Measurement and interpretation of heavy ion fusion excitation functions

Fusion excitation functions for ³²S induced reactions on ²⁴Mg, ²⁷Al, ⁴⁰Ca and ⁵⁸Ni are reported at incident ³²S ion energies of 65 to 132.5 MeV. Measurements were made using counter-telescopes with beams from Van de Graaff accelerators. From these data barrier heights and radii for fusion are extracted. These results are interpreted in terms of the nuclear diftuseness, and the nuclear attractive potentials at the fusion radii are deduced. Relative density overlaps at the fusion radius are estimated from electron scattering density distributions. Several parameterizations for the fusion radii and barrier heights are presented. Fusion cross sections are compared with reaction cross sections based on elastic scattering measurements coupled with optical model analysis. It is found that for the systems investigated, 90±10 % of the reaction cross section results in fusion.     

Effect of shell structure in the fusion reactions

The fusion reactions are studied in the central collisions ⁸²Se+ + ¹³⁴Ba and ⁸²Se+ + ¹³⁸Ba by the improved isospin-dependent quantum molecular-dynamics model, where the nucleus ¹³⁸Ba has a closed neutron shell N = 82 . Comparing the shell correction energies and fusion probabilities of these two reactions with the ones of other asymmetric or more symmetric reaction systems that form the same compound nuclei, we find the dependence of the fusion reaction on the nuclear shell structure of the colliding nuclei. The experimental data of the fusion probabilities are described well by the present model. The result suggests that the neutron shell closure N = 82 promotes fusion.     

Partial fusion of a weakly bound projectile with heavy target at energies above the Coulomb barrier

Partial-fusion cross-sections for the systems ⁶Li + ²⁰⁸Pb, ⁹Be + ²⁰⁹Bi have been determined. The effect of breakup on fusion for weakly bound projectiles ⁶Li and ⁹Be incident on ²⁰⁸Pb or ²⁰⁹Bi targets has been discussed comparing experimental fusion cross-section excitation functions to those evaluated with a semi-classical approach. It is shown that complete fusion of a weakly bound projectile with heavy target is reduced, whereas the breakup process has very little influence on the total-fusion cross-section for some of the studied systems at energies above the Coulomb barrier.     

Fusion excitation function measurements for the 16O+58Ni and 16O+62Ni systems

High precision fusion excitation functions have been measured for the ¹⁶O+⁵⁸Ni and ¹⁶O+⁶²Ni systems from which fusion barrier distributions have been evaluated. Coupled-reaction-channels (CRC) calculations, which describe elastic and quasi-elastic scattering, also satisfactorily reproduce the fusion cross sections and barrier distributions. The small value of Z₁Z₂ in this case leads to barrier distributions with relatively little structure. However, in conjunction with the detailed elastic scattering data for these systems, this allows us to elucidate the role of previously ignored states in ¹⁶O in pushing the entire distribution to lower energies. These shifts are consistent with derived magnitudes of polarization potentials for both systems.     

Large-scale collective motion in heavy-ion collisions

Heavy-ion fusion and deep inelastic reactions have been studied for symmetric systems in a classical dynamical model with deformation and necking as the collective shape coordinates. The calculated fusion excitation functions (for compound nucleus formation) are in good agreement with the experimental results from the evaporation residue measurements. It is observed that “nuclear molecules” are formed for not too heavy systems. The calculated reaction time for collisions of very heavy ions like ²³⁸U + ²³⁸U is found to be ~10⁻²¹ sec only and thus the width of the positron spectra observed in these reactions can not be explained in the light of quantum electrodynamics. The extra-extra push energies for fusion of heavy nuclei have also been studied.     

Fusion excitation functions near and below the Coulomb barrier for symmetric and asymmetric medium-mass systems

Fusion excitation functions for the systems ¹²C + ^{46, 48, 50}Ti, ^{28, 30}Si + ³⁰Si and ¹⁸O + ⁴⁴Ca have been determined at energies near and below the Coulomb barrier. Inelastic excitation functions for the targets ^{48, 50}Ti and ⁴⁴Ca have been also measured in the same energy range. The absolute cross sections were obtained by in-beam γ-ray spectroscopy technique using a rotating target. Fusion cross sections ranging in magnitude from ～ 0.3 mb to ～ 1300 mb were determined with an accuracy of 10–20%. The fusion excitation functions are analysed in the frame of a semiclassical barrier-penetration model. From the analysis, the height, the radius and the curvature of the fusion barrier for the different systems are extracted. The fusion cross sections are compared with the calculations performed using different heavy-ion potentials. The enhancement of the cross sections at sub-Coulomb energies can be reproduced with a one-dimensional barrier-penetration model taking into account the zero-point motion of the surface of the reaction partners. The fusion cross section of the system ¹⁸O + ⁴⁴Ca is well reproduced by quantum-mechanical calculations, introducing a new degree of freedom taking into account the formation of a neck during the fusion process.     

Microscopic calculation of 58,64Ni + 124,132Sn sub-barrier fusion

The fusion cross-sections of ^58,64Ni + ^124,132Sn are investigated through a coupled-channel approach using a density- and energy-dependent effective Brueckner G-matrix interaction. Microscopic Skyrme-Hartree-Fock proton and neutron density distributions are used in the calculations. A good agreement with the experimental data of the fusion cross-sections of these neutron-rich systems has been obtained, which favors the present microscopic approach for calculating the interaction potentials and fusion cross-sections.     

Nucleon states of strongly deformed nuclei and dinuclear systems in the nonoscillator two-center model

A new method for numerically solving the Schrö dinger equation for an arbitrary axisymmetric field with allowance for spin-orbit interaction is used to study neutron and proton states in strongly deformed nuclei and dinuclear systems produced at the first step of the fusion of nuclei. A quadrupole-octupole parametrization is proposed for the shape of a dinuclear system and for the potential energy of nucleons in this system. The experimentally observed deformations of the ^26,27,28Mg nuclei and the difference in the cross sections for the fusion of nuclei in the ¹⁸O + ⁵⁸Ni and ¹⁶O + ⁶⁰Ni systems are explained qualitatively.     

Fusion and peripheral reactions in the systems 16O + 148,152Sm at sub-barrier energies

Cross sections for fusion and peripheral reactions in the sub-barrier region obtained with the coupled-channel and equivalent-spheres methods are compared for the systems ¹⁶O + ^148,152 Sm. A barrier-like real potential plus a residual surface-imaginary potential is introduced as an alternative approach which allows the simultaneous fit of elastic, inelastic, fusion and peripheral reaction cross sections.     

A Langevin description of the competition between fusion and deep-inelastic collisions close to the barrier

A Langevin Monte Carlo method based on the surface friction model is used to investigate the competition between fusion and deep-inelastic collisions close to the barrier. Data for the⁵⁸Ni+^112,124Sn systems are analyzed. The measured excitation functions for fusion are well reproduced, whereas the calculated deep-inelastic cross sections deviate somewhat from experiment. Predictions for the corresponding spin distributions are made. Contrary to what is usually assumed they are not well separated in 1-space.     

Precompound charged particle emission (PCE) — a mechanism beyond element production by complete fusion

Complete fusion reactions (xn-channels) using actinide targets are observed for values of the effective fissilities x _eff ∼ 0.80 in the sub-pb range of production cross sections. The elements produced at this limit are Z = 108–112. Beyond complete fusion, heavier elements might still be produced by reaction mechanisms releasing part of the nuclear charge before an equilibrated compound system might have been reached. Precompound Charged particle Emission (PCE) is proposed as a possible mechanism following complete fusion. A scheme delivering isotopes of elements Z = 110–115 is discussed, and experimental evidence for such a process is presented. Compound systems, the atomic numbers of which are smaller than in complete fusion reactions, might be produced in ⁴⁸Ca induced reactions on actinides with larger cross sections than those at the limits of complete fusion. Besides complete fusion, the PCE-mechanism should be considered as an alternative to interpret the ⁴⁸Ca-induced reactions on actinides.     

Fusion probability of symmetric heavy,nuclear systems determined from evaporation-residue cross sections

Cross sections were measured for the formation of evaporation residues in ⁴⁸Ca-, ⁸⁶Kr- and ¹²⁴ Sn-induced reactions with Yb, Sb and Zr isotopes. For the nearly symmetric systems, the energy dependence of the fusion probability in central collisions was determined. The fusion probability at the expected fusion barrier as calculated from a one-dimensional heavy-ion potential was found to be reduced by several orders of magnitude. The fusion cross sections increase gradually, and only at energies appreciably above the expected fusion barrier the major part of the cross section of central collisions leads to fusion. The observed hindrance of the fusion process is compared with model predictions.     

Probing sub-barrier fusion and extra-push by measuring fermium evaporation residues in different heavy ion reactions

The production of fermium isotopes was attempted by complete fusion of different targets and projectiles spanning a wide range of effective entrance channel fissilities below and above the predicted threshold valuex _eff ^thr ?0.7. For the most asymmetric systems where fusion is expected to occur without dynamical hindrance we investigate to what extent the expected amount of sub-barrier fusion contributes to the production of fermium evaporation residues. For increasingly symmetric systems the experimental fusion barriers are found to exceed the fusion barriers predicted by the proximity formalism. The barrier heights are discussed in the framework of both the extra-push model and the surface friction model.     

Total fusion cross sections for 18, 17, 16O + 27Al at energies near the Coulomb barrier

Total fusion cross sections have been measured for ^{18, 17, 16}O + ²⁷Al systems at bombarding energies 27–42 MeV. The evaporation residues were detected in the angular range 4°–25° (lab) using a $\text{2E?E}$ counter telescope. Barrier radii extracted from total fusion and elastic scattering cross sections are found to increase with the projectile mass. The effect of the yrast levels on the isotopic yields in the evaporation cascade is investigated.     

Fusion using radioactive ion beams

The capture-fission cross-section is measured for the collision of the massive nucleus ¹³²Sn with ⁹⁶Zr at near-barrier energies and compared with the collision of ¹²⁴Sn with ⁹⁶Zr. This study gives insight into fusion enhancement and hindrance in systems involving neutron-rich nuclei. The dinuclear system model (DNS) calculations describe the excitation function reasonably well and if we use the barrier heights predicted by this model we can conclude that fusion hindrance (represented by extra push energy) is greater for the more neutron-rich systems.     

Mass asymmetry dependence of fusion time-scales in11B+237Np and12C,16O,19F+232Th reactions in a dynamical trajectory model

Dynamical trajectory calculations were carried out for the reactions of¹¹B+²³⁷Np and¹²C,¹⁶O and¹⁹F+²³²Th, having mass asymmetries on either side of the Businaro-Gallone critical mass asymmetry α_BG, in order to examine the mass asymmetry dependence of fusion reactions in these systems. The compound nucleus formation times were calculated as a function of the partial wave of the reaction for all the systems. This study brings out that for systems with α<α_BG, the formation times are significantly larger than for α>α_BG, which is caused by the dynamical effects involved in the large scale shape changes taking place in the fusion process as well as due to the interplay between the thermal and the collective motion during the collision process. The calculated time scales are comparable to the experimental values derived from the pre-fission neutron multiplicity measurements.     

Light ion fusion in deformation model

Experiments with heavy ions at moderate energies show the importance of deformation in heavy ion collisions. A deformation model which takes deformation dynamically into account is developed. Having described fusion and deep inelastic collision for a very heavy system (Xe + Bi) and a medium heavy system (Ar + Th) at various energies successfully, we turn to some comparatively lighter heavy ions where fusion is the most dominant feature. Fusion cross-sections for six pairs of lighter systems (³⁵Cl +¹¹⁶Sn,⁵⁸Ni+⁶²Ni,³⁵Cl+⁶²Ni,³²S+²⁴Mg,²⁴Mg+²⁴Mg and¹²C+²⁷Al) have been obtained using our deformation model which agree well with experiment. The two-slope-behaviour of fusion excitation function which is an important feature of light ion fusion systematics is also obtained, in our model calculations for all the systems studied.     

Fusion-fission and neutron-evaporation-residue cross-sections in 40Ar- and 50Ti-induced fusion reactions

For the reactions ⁴⁰Ar+ ¹⁶⁵Ho. ¹⁶⁹Tm, ¹⁷⁴Yb. ¹¹⁵Lu, ^176–180Hf, ¹⁸¹Ta, ²⁰⁸Pb and ⁵⁰Ti + ²⁰⁸Pb, ²⁰⁹Bi the cross sections for the fusion-fission process were determined by measuring energy and time-of-flight of the reaction products. In addition, the neutron-evaporation-residue cross sections were measured by using the velocity filter SHIP. A σ versus 1/E analysis of the fusion-fission cross sections is used to determine fusion barriers and fusion radii. The evaporation- residue cross sections are used to extract in an approximate way barriers for compound-nucleus formation. These barriers are found to agree with the fusion barriers determined from the fission cross sections. For all systems investigated the neutron-evaporation-residue cross sections reach their maximum close to the fusion barrier as calculated from the Bass potential.     

Theory of fusion for superheavy elements

A new model is proposed for fusion mechanisms of massive nuclear systems, where so-called fusion hindrance exists. The model describes the whole process in two steps: two-body collision processes in an approaching phase and shape evolutions of an amalgamated system into the compound nucleus formation. It is applied to ⁴⁸Ca-induced reactions and is found to reproduce the experimental fusion cross sections extremely well, without any free parameter. A schematic case is solved in an analytic way, the results of which shed light on fusion mechanisms. Combined with statistical decay theory, residue cross sections for superheavy elements can be readily calculated. Examples are given.     

Effect of nuclear shell structure on fusion reaction

The dependence of the fusion reaction on the nuclear shell structure was investigated for the two reaction systems ⁸²Se + ¹³⁸Ba and ⁸²Se + ¹³⁴Ba, where the nucleus ¹³⁸Ba has a closed neutron shell N=82, while the nucleus ¹³⁴Ba has a neutron number 78. Evaporation residues for these fusion reactions were measured near the Coulomb barrier region. The measured evaporation residue cross sections for the reaction system ⁸²Se + ¹³⁸Ba were two orders of magnitude larger than those for the reaction system ⁸²Se + ¹³⁴Ba in the excitation energy region of 20–30 MeV. The evaporation residue cross sections were compared with those of the other reaction systems that produce the same compound nucleus as the present systems. It was found that the fusion reaction ⁸²Se + ¹³⁸Ba occurs without hindrance, while that of ⁸²Se + ¹³⁴Ba is considerably hindered, as commonly observed in the massive reaction system with the charge product Z _p Z _t >1800 of projectile and target. This suggests the importance of the shell closure N=82 in the heavy-ion fusion reaction.