Calculation of the Coulomb potential between spherical-deformed and deformed-deformed nuclei using the Monte Carlo method

The Coulomb potential between spherical-deformed and deformed-deformed nuclei has been calculated using the Monte Carlo simulation. The results obtained for the Coulomb potential in the ¹⁶O$ + $²³⁸U and ²⁷Al$ + $⁷⁰Ge reactions are in good agreement with those obtained using the double-folding method. The simulation technique employed here has the ability of calculating the Coulomb potential taking into account the finite diffuseness parameter, all the possible deformation degrees of freedom, and different orientations of the symmetry axes of the target and the projectile nuclei with respect to each other. The accuracy of this simulation technique is high and the computer time taken to do these calculations is much less than those of the double-folding method.     

Analytical calculation of fusion cross-sections

We analyse the fusion cross-sections, calculated by using two different analytical parameterisations and compare them with the experimental data. Both the parameterisations are based on ion-ion potentials calculated within the framework of Skyrme energy density formalism. In the first case, the ion-ion potential (including the spin-density term) was parameterised and then, by adding the Coulomb potential, one could compute the fusion barrier analytically. In the second case, the calculated fusion barrier heights and positions were parameterised directly. Both of these (previously) reported parameterisations are used here to calculate the fusion barriers and fusion excitation functions for more than 50 reactions belonging to the s-d and f-shell nuclei. A detailed comparison of these parametrisations with the experimental and several other theoretical results shows that both of these parameterisations are able to reproduce the experimental data equally well. As the (second) direct parameterisation depends only on the charges and masses of colliding nuclei, it is very useful for predicting/ understanding the fusion process in low energy heavy-ion reactions. Received: 24 February 1999 / Accepted: 16 March 2000     

A semiclassical formula for the reaction cross-section of heavy ions

We discuss an improvement to the expression for the heavy-ion reaction cross-section proposed long ago by Karol (P.J. Karol, Phys. Rev. C 11, 1203 (1975)) which describes the two-ion interaction through the nucleon-nucleon collisions in their overlap region when they come into contact. This improvement consists in considering the limitations due to the Pauli principle, which at low incident energies forbids many nucleon interactions, and in including a Coulomb factor. The final expression for which we also propose a new parametrization reproduces satisfactorily the known reaction cross-sections for many ion combinations using a global set of parameters. A decomposition of the reaction cross-section into the contributions of peripheral interactions, as a function of the impact parameter, is also discussed.     

Production cross sections for the heaviest nuclei in complete fusion reactions induced by heavy ions

Production of the heaviest nuclei in complete fusion reactions induced by heavy ions has been considered in a systematic way in the framework of the conventional barrier passing model coupled with the statistical model. Available data on excitation functions for fission and production of evaporation residues (ER) in very asymmetric combinations induced by ions lighter than Ne on actinide target nuclei are described rather well in the framework of these models. The data allow one to adjust model parameters and to reveal the quasi-fission effect caused by the interaction with deformed target nuclei, which is manifested in the suppression of the ER production at sub-barrier energies. For reactions induced by Mg and heavier projectiles, quasi-fission is starting to suppress fusion (ER production) at energies above the Coulomb barrier. One has to introduce empirically the quantity of the fusion probability P_fus to reproduce the ER excitation functions in the framework of the conventional approach. The exponential dependence of P_fus on the combined fissility parameter (a similar parameter that was introduced for the extra-push energy scaling) was found in search for scaling for the P_fus values resulting from the data analysis.     

Nuclear Potential and Fusion Cross Sections for Synthesizing Super-Heavy Elements in Di-nuclear Systems

A double foMing method with simplified Skyrme-type nucleon nucleon interaction is used to calculate the nuclear interaction potential between two nuclei. The calculation is performed in tip-to-tip orientation of the two nuclei if they are deformed. Based on this method, the potential energy surfaces~ the fusion probabilities and the evaporation residue cross sections for some cold fusion reactions leading to super-heavy elements within di-nuclear system model are evaluated. It is indicated that after the improvement, the exponential decreasing systematics of the fusion probability with increasing charge number of projectile on the Pb based target become better and the evaporation residue cross sections are in better agreement with the experimental data.     

A Possible Origin of e+e- Peaks in Heavy-Ion Collisions

The behavior of a neutral composite particle composed of a positron and an electron in the strong Coulomb field of a nuclear uquasimolecule formed in low-energy heavy-ion collions is examined. A strong attractive interaction between the composite particle and the Coulomb field is found when the combined nuclear charge of two nuclei is sufficiently large, which is possible to lead to the e⁺e^- pair production in heavy-ion collisions.     

Nuclear dynamical octupole deformation in heavy-ion reactions

Within the quantum molecular dynamics (QMD) model, the dynamical octupole deformation is studied as a function of the central distance between the projectile and target in the approaching process of heavy-ion fusion reactions. The dependence of the maximum dynamical octupole deformations on the incident energies is also investigated. The dynamical octupole deformations can be observed during the approaching process, and the maximum dynamical octupole deformations become more significant with decreasing incident energies. The distributions of the proton and neutron centers in the projectile and target are also investigated, respectively. In the approaching process of heavy-ion fusion reactions, the separation between proton centers for two nuclei is larger than that between neutron centers because of the strong Coulomb potential.     

Vacuum polarization and the nuclear mass formula

The effect of vacuum polarization on the ground-state mass systematics of nuclei is considered. Its contribution to the ground-state energy of heavy nuclei is found to be a few MeV and can be simulated by using an effective nuclear radius in the Coulomb energy term of the mass formula. Its effect on the fission barriers seems to be small. The sub-barrier heavy-ion fusion cross section is found to be reduced at most by an order of magnitude with the inclusion of vacuum polarization in the interaction potential.     

Nuclear Hexadecapole Deformation Effects on the Production of Super-Heavy Elements

We investigate the effects of the nuclear hexadecapole deformations on the interaction potentials between nuclei, the driving potentials and the fusion probabilities for some cold fusion reactions leading to super-heavy elements. It is found that nuclear hexadecapole deformations change significantly the structure of the driving potentials and the fusion probabilities for some reaction channels.     

Transfer reactions and sub-barrier fusion in 40Ca + 90, 96Zr

The two systems ⁴⁰Ca + ^90,96Zr have been studied by measuring nucleon transfer reactions at two energies near the Coulomb barrier, thus complementing the available sub-barrier fusion cross-sections. Angular distributions for various transfer channels have been determined. Significantly larger neutron transfer cross-sections are found for the target ⁹⁶Zr that exhibits the larger enhancement in the sub-barrier fusion cross-sections. All data have been analyzed with a new model for heavy-ion collisions that calculates simultaneously transfer cross-sections, fusion excitation functions and barrier distributions. The model gives a good account of both transfer and fusion data. Received: 2 May 2002 / Accepted: 4 June 2002 / Published online: 26 November 2002 RID="a" ID="a"e-mail: montagnoli@pd.infn.it, Fax +39049 8277102, Tel. +39049 8277117. RID="b" ID="b"On leave from the China Institute for Atomic Energy, 102413 Beijing, China. Communicated by C. Signorini     

Energy dependence of Lévy stability and multi-fractal spectrum in e<Superscript>+</Superscript>e<Superscript>−</Superscript> collisions

Various nuclear reactions like quasi-fission, fusion-fission or particle and cluster evaporation from excited compound nuclei were studied in heavy-ion reactions at the velocity filter SHIP of GSI. The velocity filter offers the possibility to detect all reaction products under zero degree relative to the beam direction. Together with the measurement of the product velocity distribution this allows for an identification of the underlying reaction mechanism. This article is focussed on reactions of ²⁵Mg and ⁶⁴Ni beams on ^{206, 207}Pb targets at energies of 5.9×AMeV and 8.7×AMeV . Besides evaporation residues from ²⁵Mg + ²⁰⁶Pb collisions we found evidence for rotation and quasi-fission of nuclear molecules formed in the entrance channel after the capture stage. The break-up of the systems showed a preferred clustering leading to isotopes in the region 84 ⩽ Z ⩽ 88 and 122 ⩽ N ⩽ 127 of the chart of nuclei.     

Influence of entrance channel properties on heavy-ion reaction dynamics

In the fusion of heavy nuclei, there is a distribution of fusion barrier energies resulting from coupling between intrinsic motion and internal degrees of freedom. Precise experimental measurements of excitation functions have allowed the extraction of the distributions by taking the second derivative using a point-difference method. In the case of statically deformed nuclei, experimental data shows that the different fusion barrier energies correspond to different physical configurations of the colliding nuclei, the latter affecting the subsequent dynamical trajectories over the potential energy surface, influencing the ultimate reaction products, as for example in quasi-fission. The fusion barrier distribution is also valuable in understanding the fusion of weakly bound nuclei, enabling a reliable prediction of the expected fusion cross-sections, and thus the determination of fusion suppression factors at above-barrier energies. Received: 1 May 2001 / Accepted: 4 December 2001     

Modified Glauber model for the total reaction cross-section of 12C + 12C collisions

Glauber's theory has been adopted to calculate the total heavy-ion reaction cross-sections at high energies. At relatively low energies, Glauber's total reaction cross-section has been modified in order to take into account the Coulomb field effect and is called modified Glauber model I. In addition to the Coulomb field effect, the nuclear effect has also been taken into account in the Glauber model and is called modified Glauber model II. An analytical expression for the transparency function for heavy-ion reactions, involving the nuclear densities of the colliding ions and the nucleon-nucleon cross- section, has been obtained within the framework of the modified Glauber models I and II. The transparency and the total reaction cross-sections of the ¹²C + ¹²C collisions are calculated at different bombarding energies. The obtained results are in good agreement with the experimental data and with previous theoretical calculations. Received: 26 January 2001 / Accepted: 15 October 2001     

Entrance channel potentials in the synthesis of the heaviest nuclei

Entrance channel potentials in nucleus-nucleus collisions, relevant for the synthesis of superheavy elements, are systematically studied within a semi-microscopic approach, where microscopic nuclear densities of the colliding spherical or deformed nuclei are used in semi-classical expressions of the energy-density functional. From experimental data on fusion windows evidence is found that the existence of pockets in the entrance channel potentials is crucial for fusion. Criteria for the choice of the best collision systems for the synthesis of superheavy elements are discussed. Received: 14 May 2002 / Accepted: 6 June 2002 / Published online: 26 November 2002 RID="a" ID="a"e-mail: v.denisov@gsi.de, denisov@kinr.kiev.ua RID="b" ID="b"e-mail: w.nrnbrg@gsi.de Communicated by P. Schuck     

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.     

Enhanced neutron pair transfer and collective excitations in the system 206Pb + 118Sn at barrier energies

At energies below the Coulomb barrier, neutron transfer and Coulomb excitation have been measured in a very heavy asymmetric nuclear system, in ²⁰⁶Pb + ¹¹⁸Sn. These are semi-magic nuclei showing super-fluid properties. Particle-γ coincidence techniques using 5 Euroball Cluster detectors (EB), combined in a set-up with the Heidelberg-Darmstadt NaI Crystal Ball (CB), have been used. Position-sensitive detectors allowed the observation of scattering processes covering angles from 110 up to 150 degrees. The fragments are identified via the known γ-decays of the lowest excited states using the high resolution of EB. Using the unique feature of the set-up with the CB, transfer to well-defined final channels with known quantum numbers is selected using the high-efficiency multiplicity filter of the CB with no second γ-ray, i.e. without feeding. The data are analysed using the semi-classical approach and transfer probabilities are obtained. Coulomb excitation has been analysed using known transition probabilities. The enhancement is deduced for the two-neutron transfer populating the low-lying super-fluid 2⁺ states in ¹²⁰Sn and ¹¹⁶Sn, while the 2n transition remains in the ground state for the ^20NPb nuclei. Large enhancements up to EF ≃ 10³ are observed. This is the first observation of neutron pair transfer enhancement for a heavy nuclear binary system with super-fluid properties with experimentally separated levels. The calculations with microscopic 2-neutron wave functions, with configuration mixing over six shell model configurations and using the coupled reaction channels approach, reproduce well the observed probabilities and the enhancement. Received: 27 August 2002 / Accepted: 9 December 2002 / Published online: 25 March 2003 RID="a" ID="a"e-mail: oertzen@hmi.de Communicated by D. Schwalm     

Evolution of the fusion cross-section for light systems at intermediate energies

Ar + Ni and Ni + Ni collisions are investigated between 32 and around 100A MeV incident energy with the 4π multidetector INDRA. Fusion cross-sections are found to decrease from ∼ 180mb at 32A MeV to zero above 50A MeV. Other experimental results, for light systems, are compared. Moreover, theoretical works are discussed and fusion cross-sections, calculated from two dynamical simulations based on nuclear Boltzmann equation (Boltzmann-Nordheim-Vlasov and Landau-Vlasov models), are also compared to experimental results.     

Strong enhancement of two-neutron transfer in the system 206Pb+118Sn

One and two neutron transfer has been measured in the heaviest asymmetric nuclear system with semi magic nuclei showing superfluid properties, in ²⁰⁶Pb+¹¹⁸Sn collisions at an energy well below the Coulomb barrier with scattering orbits covering the largest angles. Particle-γ coincidence techniques using 5 Euroball-Cluster detectors (EB) combined in a set-up with the Heidelberg-Darmstadt NaI-Crystal Ball (CB) have been used. Transfer channels are identified with EB via their known γ-decays of the lowest excited states. Using the unique feature of the set-up with the CB, transfer to well defined final states with known quantum numbers (without feeding) are selected using the high efficiency multiplicity filter of the CB (no second γ-ray). The data are analysed using the semiclassical approach and transfer probabilities are obtained. The enhancement for the two-neutron transfer populating the low lying superfluid 2⁺ state in ¹²⁰Sn (and ¹¹⁶Sn), while the Pb-branch is in the groundstate is deduced by comparison with the strongest single neutron transfer transition. Large enhancements (EF ≃ 10³) are observed. This is the first direct measurement of enhancement for a heavy nuclear binary system with experimentally separated levels suggesting a strong contribution from superfluid pair transfer. Received: 18 December 1998 / Revised version: 23 February 1999     

Neck dynamics

Intermediate-energy heavy-ion reactions produce a mid-rapidity region or neck, mostly in the semiperipheral collisions. Brief theory and experiment surveys are presented. General properties of the mid-rapidity zone are reviewed and discussed in the framework of reaction dynamics. Hierarchy effect, neutron enrichment, isospin diffusion are all new neck phenomena which are surveyed. The main neck observables are also examined, mainly in the context of the symmetry term of the nuclear equation of state.