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Systematics of incomplete and/or complete fusion cross sections in heavy ion reactions at intermediate energies. Fusion excitation function revisited. Ph.Eudes 1 , Z. Basrak 2 , V. de la Mota 1 , G.Royer 1 , F. Sébille 1 and M. Zoric 1,2.

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Systematics of incomplete and/or completefusion cross sections in heavy ion reactions at intermediate energies

Fusion excitation function revisited

Ph.Eudes1, Z. Basrak2, V. de la Mota1,

G.Royer1, F. Sébille1 and M. Zoric1,2

1Subatech, EMN-IN2P3/CNRS-Universite de Nantes, F-44037 Nantes, France

2Ruđer Bošković Institute, HR-10002 Zagreb, Croatia

NN2012, May 27 – June 1, San Antonio [email protected]


1 – The raw Fusion Cross Sections (FCS)

168 points 57 systems

Colour code

FUSaround Coulomb barrier are not considered.

Fusion-evaporation and fusion-fission cross sections plotted as a function of incident energy per nucleon

BLUE andGREENsymbols:

Light systems

26 ≤ Asyst≤116

REDandPINKsymbols:

Heavysystems

146 ≤Asyst≤ 246

BLACKsymbols:

Overestimation of the fission cross sections

OR

Only ER measurements

E > 4 A.MeV


Entrance channel parameter ranges

For the 57 collected reactions:

~4 < Einlab < 155 A.MeV

26 < Asyst < 246

0 < μ< 0.886

1 < N/Z < 1.536

μ= (AT - AP) / (AT + AP)

According to the colour code

  • Blue andgreensymbols: 97 points

  • 26 < A < 116

  • For most of these points:

  • 63 points with μ < 0.30 < μ < 0.5

    • 1 < N/Z < 1.25

  • Red and pink symbols:

  • 146 < A < 246

  • For most of these 71 points:

  • 0.75 < μ < 0.886

    • 1.3 < N/Z < 1.536

The lightest systems are rather symmetric both in mass and isospin

The heaviest systems are rather asymmetric both in mass and isospin

 Light Symmetric Systems

 Heavy Asymmetric Systems


Comments

Available datacan be clearly divided into two sets :

Light symmetric systems:FCS regularly decreasewith incident energy and disappear around 40-50 A.MeV

Heavy asymmetric systems:FCS increase up to 20 A.MeV, then decrease. It seems to persist up to 155 A.MeV that is rather surprising…

  • Available data show anevidentlack of data :

  • Heavy asymmetric systemsbetween 30 and 100 MeV/A

  • Medium mass asymmetries on the entire energy range

  •  New data would be welcome


2 – Normalization of the fusion cross sections

To ease comparison of various systems, it is convenient to normalise fusion cross sectionsfus to reaction cross sectionsR and to express them relative to the AVAILABLE ENERGY (A.MeV).

There exist at least four parameterizations to calculate reaction cross sections (see GEANT4).

Sihver formula

Kox formula

R : interaction radius B : interaction barrier

Shen formula

Tripathi formula

Tripathi formula

L. Sihver et al., Phys. Rev. C47, 1225 (1993)

Kox et al. Phys. Rev. C35, 1678 (1987)

Shen et al. Nucl. Phys. A491, 130 (1989)

Tripathi et al, NASA Technical Paper 3621 (1997)


  • The Tripathi’s formula is supposed to work from a few AMeV to a few AGeV for any system of colliding nuclei…


2 – Normalization of the fusion cross sections

To ease comparison of various systems, it is convenient to normalise fusion cross sectionsfus to reaction cross sectionsR and to express them relative to the so-called available energy (corrected for Coulomb barrier).

There exist at least four parameterizations to calculate reaction cross sections (see GEANT4).

Sihver formula

Kox formula

Kox formula

R : interaction radius B : interaction barrier

Shen formula

Tripathi formula

Tripathi formula

quite compatible except at lowest energies

In next figures, a yellow zone recalls the energy domain of incompatibility


Normalization with Tripathi formula : all data

Contrary to what one could expect, there is no universal law

The two sets still exist…

 More evident by separating data


When only LS systems are considered

Landau-Vlasov model simulations

Apart from a few points, very nice correlation between σF/σR and ECM

36Ar+36Ar -40 A.MeV – b = 2 fm

Exponential fit

Hyperbolic fit

Fusion excitation function tends to zero around 12 A.MeV

  • The projectile and the target cross each other. A PLF and a TLF are formed in the exit channel.

Transparency effect could explain the vanishing of fusion


When only HA systems are considered

It’s less clear ! But…

If we remove these points for which fission components were not included

If we forget low energy FCS due to normalization uncertainty

 The few remaining data suggest the existence of a second branch tending towards a constant value

Observation strongly based on high energy points

How does one explain the persistence of fusion above 100 A.MeV?


Fusion cross section at high energy

Supposing the simple formula :

One gets:

Or as a function of

s

=

F

s

R

In a very simple picture, it can be parameterized as:

Rp

Rt

14.4% for N + Sm

17% for N + Au

In agreement with experimental data

Landau-Vlasov model simulations corroborate this scenario


29 points 10 systems

Complete Fusion : light sym. systems

Again, nice correlation is observed

Again, average behaviour reproduced by a hyperbolic function

Disappearance of CF around 6 A.MeV

= Maximum excitation energy deposited into light compound nuclei

Superposition of the two fits  overview of the average weight of CF and IF mechanisms

What happens for heavier systems? For more asymmetric systems? Need new data…

Light systems

40 ≤ Asyst≤ 68


CONCLUSIONS

Available experimental data allowed building fusion excitation functions. One may draw the following main conclusions:

  • For lightsymmetricsystems:

  • 1 – CF component rapidly decreases and disappears around 6 MeV/A. Opened question for heavier and/or more asymmetric systems...

  • 2 – IF component appears around 1 MeV/A, increases up to 6 MeV/A where it is maximum and disappears around 12 MeV/A.

  • 3 – IF+CF excitation function shows a universal trend.

  • For heavy asymmetricsystems:

  • 4 – Above 20 MeV/A, a decrease along a second branch is observed leading to a constant value depending on the system mass asymmetry.

  • 5 – Additional experimental data would be required to confirm the point 4 and to extent our knowledge to medium mass asymmetries.

  •  Describing all the observed trends of these fusion excitation functions could be a real challenge for all transport models intending to describe heavy ion collision properties in this energy range.


About fission cross section components

Comments about the caption :

Blue symbols: Very small fission component for *.

Green symbols: A fission component exists and reaches about 50% of the fusion cross section

Red and pink symbols: If not unique, fission component plays a leading role.

Black symbols: the fission component could contain quasi-fission contribution.


When only LS systems are considered

Apart from a few points, very nice correlation between σF/σR and ECM

Exponential fit

Hyperbolic fit

Fusion excitation function tends to zero around 12 A.MeV

Landau-Vlasov Simulations

Transparency effect could explain the vanishing of fusion


Nature of fusion disappearance?

Simulations undertaken with the Landau-Vlasov model

36Ar+36Ar at 40 A.MeV

40 A.MeV – b = 2 fm

  • The projectile and the target cross each other. A PLF and a TLF are formed in the exit channel.

  • Fusion vanishes due to transparency effect

  • Above this energy limit, all the reaction cross section is of binary nature

C. Grégoire et al. Nucl. Phys. A465, 317 (1987)

F. Sébille et al., Nucl. Phys. A501, 137 (1989)


Persistence of fusion?

14N+154Sm simulations at 150 A.MeV

Peripheral collision

b = 7 fm

Time evolution of the contour plots of the density projected onto the reaction plane

 pre-equilibrium emission from the overlapping zone and 2 nuclei are formed in the exit channel


Persistence of fusion?

Central collision

b = 3 fm

14N+154Sm simulations at 150 A.MeV

 Complete overlap  Formation of a massive incomplete fusion nucleus

Fusion cross section is then comparable to the cross section corresponding to a complete overlap


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