1. 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

2. 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

3. Weakly bound projectilesα 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 Be + p (Sthre = MeV)
6He n + α (S2n = 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

4. 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+ , , 1+
l – 1 < j < l + 1, π = (-1) l
Eres (MeV) Γres (MeV)

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

6. 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

7. 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

8. 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

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

10. 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, (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, (20016)

11. 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

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

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

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

19. 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

20. Grazie Tante

21. Elastic scattering angular distributions

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

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

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

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

26. 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

27. 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,

28. 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

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

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

31. 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/ fm ; a = 0.56 fm
L.C. Chamon, et al., Phys. Rev.C 66 (2002)

32. 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)

33. d+α interaction. Ψ ( 6 Li ) is determined from the potentials
Woods-Saxon shape,
Ground state V0 = MeV, r0=1.15 fm, a0 = 0.7 fm
Resonant states V0 = 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+ , , 1+
l – 1 < j < l + 1, π = (-1) l
Eres (MeV) Γres (MeV)
A. Diaz Torres, Phys. Rev. C 68 (2003)

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

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

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

37. 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

38. 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

39. 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 = 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

40. Effect of breakup couplings on elastic scattering for 8B+58Ni. CDCC
8B Be + 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 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 { , 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)

41. 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, (2009)
One-channel calculation
No inelastic excitations of the target considered
No continuum-continuum couplings

42. 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

43. 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