCC calculations Ψ JM ( R, r, ξ ) = Σ i ( F ji ( R ) / R ) ϒi JM( R, r, ξ) ; i={ εi li ji Ii }, L ^ Radial coupled-channel equations, ( elastic channel i=0 ε0 = 0, l0 = 1, j0 = 3/2, I0 =0 ), i > 0 associated to continuum proyectile and/or target excited states ) [ TL + U Ji i ( R ) – E + ϵi ] F J i ( R ) = - Σj U Ji j ( R ) F Jj( R ) Projectile-target interaction, V( R, r, ξ ) = VCT ( R, r, ξ ) + VpT ( R, r, ξ ) Matrix-elements, r internal coordinates of projectile. ξ intrinsic coordinates of target. Index i stands for elastic channel in 2 eq, i>0 are associated to continuum staes of the proyectiles or excited staes of the target ^ ^ ^ U Ji j ( R )= ∫ dR d 3r dξ ϒiJM*( R, r, ξ) V( R, r, ξ) ϒjJM( R, r, ξ ) r V( R, r, ξ) up to quadrupole terms p 7Be E = 0.137 MeV ϵi total excitation energy of channel i εi relative energy of projectile 7Be+p ei excitation energy of target ξ internal coordinates of target R ϵi = εi + ei ξ T

Effect of breakup couplings on elastic scattering for 8B+58Ni. CDCC 8B 7Be + p p in the continuum Continuum states of projectile are approximated by a discrete finite number of channels 8B is described as an inert 7Be core plus a proton In the entrance channel, the two fragments are bound with separation energy of 0.137 MeV in the 1p3/2 ground state The remaining projectile´s states are in the continuum, which are approximated by a set of square integrable bin wave functions which are linear combinations of 7Be+p scattering states. Spin of core 7Be neglected. The 7Be+p interaction is not spin dependent. Target excitations considered I = 21+ and { 22+ , 41+ , 02+} Multipole expansion of the interaction up to quadrupole terms Continuum states Up to energy=5 and 8 MeV, Rbin max=60 fm, for each bin l=0,1,2,3 Elastic scattering data of E.F. Aguilera, A. Gomez Camacho et al., Phys. Rev. C 79 (2009) 021601

Results elastic scattering angular distributions 8B+58Ni The inclusion of continuum-continuum couplings is essential to reproduce the data E.F. Aguilera et al., Phys. Rev. 79, 021601 (2009) One-channel calculation No inelastic excitations of the target considered No continuum-continuum couplings

Importance of Coulomb and Nuclear couplings Couplings between elastic and inelastic excitations of target are considered Polarization potential associated to the Coulomb breakup is repulsive Polarization potential associated to the nuclear breakup is attractive Coulomb nuclear

Importance of different multipoles of the interaction Multipoles higher that λ=2 have negligible effect λ=2 Monopole leads to an attractive polarization potential λ=1 Dipole to a repulsive polarization potential λ=0,1,2 Quadrupole to a repulsive polarization potential λ=0,1 λ=0