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 1 Subatech, EMN-IN2P3/CNRS-Universite de Nantes, F Nantes, France 2 Ruđer Bošković Institute, HR Zagreb, Croatia NN2012, May 27 – June 1, San Antonio Systematics of incomplete and/or complete fusion cross sections in heavy ion reactions at 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 OR Only 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 < < μ < < N/Z < 1.25 For the 57 collected reactions: ~4 < E in lab < 155 A.MeV 26 < A syst < < μ < < N/Z < Red and pink symbols: 146 < A < 246 For most of these 71 points: 0.75 < μ < < N/Z < The lightest systems are rather symmetric both in mass and isospin The heaviest systems are rather asymmetric both in mass and isospin μ = (A T - A P ) / (A T + A P ) According to the colour code  Light Symmetric Systems  Heavy Asymmetric Systems

Comments vailable data Available data can be clearly divided into two sets :  Light symmetric systems: FCS regularly decrease with incident energy and disappear around 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… vailable data show an Available data 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 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 Shen formula Tripathi formula R : interaction radius B : interaction barrier 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) Tripathi formula

 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 Shen formula Tripathi formula R : interaction radius B : interaction barrier Tripathi formula Kox 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 The two sets still exist… Contrary to what one could expect, there is no universal law  More evident by separating data

Apart from a few points, very nice correlation between σ F /σ R and E CM LS When only LS systems are considered Exponential fit Fusion excitation function tends to zero around 12 A.MeV Transparency effect could explain the vanishing of fusion Hyperbolic fit - 40 A.MeV – b = 2 fm 36 Ar+ 36 Ar - 40 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 HA When only HA 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 One gets: Supposing the simple formula : Or as a function of    R F 14.4% for N + Sm 17% for N + Au In agreement with experimental data RtRt RpRp 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? For more asymmetric systems? Need new data… Superposition of the two fits  overview of the average weight of CF and IF mechanisms

CONCLUSIONS Available experimental data allowed building fusion excitation functions. One may draw the following main conclusions: For light symmetric 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 heavy asymmetric systems: 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  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.

Apart from a few points, very nice correlation between σ F /σ R and E CM LS When only LS systems are considered Exponential fit Fusion excitation function tends to zero around 12 A.MeV Transparency effect could explain the vanishing of fusion Hyperbolic fit Landau-Vlasov Simulations

40 A.MeV – b = 2 fm Nature of fusion disappearance? 36 Ar+ 36 Ar 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 fm Persistence of fusion? 14 N+ 154 Sm 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 density 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 fm 14 N+ 154 Sm simulations at 150 A.MeV