FusionFusion excitation excitation function revisitedfunction 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, France2Ruđer Bošković Institute, HR-10002 Zagreb, Croatia

NN2012, May 27 – June 1, San Antonio eudes@subatech.in2p3.fr

Systematics of Systematics of incompleteincomplete and/or and/or completecomplete fusionfusion cross cross

sections in heavy ion reactions sections in heavy ion reactions at intermediate energiesat intermediate energies

Colour code

168 points 57 systems

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

BLUE and GREEN symbols:

Light systems

26 ≤ Asyst ≤ 116

RED and PINK symbols:

Heavy systems

146 ≤ Asyst ≤ 246

FUS around Coulomb barrier are not considered.

BLACK symbols:Overestimation of the fission cross sections

OROnly ER measurements

1 – The raw Fusion Cross Sections (FCS)

E > 4 A.MeV

Entrance channel parameter ranges

Blue and green symbols: 97 points

26 < A < 116

For most of these points:

63 points with μ < 0.3 0 < μ < 0.5

1 < N/Z < 1.25

For the 57 collected reactions:

~4 < Ein lab < 155 A.MeV

26 < Asyst < 246

0 < μ < 0.886

1 < N/Z < 1.536

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 The lightest systems are rather symmetric both in rather symmetric both in

mass and isospinmass and isospin

The heaviest systems are The heaviest systems are rather asymmetric both in mass rather asymmetric both in mass

and isospinand isospin

μ = (AT - AP) / (AT + AP)According to the According to the

colour codecolour code

Light Symmetric Systems Light Symmetric Systems Heavy Asymmetric SystemsHeavy Asymmetric Systems

CommentsAvailable datavailable data can be clearly divided into two sets : Light symmetric systems: FCS regularly decrease with 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 datavailable data show an show an evident lack of data : Heavy asymmetric systems between 30 and 100 MeV/A Medium mass asymmetries on the entire energy range

New data would be welcome

2 – Normalization of the fusion cross sectionsTo 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 There exist at least four parameterizations to calculate reaction cross sections (see GEANT4)sections (see GEANT4). .

Sihver formulaSihver formula

Kox formulaKox formula

Shen formulaShen formula

Tripathi formulaTripathi formula

R : interaction radius R : interaction radius B : interaction barrierB : interaction barrier

L. Sihver et al., Phys. Rev. C47, 1225 (1993) L. Sihver et al., Phys. Rev. C47, 1225 (1993) Kox et al. Phys. Rev. C35, 1678 (1987)Kox et al. Phys. Rev. C35, 1678 (1987)Shen et al. Nucl. Phys. A491, 130 (1989)Shen et al. Nucl. Phys. A491, 130 (1989)Tripathi et al, NASA Technical Paper 3621 (1997)Tripathi et al, NASA Technical Paper 3621 (1997)

Tripathi formulaTripathi formula

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

2 – Normalization of the fusion cross sectionsTo 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 There exist at least four parameterizations to calculate reaction cross sections (see GEANT4)sections (see GEANT4). .

Sihver formulaSihver formula

Kox formulaKox formula

Shen formulaShen formula

Tripathi formulaTripathi formula

R : interaction radius R : interaction radius B : interaction barrierB : interaction barrier

Tripathi formulaTripathi formula

Kox formulaKox formula

quite compatible except at lowest energiesquite compatible except at lowest energies

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

Normalization with Tripathi formula : all data

The two sets still exist…The two sets still exist…

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

More evident by separating data

Apart from a few points, very nice

correlation between σF/σR and ECM

When only LLSS systems are considered

Exponential fit

Fusion excitation function tends to zero

around 12 A.MeV

Transparency effect could explain the vanishing of fusionTransparency effect could explain the vanishing of fusion

Hyperbolic fit

36Ar+36Ar - - 40 A.MeV – b = 2 fm40 A.MeV – b = 2 fm

Landau-Vlasov model simulations

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

If we remove these points for which fission components

were not included

When only HHAA systems are considered

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

If we forget low energy FCS due to normalization

uncertainty

It’s less clear ! But…

The few remaining data suggest the existence of a

second branch tending towards a constant value

Observation strongly based on high energy points

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

Fusion cross section at high energy

23/1p0

3/1t0

2pt

2maxF )ArAr()RR(b

One gets:

23/1p

3/1t

20

23/1p

3/1t

20

R

F

)AA(r

)AA(r

2

3/1

p

t

23/1p

3/1t

23/1p

3/1t

A

A1

21

)AA(

)AA(

R R

Supposing the simple formula : 23/1p

3/1t

20 )AA(r R

Or as a function of

pt

pt

AA

AA

2

31

11

1

21

/

R

F

14.4% for N + Sm14.4% for N + Sm

17% for N + Au17% for N + Au

In agreement with In agreement with experimental dataexperimental data

Rt

Rp

Landau-Vlasov model simulations corroborate this scenario

Complete Fusion : light sym. systems

29 points

10 systems

Light systems

40 ≤ Asyst ≤ 68

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

What happens for heavier systems? What happens for heavier systems? For more asymmetric systems? For more asymmetric systems?

Need new data…Need new data…

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

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

For For lightlight symmetricsymmetric systems:systems: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 For heavy heavy asymmetricasymmetric systemssystems::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 trendsDescribing 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 componentsComments 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.

Apart from a few points, very nice

correlation between σF/σR and ECM

When only LLSS systems are considered

Exponential fit

Fusion excitation function tends to zero

around 12 A.MeV

Transparency effect could explain the vanishing of fusionTransparency effect could explain the vanishing of fusion

Hyperbolic fit

Landau-Vlasov Simulations

40 A.MeV – b = 2 fm40 A.MeV – b = 2 fm

Nature of fusion disappearance?

36Ar+36Ar at 40 A.MeV

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

Simulations undertaken with the Landau-Vlasov model

C. Grégoire et al. Nucl. Phys. A465, 317 (1987)F. Sébille et al., Nucl. Phys. A501, 137 (1989)

Peripheral collision b = 7 fmb = 7 fm

Persistence of fusion?

14N+154Sm simulations at 150 A.MeV

pre-equilibrium emission from the overlapping zone and 2 nuclei

are formed in the exit channel

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

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

complete overlap

Persistence of fusion?

Complete overlap Formation of a massive incomplete fusion nucleus

Central collision b = 3 fmb = 3 fm

14N+154Sm simulations at 150 A.MeV