Classical simulations of heavy-ion fusion reactions and weakly-bound projectile breakup reactions

Heavy-ion collision simulations in various classical models are discussed. Heavy-ion reactions with spherical and deformed nuclei are simulated in a classical rigid-body dynamics (CRBD) model which takes into account the reorientation of the deformed projectile. It is found that the barrier parameters not only on the initial orientations of the deformed nucleus, but also on the collision energy and the moment of inertia of the deformed nucleus. Maximum reorientation effect occurs at near- and below-barrier energies for light deformed nuclei. Calculated fusion cross-sections for ²⁴Mg + ²⁰⁸Pb reaction are compared with a static-barrier-penetration model (SBPM) calculation to see the effect of reorientation. Heavy-ion reactions are also simulated in a 3-stage classical molecular dynamics (3S-CMD) model in which the rigid-body constraints are relaxed when the two nuclei are close to the barrier thus, taking into account all the rotational and vibrational degrees of freedom in the same calculation. This model is extended to simulate heavy-ion reactions such as⁶Li + ²⁰⁹Bi involving the weakly-bound projectile considered as a weakly-bound cluster of deuteron and ⁴He nuclei, thus, simulating a 3-body system in 3S-CMD model. All the essential features of breakup reactions, such as complete fusion, incomplete fusion, no-capture breakup and scattering are demonstrated.     

Neutron transfers and collective excitations at frontal and grazing low-energy collisions of heavy ions

Semi-classical models have been used to investigate the evolution of the wave functions of external neutrons and rotation of a deformed ¹⁵⁴Sm nucleus upon collision with a spherical ¹⁶O nucleus with energies near the Coulomb barrier. The probabilities of neutron transfers in the ⁴⁰Ca + ⁹⁶Zr reaction are determined.     

Quantum calculations of Coulomb reorientation for sub-barrier fusion

Classical mechanics and time dependent Hartree-Fock (TDHF) calculations of heavy ions collisions are performed to study the rotation of a deformed nucleus in the Coulomb field of its partner. This reorientation is shown to be independent of the charges and relative energy of the partners. It only depends upon the deformations and inertias. TDHF calculations predict an increase by 30% of the induced rotation due to quantum effects while the nuclear contribution seems negligible. This reorientation modifies strongly the fusion cross section around the barrier for light deformed nuclei on heavy collision partners. For such nuclei a hindrance of the sub-barrier fusion is predicted.     

Heavy ion reactions around the Coulomb barrier

The angular distributions of fission fragments for the ³²S+¹⁸⁴W reaction near Coulomb barrier energies are measured. The experimental fission excitation function is obtained. The measured fission cross sections are decomposed into fusion-fission, quasi-fission and fast fission contributions by the dinuclear system (DNS) model. The hindrance to completing fusion both at small and large collision energies is explained. The fusion excitation functions of ³²S+^90,96Zr in an energy range from above to below the Coulomb barrier are measured and analyzed within a semi-classical model. The obvious effect of positive Q-value multi-neutron transfers on the sub-barrier fusion enhancement is observed in the ³²S+⁹⁶Zr system. In addition, the excitation functions of quasi-elastic scattering at a backward angle have been measured with high precision for the systems of ¹⁶O + ²⁰⁸Pb, ¹⁹⁶Pt, ¹⁸⁴W, and ^154,152Sm at energies well below the Coulomb barrier. Considering the deformed coupling effects, the extracted diffuseness parameters are close to the values extracted from the systematic analysis of elastic and inelastic scattering data. The elastic scattering angular distribution of ¹⁷F+¹²C at 60 MeV is measured and calculated by using the continuum-discretized coupled-channels (CDCC) approach. It is found that the diffuseness parameter of the real part of core-target potential has to be increased by 20% to reproduce the experimental result, which corresponds to an increment of potential depth at the surface region. The breakup cross section and the coupling between breakup and elastic scattering are small.     

Heavy-ion fusion at low energies

Through precision measurements of fusion cross sections at energies close to the Coulomb barrier and through the application of the method of “experimental barrier distributions” which these permit, many recent advances have been made in our understanding of the dynamical processes occurring during a heavy-ion collision. It is now clear that the target and projectile reach one another in superpositions of states which correspond to different orientations for rotational nuclei or to different induced deformations for vibrational nuclei. The creation of a neck of neutron matter has also long been postulated and by studying the isotopic dependence of the fusion reaction, some recent results in the ¹⁰Ca+^90,96Zr systems appear to confirm this result. For large Z ₁ Z ₂ a type of extra-push effect can arise from the same inelastic entrance-channel effects which enhance the fusion of lighter systems, though this will be absent in cases where the enhancement arises from neutron transfers. The existence of different barriers will of course influence all other reaction channels. Fusion simply allows one to visualise the barriers most easily, since for this process, the total cross section is an incoherent sum of the contributions from all relevant eigenchannels. Some effects in other channels have already been observed. Other possible effects will be discussed. These include; the exploitation of the lowest-energy barrier to produce exotic evaporation residues and strongly deformed high-spin states at low excitation energy.     

Allowance for the shell structure of colliding nuclei in the fusion-fission process

The motion of two nuclei toward each other in fusion-fission reactions is considered. The state of the system of interacting nuclei is specified in terms of three collective coordinates (parameters). These are the distance between the centers of mass of the nuclei and the deformation parameter for each of them (the nose-to-nose orientation of the nuclei is assumed). The evolution of collective degrees of freedom of the system is described by Langevin equations. The energies of the Coulomb and nuclear (Gross-Kalinovsky potential) interactions of nuclei are taken into account in the potential energy of the system along with the deformation energy of each nucleus with allowance for shell effects. The motion of nuclei toward each other are calculated for two reaction types: reactions involving nuclei that are deformed (₄₂¹⁰⁰Mo + ₄₂¹⁰⁰Mo → ₈₄²⁰⁰Po) and those that are spherical (₈₂²⁰⁸Pb + ₈¹⁸O → ₉₀²²⁶Th) in the ground state. It is shown that the shell structure of interacting nuclei affects not only the fusion process as a whole (fusionbarrier height and initial-reaction-energy dependence of the probability that the nuclei involved touch each other) but also the processes occurring in each nucleus individually (shape of the nuclei and their excitation energies at the point of touching).     

Elastic and inelastic scattering of 16O and 12C by 24Mg near the Coulomb barrier

Angular distributions for ¹⁶O + ²⁴Mg and ¹²C + ²⁴Mg elastic and inelastic (2⁺, 1.37 MeV state in ²⁴Mg) scattering have been measured at energies spanning the Coulomb barrier. Apart from the structure typical of strong destructive Coulomb-nuclear interference, the data exhibit some additional specific features. Coupled channel calculations were performed, along with DWBA calculations to analyse the data using fixed coupling strengths deduced from the results of Coulomb excitation work. The importance of higher-order effects such as reorientation, is demonstrated.     

Proton and alpha evaporation spectra in low energy 12C and 16O induced reactions

Proton and alpha particle spectra have been measured in the ¹²C+⁹³Nb and ¹²C+⁵⁸Ni reactions at E(¹²C)=40 and 50 MeV and in the ¹⁶O + ⁹³Nb reaction at E(¹⁶O) =75 MeV. The spectra are compared with the statistical model calculations. The shapes of the calculated spectra are in agreement with experimental data except for the alpha spectrum in the ¹²C + ⁹³Nb reaction at 40 MeV. The observed evaporation bump is at ∼2 MeV lower energy compared to the calculated one. This discrepancy could imply alpha particle emission from a deformed configuration before compound nucleus formation at this near Coulomb barrier beam energy.     

Comparison of the fusion reactions12C+20Ne and16O+16O near the Coulomb barrier

The partial cross sections of heavy residual nuclei produced in the heavy ion fusion of¹²C+²⁰Ne have been measured atE _c.m.=6–15 MeV viaγ-ray spectroscopy with a Ge(Li) detector. Windowless and recirculating gas target systems have been used. The dominant residual nuclei are²⁴Mg,²⁷Al,²⁸Si,³⁰Si,³⁰P and³¹P, which arise from two- and three-body breakups in the exit channels. The observed excitation functions are smooth in their energy dependence and give no indications for the existence of pronounced resonance structures, in contrast to theoretical predictions. The Coulomb excitation of²⁰Ne served as an intrinsic calibration standard in the determination of absolute partial and total fusion cross sections. The same experimental set-up was also used in the reaction studies of¹⁶O+¹⁶O atE _c.m.=7–14 MeV, going through the same compound nucleus³²S at similar excitation energies. The observed energy dependence in the excitation functions is in good agreement with previous work. The total fusion cross section agrees fairly well with two sets of values reported previously, but deviates significantly from other reported absolute cross section values. The relative evaporation distributions of the residual nuclei are similar for both heavy ion reactions. However, the ratio of their total fusion cross sections deviates from model predictions and suggests that compound nucleus formation does depend on the microscopic structure of the colliding nuclei in the entrance channel. From the observed energy dependence of the above ratio, particularly at subcoulomb energies, geometrical effects in the entrance channel (due to deformed and spherical nuclei) appear to be weak. The astrophysical aspects of the data in the context of late stellar nucleosynthesis are discussed.     

Comparison of cross sections for C + O reactions in the second regime of complete fusion

Kinetic energy spectra, angular distributions, and elemental yield distributions have been measured for the ¹² C + ¹⁶ O, ¹² C + ¹⁸ O and ¹³ C + ¹⁷ O reaction products over an energy range from 2 to 7 times the Coulomb barrier energy. A careful kinematic analysis of the evaporation residues and comparisons with statistical-model calculations show that fusion proceeds with full momentum transfer followed by a statistical decay of the compound nucleus. The competition between complete fusion process and peripheral reactions in the ¹² C + ¹⁶ O system is less important than for the ¹² C + ¹⁸ O and ¹³ C + ¹⁷ O reactions. The unexpectedly high ¹² C + ¹⁶ O complete fusion cross sections are related to the possible occurrence of a superdeformation of the ²⁸ Si compound nucleus.     

Evaporation residue formation competing with the fission process in the197Au+16O,12C reactions and fission barriers at a specifiedJ window

Evaporation-residue excitation functions for¹⁶O and¹²C+¹⁹⁷Au reactions were measured by means of the activation technique. The competition between evaporation and fission of the compound nuclei was studied by comparing the observed evaporation-residue data with the published fission excitation functions. A newly devised analysis was applied in order to deduce a fission barrier height at a specified angular momentum and determine the relevant fissioning nucleus as well. We found the fission barriers to be 8.2 MeV for the²¹¹Fr nucleus at 16? and 8.2 MeV for the²⁰⁷At nucleus at 27?.     

A time-dependent quantum description of nucleon transfer in reactions with deformed nuclei

Time-dependent Schrödinger equations are solved by difference method for external neutrons of a spherical ⁴⁸Ca nucleus and a deformed ²³⁸U nucleus experiencing grazing collisions. At energies in the vicinity of the Coulomb barrier, the probabilities for the transfer of neutrons are determined as a function of the minimum internuclear distance, deformed nucleus orientation, and quantum numbers of the initial neutron states.     

Nuclear shapes and interaction potentials of two colliding208Pb nuclei at finite temperature

The effect of nuclear and Coulomb interactions on the shapes of two colliding²⁰⁸Pb nuclei at finite temperature is investigated. The complex potential energy density derived by Faessler and collaborators and the kinetic energy density and entropy density for two Fermi spheres at finite temperature are used to calculate the free energy of the²⁰⁸Pb +²⁰⁸Pb system in the energy density formalism. Shell corrections are added to the free energy in the framework of the Strutinsky method. The total free energy is minimized with respect to the quadrupole deformation and the diffuseness to determine the density distribution of²⁰⁸Pb nucleus at certain distanceR and temperatureT assuming the deformed Woods-Saxon shape for each nucleus. It is found that the nucleus acquires larger deformation and diffuseness as the temperature increases. The interaction potential between two²⁰⁸Pb nuclei is calculated from the minimized free energy. The total (nuclear + Coulomb) potential is found to decrease with increasing temperature, whereas the real part of the nuclear potential becomes more repulsive as the temperature increases.     

Correlations between emerging alpha-particles and heavy ions in 32S+197 Au collisions

In-plane angular correlations between α-particles and projectile residue nuclei (Z=5?22) emerging from ³²S+¹⁹⁷Au collisions have been measured at 373 MeV incident energy. In the combined Coulomb field of the projectile residue and the target nucleus coincident α-particles are strongly focused into a narrow cone. Model calculations indicate that α-emission originates from the region of the projectile which is facing the target nucleus during the collision.     

Coulomb contributions to nuclear radii and energies

The effect of including the isospin non-scalar Coulomb interaction (also taking into account the finite proton size) in binding energy and rms radius calculations for the closed shell nuclei ⁴He, ¹⁶O and ⁴⁰Ca is discussed in detail. Using the saturating Sussex matrix elements and including 0 + 2?ω excitations it is found that the major Coulomb contribution is in the first order, and that the pure second-order Coulomb contribution to the energy, near the saturation value of the size parameter for each nucleus, is more important than the mixed second-order contribution. Finally, for third order and above one can safely neglect the pure Coulomb contributions compared to the mixed ones which are in turn small compared to the pure nuclear contribution, in that order.     

Effect of octupole and higher deformations on Coulomb barrier

The effect of octupole deformation, β₃, and the higher multipole deformations, β₆ and β₈, on the distribution of barriers in orientation degrees of freedom, is studied. Coulomb barriers are derived in the frame work of the double folding model with the realistic M3Y nucleon–nucleon interaction and its density dependent version. ⁴⁸Ca + ²⁴⁴Pu spherical-deformed nuclear interacting pair is considered, as an example, to study the effect of deformation parameters on the height and position of the Coulomb barrier. Although the dependence of the Coulomb barrier parameters on the higher deformations is complicated, numerically, if it is treated in microscopic way, we found a simple linear variation of barrier parameters with the different orders of deformation. We found that the variation of the barrier parameters with deformation exactly follows the change of the half density radius of the deformed nucleus in the direction of the separation vector between the centers of mass of the interacting nuclei. This suggests a simple and straightforward way to predict the behavior of the barrier parameters with different orders of deformations. We compared the results of present work with a similar recent study of the same quantities based on simple expression derived by Wong for the Coulomb part and proximity approach for the nuclear part of heavy ion potential, respectively.     

IWB analysis of scattering and fusion cross sections for the 12C+12C, 13C+16O and 16O+16O reactions for energies near and below the Coulomb barrier

Calculations are presented of the elastic scattering and fusion cross sections for the astrophysically interesting reactions ¹²C+¹²C, ¹²C+¹⁶O and ¹⁶O+¹⁶O. The calculations are performed using the incoming wave boundary condition (IWB) and a real ion-ion interaction potential. The results are compared with the available experimental data for the energy region near and below the Coulomb barrier. With values of two adjustable potential parameters (the radial position of the l = 0 barrier and the diffuseness) determined by fitting elastic scattering data, good agreement is obtained for the average energy dependence of the ¹²C+¹²C and ¹²C+¹⁶O fusion cross sections. In the case of ¹⁶O+¹⁶O, both the calculated absolute magnitude and the energy dependence of the fusion cross section are inconsistent with the data and this discrepancy is discussed.     

High-energy approach for heavy-ion scattering with excitations of nuclear collective states

A phenomenological optical potential is generalized to include the Coulomb and nuclear interactions caused by the dynamical deformation of its surface. In the high-energy approach, analytical expressions for elastic and inelastic scattering amplitudes are obtained, where all the orders in the deformation parameters are included. The multistep effect of the 2⁺ rotational state excitation on elastic scattering is analyzed. Calculations of inelastic cross sections for the ¹⁷O ions scattered on different nuclei at about tens of MeV/nucleon or higher are compared with experimental data, and the important role of the Coulomb excitation is established. The text was submitted by the authors in English.     

Coulomb Potentials between Spherical and Deformed Nuclei

Coulomb potentials for spherical-deformed reaction as well as the conventional formulas. Our results approaches have quite different behaviours in the partners are calculated in terms of the double folding model show that the Coulomb potentials calculated with different internal region of the potential. Because fusion process is sensitive to the barrier height and the internal part of the potential, the fusion excitation function, especially the fusion barrier distribution, should provide a strict test of the interaction potentials. Therefore, we calculate the fusion excitation function and barrier distribution for the ^16O ＋^154Sm system with different versions of the Coulomb potentials, in comparison with the experimental results. It is found that the fusion excitation function and barrier distribution of ^16O＋^154Sm are obviously different for the different versions of the Coulomb potentials. By means of this comparison, we may conclude that the double folding model with the accurate approximate form can provide rather reasonable Coulomb potentials.