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A new framework is proposed for the study of collisions between very heavy ions which lead to the synthesis of Super-Heavy Elements (SHE), to address the fusion hindrance phenomenon. The dynamics of the reaction is studied in terms of collective degrees of freedom undergoing relaxation processes with different time scales. The Nakajima–Zwanzig projection operator method is employed to eliminate fast variables and derive a dynamical equation for the reduced system with only slow variables. There, the time evolution operator is renormalised and an inhomogeneous term appears, which represents a propagation of the given initial distribution. The term results in a slip to the initial values of the slow variables. We expect that gives a dynamical origin for the so-called “injection point $s$” introduced by Swiatecki et al. in order to reproduce absolute values of measured cross sections for SHE. A formula for the slip is given in terms of physical parameters of the system, which confirms the results recently obtained with a Langevin equation, and permits us to compare various incident channels.

International audience

International audience; Synthesis of superheavy elements beyond oganesson is facing new challenges as new target-projectile combinations are necessary. Guidance from models is thus expected for future experiments. However, hindered fusion models are not well established and predictions in the fission barriers span few MeVs. Consequently, predictions are not reliable. Strategies to constrain both fusion hindrance and fission barriers are necessary to improve the predictive power of the models. But, there is no hope to get an accuracy better than one order of magnitude in fusion-evaporation reactions leading to superheavy elements synthesis.

For the synthesis of superheavy elements, it is indispensable to divide the fusion process into two steps : Overcoming the Coulomb barrier and passing over the conditional saddle or the ridgeline. To facilitate the understanding of the mechanism which explains the fusion hindrance, we first employ an analytic model with an inverted parabola for the saddle. Then, results by realistic calculations are given for the cold fusion. Ambiguities of the model are also discussed for future investigations. Since the model is general, it is applied to incident channels with neutron-rich projectiles and/or targets. These are necessary for synthesis of nucleides in so-called superheavy island around Z =114 and N =184.

Recent developments on reaction mechanisms in fusion of massive systems are briefly reviewed, especially on a two-step model. The fusion hindrance observed experimentally is simply explained in connection with the fluctuation-dissipation dynamics of shape evolutions and of two-body collisions. Examples are given for fusion excitation functions and compared with the available data on 48Ca+actinide target systems.

Based on the theory of the compound nucleus reaction, a brief review is given on the special aspects of the reaction dynamics in the synthesis of the superheavy elements (SHE), where the fusion probability is the most unknown factor. A new viewpoint of the fusion reaction is proposed that it consists of two processes; the first process up to the contact of two nuclei of the incident channel and the second one of a dynamical evolution to the spherical compound nucleus from the contact configuration. The fusion probability is, thus, given as a product of a contact probability and a formation probability. Analytic studies of the latter probability are discussed in the one-dimensional model, where a simple expression is given to the so-called extra-push energy in terms of the reduced friction, the curvature parameter of the conditional saddle point and the nuclear temperature. Preliminary results of numerical analyses of the contact probability are given, using the surface friction model (SFM). Remarks are given on the present status of our knowledge and for future developments.

We study the dynamical process of synthesizing superheavy elements on the basis of the fluctuation-dissipative dynamics with a multi-dimensional diffusion model. A multi-dimensional Langevin equation is used from the contact of two nuclei. After the dissipation of relative kinetic energy, a two-dimensional Smoluchowski equation is adopted taking into account the temperature dependent shell correction energy. The fusion probability is large for high incident energies. For low incident energies, owing to the restoration of the shell correction energy, the survival provability against fission is large. The competition between the two factors results in the optimum energy at 20–30 MeV. A three-dimensional model is used for the treatment of mass-asymmetric incident channels.

A dynamical theory is proposed for nuclear reactions leading to residues of superheavy elements. Fusion and fission processes are treated consistently by a diffusion equation which describes a time-dependent probability distribution in the collective coordinate or deformation space. The potential energy in the equation is time-dependent, because cooling due to particle evaporation gradually restores the shell correction energy which gives rise to a potential pocket essential for the stabilization of the superheavy elements around Z = 114 and N = 184. It is shown that there is an optimum initial excitation energy or incident energy of reactions as the result of a compromise between two conflicting requirements; higher energies which favour larger fusion probabilities and lower energies which favour larger residue probabilities or a quicker restoration of the shell-correction energy. A promising experimental direction is suggested.

The fusion-fission process in heavy systems is analyzed by the Smoluchowski equation with the finite-range droplet model potential of no pocket and the temperature-dependent shell correction energy which generates the pocket around the spherical shape. The evaporation residue cross sections of superheavy elements have been shown to have an optimum value at a certain initial temperature, due to the balance between the diffusibility for fusion at high temperature and the restoration of the shell correction energy against fission at low temperature. {copyright} {ital 1997} {ital The American Physical Society}

The fusion hindrance in heavy-ion collisions is studied in the framework of the two-center liquid drop model. It appears that the neck and the radial degrees of freedom might both be hampered by an inner potential barrier on their path between the contact configuration to the compound nucleus. Heavy-ion reactions with and without the two kinds of fusion hindrance are classified through systematic calculations. It is found that the number of reactions without radial fusion hindrance is much smaller than that without neck fusion hindrance, and for both kinds of fusion hindrance the number of reactions without fusion hindrance at small mass-asymmetry parameter $\ensuremath{\alpha}$ is smaller than that at large $\ensuremath{\alpha}$. In the formation of a given compound nucleus, if a reaction with ${\ensuremath{\alpha}}_{c}$ is not hindered, then other reactions with $\ensuremath{\alpha}g{\ensuremath{\alpha}}_{c}$ are also not hindered, as is well known experimentally.

3 figures; International audience; It is well known that there is a hindrance of fusion in collisions with heavy nuclei that plays a decisive role in synthesis of the super-heavy elements (SHE). The origin of the fusion hindrance is nowadays qualitatively understood but there are still quantitative ambiguities on the dynamics of the fusion mechanism and the predictions need to be assessed. In this communication, we stress the fact that dynamical effects play a crucial role in the amplitude of the reduction of the fusion probability.We found that the fast evolution of the neck degree of freedom affects the slow radial motion, i.e., the fusioning motion, through a dynamical coupling. We showed that we could do a so-called adiabatic elimination of the fast variable in the coupled equation, resulting in an effective one-dimensional equation for the radial motion with a shift of the starting point. This treatment of the dynamical coupling leads to a larger hindrance.

International audience; Present status of theory for synthesis of Super-Heavy Element (SHE) is reported, based on the compound nucleus reaction. Specific aspects are discussed in fusion and survival probabilities. They are the hindrance of fusion in the former and the fragility of the compound nucleus in the latter. Examples of applications are given.

International audience; Fusion process is shown to firstly form largely deformed mono-nucleus and then to undergo diffusion in two-dimensions with the radial and mass-asymmetry degrees of freedom. Examples of prediction of residue cross sections are given for the elements with Z=117 and 118.

The fusion hindrance, which is also denominated by the term extra-push, is studied on mass-symmetric systems by the use of the liquid drop model with the two-center parameterization. Following the idea that the fusion hindrance exists only if the liquid drop barrier (saddle point) is located at the inner side of the contact point after overcoming the outer Coulomb barrier, the reactions in which two barriers are overlapped with each other are determined. It is shown that there are many systems where the fusion hindrance does not exist for the atomic number of projectile or target nucleus $Z\leq43$, while for $Z>43$, all of the mass-symmetric reactions are fusion-hindered., 6 pages, 4 figures. to be published in Sci. in China G

Dynamics of the neck degree of freedom during fusioning process between heavy ions is studied. Time scales of the three degrees of freedom (the relative distance, the neck and the mass-asymmetry) are studied, showing an early equilibration of the neck. This means that a di-nucleus formed by the incident combination of ions quickly forms a mono-nucleus with a superdeformation during the fusion process and that the other two degrees of freedom have to be solved in a coupled way. A brief introduction of Langevin approach and dissipation-fluctuation dynamics is also given and of the application to the synthesis of the superheavy elements., 6 pages, 3 figures, Proceedings of the Japanese French Symposium - New paradigms in Nuclear Physics, Paris, 29th September - 2nd October, to be published in Int. J. of Modern Physics E

This work has been first presented at : First Workshop on State of the Art in Nuclear Cluster Physics, 13/16-05-2008 Strasbourg (France); International audience; The Two-Step Model for fusion of massive systems is briefly recapitulated, which clar- ifies the mechanism of so-called fusion hindrance. Since the neck changes the potential landscape, especially the height of the conditional saddle point, time evolution of the neck degree of freedom plays a crucial role in fusion. We analytically solve time-evolution of nuclear shape of the composite system from di-nucleus to mono-nucleus. The time- dependent distribution function of the neck is obtained, which elucidates dynamics of fusion processes in general, and thus, is useful for theoretical predictions on synthesis of the superheavy elements with various combinations of incident heavy ions.

The fusion process of $^{48}$Ca induced reactions is studied with the two-step model. In this model, the fusion process is devided into two stages: first, the sticking stage where projectile and target come to the touching point over the Coulomb barrier from infinite distance, and second, the formation stage where the di-nucleus formed with projectile and target evolve to form the spherical compound nucleus from the touching point. By the use of the statistical evaporation model, the residue cross sections for different neutron evaporation channels are analyzed. From the results, optimum reactions are given to synthesize $Z$ = 117 element with $^{48}$Ca induced reactions., Comment: 14 pages, 8 figures

A mechanism for fusion hindrance is clarified, based on the observation that the sticking configuration of projectile and target is located outside of the conditional saddle point. Accordingly, the fusion process is described by two sequential steps of passing over the Coulomb barrier and shape evolution toward the spherical compound nucleus. The latter one is indispensable in massive systems. With the use of a two-step model, excitation functions of fusion reaction are calculated for various combinations of projectiles and targets which lead to superheavy elements. The hindered fusion excitation measured is reproduced precisely without any adjustable parameter. Combined with survival probabilities calculated by the statistical theory of decay, excitation functions for residues of superheavy elements are calculated to compare with the systematic data measured for the cold fusion path. The peak positions and the widths are correctly reproduced, though it is necessary to reduce the shell correction energies of the compound nuclei predicted by the structure calculations in order to reproduce their absolute values. Predictions are made for a few unknown heavier elements. Furthermore, a preliminary attempt toward the shell closure N = 184 is also presented using a neutron-rich secondary beam.

The two-step model and the statistical decay code KEWPIE is applied to the 1n reactions producing Z=110, Z=111 and Z=112. The experiments are well reproduced with a single scaling parameter to the shell correction energy. Thus, an extension of the recipe is expected to predict reliable values of the residue cross-sections of the elements Z=113 and Z=114 which will be a useful guide for planning of experiments.

The two-step model for fusion reactions of massive systems is briefly reviewed.By the use of fusion probabilities obtained by the model and of survival probabilities obtained by the new statistical code, we predict residue cross sections for 48Ca+actinide systems leading to superheavy elements with Z=114, 116 and 118., 7 pages, 4 figures, Halong Bay meeting proceeding

A new model is proposed for fusion mechanisms of massive nuclear systems where so-called fusion hindrance exists. The model describes two-body collision processes in an approaching phase and shape evolutions of an amalgamated system into the compound nucleus formation. It is applied to $^{48}$Ca-induced reactions and is found to reproduce the experimental fusion cross sections extremely well, without any free parameter. Combined with the statistical decay theory, residue cross sections for the superheavy elements can be readily calculated. Examples are given., Comment: 4 pages, 4 figures

A two-step model is proposed for the fusion mechanism of massive heavy-ion systems which is the most unknown part in the reaction theory for the synthesis of the superheavy elements. It consists of the approaching phase of incident ions and the dynamical shape evolution of the amalgamated system toward the spherical compound nucleus. Preliminary results are presented.

The dynamical process of synthesizing superheavy elements is studied on the basis of the fluctuation-dissipative dynamics. The whole process is divided into three stages, i.e., the approaching stage, the formation stage, and the surviving stage. For the study of the formation stage, a three-dimensional Langevin equation is used from the contact of two nuclei to calculate the competition between the complete fusion and the quasi-fission. We estimate the effects of the nuclear deformation on the fusion probability. The results are consistent with the experimental results. For the description of the surviving stage, a statistical model is used to estimate the competition between the fission and the evaporation process. In this stage, it is very important to include the temperature-dependent shell correction to the fission barrier and the collective enhancement of the level density. From the study of the isotope dependence of the production cross section, the survival probability is found to be very sensitive to the separation energy of neutron. The results show the importance of the use of neutron-rich beams and targets.

A dynamical framework for synthesis of the superheavy elements is presented, where fusion process is treated by fluctuation-dissipation dynamics with Langevin equations in two steps; approaching phase and shape evolution. Preliminary results are given for residue cross sections of Z=114 with 48Ca+244Pu entrance channel, combining the statistical decay calculations.

Starting from the contact configuration of colliding nuclei, the evolution of the nuclear shape is simulated in two-dimensional deformation space fixing the asymmetry by means of the Langevin equation under the influence of the one body dissipation. It is quantitatively shown that, in heavy mass region, the asymmetry of the entrance channel plays an important role to get an optimum fusion probability in connection with the extra barrier to be overcome. It is necessary to take into account this effect together with the excitation energy at Bass barrier to consider the optimum target-projectile combination for the synthesis of super heavy elements.

We have investigated the fusion-fission process of massive nuclei using the multi-dimensional Langevin equation and the statistical model. The compound nucleus formation probability has been obtained as a function of extra energy above incident barrier. The excitation function of evaporation residue cross section for three systems; 100Mo+100Mo, 100Mo+110Pd, and 110Pd+110Pd, were calculated using the survival probability against fission obtained with the statistical model and were compared with experimental data. Our results reproduce the phenomenon of fusion hindrance for the fusion reactions of massive nuclei in the case of nearly symmetric projectile-target combinations.

Over a wide energy range, the limitation on fusion is interpreted from the properties of the compound nuclear level density obtained by taking into account the finiteness of the nucleus. The essential point is to use the shell model single particle bound states in the calculation of the grand partition function. The critical angular momentum for fusion is estimated in connection with the critical temperature for fusion.

A band crossing model is shown to be successful in the explanation and the prediction of resonance phenomena in heavy ion scatterings, which is exemplified in the 12C+16O system. Oscillations observed in fusion cross sections, for example in the 12C + 12C system, are shown not to be resonances or a reflection of resonances, but to be signals of strong absorption suffered by particular partial waves. Molecular resonances, possible existences of which are predicted by the band crossing model, are investigated whether they survive or are smeared out under strong absorption. The absorption is treated by smooth‐cut‐off model. The intermediate structure observed in the elastic excitation function of the 12C + 12C system is reproduced excellently well and shown to be a direct reflection of existence of molecular resonances.