Fusion radial potential barriers for 8B+58Ni from asimultaneous optical model analysis of elastic scattering

and fusion data

Arturo Gómez CamachoInstituto Nacional de Investigaciones Nucleares, México

Reduced energy ERed = E / γ where γ = ZpZT / RpT with RpT = A1

1/3+A21/3

Reduced cross section

σRed = σ / R2pT

Wong

Reduced reaction cross section

V0 = γ VRed , R0 = RpT rob , ℏ ω0 = ε0 γ

Contents.Fusion potential barriers are determined from a simultaneous analysis of the elastic

scattering and fusion cross section data for 8B+58Ni.

It is found that the calculated fusion polarization potential VF “pushes” the barriers to larger distances respect to the Coulomb barrier. 8B-Halo structure

The net effect of break-up couplings on the fusion cross section is studied by analyzing the separate effect of VDR and WDR.

The analysis uses fusion and direct reaction Woods-Saxon polarization potentials, UF = VF + iWF and UDR = VDR + iWDR, that respectively account for fusion and direct reaction couplings. VF and WF , as well as , VDR and WDR are linked by the dispersion relation.

The potential parameters of VF , WF and VDR , WDR are calculated during the simultaneous fitting of the data.

VDR results a repulsive potential that hinders fusion, particularly at the lowest energies.

Polarization Potential

Dynamic Polarization potential Δ U ( E ) = Δ V ( E ) + iW ( E )

Represents the effects on elastic scattering of couplings between the elastic and non-elastic channels.

Δ V ( E ) arises from virtual excitations to these non- elastic channels

W describes the actual loss of flux into them

V (E) = V0 + ΔU (E)

[ T + V ( E ) ] χ(+) = E χ(+)

V0 ( r ) Static average nuclear potential

Energy dependence of polarization potential. Threshold Anomaly ( Tightly bound systems )

Δ V(E)

W(E)

16O+208PbPolarization potential

Δ U (E) = Δ V (E) + iW(E)

with V (E) = V0 + ΔV (E)

Dispersion relation

VB

attractive

←closing of reaction channels

Fusion and direct reaction polarization potentials

Polarization potential Upol ( r, E ) = UF ( r, E ) + UDR ( r, E )

Fusion polarization potential UF ( r,E ) = ΔVF ( r,E ) + i WF ( r,E )

Direct reaction polarization potential UDR ( r,E ) = Δ VDR ( r,E ) + i WDR ( r,E )

[ T + V ] χ(+) = E χ(+)

V ( r, E ) = Vcoul ( r ) - Vbare ( r ) - Upol (r, E)

Polarization potentials for 8B+58Ni from simultaneous χ2-analysis to elastic and fusion data

E.F. Aguilera et al., Phys. Rev. C79, 021601 (2009)E.F. Aguilera et al., Phys. Rev. Lett, 107, 092701 (2011)

Fusion and direct reaction polarization potentials

Fusion polarization potential

UF ( r,E ) = ΔVF ( r,E ) + i WF ( r,E )

Woods-Saxon volume shape

Ri = ri (A11/3+A2

1/3)

DR polarization potential UDR (r,E) = ΔVDR (r,E) + iWDR (r,E)

Δ

Woods-Saxon surface shape

Δ

Fusion and direct reaction cross sections

Fusion and direct reaction cross sections

σi (E) = 2 / ( v ) < χ ℏ (+) | Wi ( E ) | χ (+)

> ; i = F, DR

Total reaction cross section

σ R ( E ) = 2 / ( v ) < χℏ (+) | WF ( E )+WDR ( E ) | χ (+) >

Potential parameters

Fusion potential barriers (l = 0)

RB = 9.3 fm

VB ≈ 20.8 MeV

Nominal barrier →

Parabolic approx.V( r )=VB - ( ½ ) μ ω2 ( r – RB ) 2

ℏ ω0 = ( / ℏ μ ) [ d 2V ( r ) / dr 2 ]1/2R0 = 5.3 MeV

V( r, E ) = Vcoul ( r ) – Vbare ( r ) - VF ( r, E ) - VDR ( r, E )

Barrier position

{ d V / dr }RB = 0

barrier height

VB = V ( RB )

Effects of barrier lowering and rising due to VF and VDR real polarization potentials

Effect of breakup on fusion cross section

→ VDR = 0, ≠ 0

→ Nominal barrier

Effect of breakup on fusion cross section

Blanco Effects of barrier rising ( VDR ) and loss of flux ( WDR ) into DR reactions

→ VDR = 0, WDR ≠ 0 VDR ≠ 0, WDR = 0 VDR ≠ 0, WDR ≠ 0

The parameters of the Wodds-Saxon fusion and direct reaction polarization potentials are determined from a simultaneous analysis of elastic scattering and fusion data

Mainly by the action of the fusion polarization potential, the barrier is displaced to larger distances from the nominal barrier.

The effect on the barrier from the attractive fusion and repulsive direct reaction polarization potentials has been studied

The net effect of breakup reactions on fusion cross sections is obtained from the individual effects from barrier raising produced by VDR and the loss of flux into direct reactions accounted for by WDR

Summary

Fusion potential barriers for 8B+58Ni have been obtained from fusionFusion potential barriers for 8B+58Ni have been obtained from fusionand direct reaction polarization potentials.and direct reaction polarization potentials.