Analytical calculation of fusion barriers and cross-sections for spin-saturated colliding nuclei

We present an analytical formula for calculating the fusion barriers and fusion cross-sections. This is based on Skyrme energy density formalism and the Proximity force theorem. Comparisons with experimental data are also presented.     

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     

Fusion barriers,empirical and theoretical: Evidence for dynamic deformation in subbarrier fusion

Semiempirical fusion barriers have been obtained by analysis of 87 excitation functions for complete fusion. These barriers are used as a test for several theoretical potentials. For most of the reactions at energies below the operational s-wave fusion barrier, the cross sections are greater than expected from considerations of simple barrier penetration alone. We ascribe this to dynamic effects, such as neck formation which may be expected as precursors of fusion. The effects of static deformation do not appear to be sufficient to explain the discrepancies.     

Imploding shocks in laser-driven fusion

The role of a sequence of imploding spherical and cylindrical shocks is investigated in the context of laser-driven fusion in deuterium-tritium pellets. An approximate analytical treatment of a convergent sequence of shocks is presented within the framework of gas-dynamic equations for self-similar motion. These analytical solutions are compared with the exact numerical solutions. The solutions display an explicit dependence on the relative strength between the successive shocks and the ratio of the final to the initial pressure in the shocks. These solutions are employed to estimate the fusion yield for a given input shock energy.     

Distribution of fusion barriers

Heavy ion fusion cross sections and compound nucleus average spin values obtained from distribution of fusion barriers are discussed. Various shapes of distribution functions are studied using a truncated Gaussian distribution function (TGD). It is shown that fusion cross section and average spin values are less sensitive to different parametrization of TGD function, whereas the second derivative of the product of energy and fusion cross sections (w.r.t. energy), obtained from the corresponding TGD functions are significantly different depending on the shape of the barrier distribution function. It is also shown byχ ² analysis of fusion cross section data that some systems favour a narrow Gaussian distribution function whereas others, for which the vibrational and rotational collective states are less important, favour a flat barrier distribution. A physical interpretation of the dynamical process that gives rise to different barrier distribution is given in the framework of microscopic coupled channel calculations.     

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.     

A pocket formula for fusion barriers using proximity-type potentials

By using three potentials parameterized within the proximity concept, namely, Bass 80, AW 95, and Denisov DP, a pocket formula for fusion barriers is presented. This formula was obtained by analyzing as many as 400 reactions with mass between 15 and 296 units. Further, this pocket formula can reproduce the exact barriers within an accuracy of ±2%. A comparison with experimental data yields close agreement.     

Accuracy in the detemination of fusion barriers

Although there is no proof of the validity of the method, in a lot of papers, one computes the Second derivative of the the experimental fusion cross section multiplied by the energy to extract the fusion barriers directly from the experiment. The purpose of this paper is to check the consistency of the method, as experimental points are sandwiched inside their error bars. We therefore analysed the validity, by giving ourselves a cross section σ(E), resulting from the coupled channel calculation code ECIS, upon which, in spite of the complexity of the calculations, we have full control. We took these calculated points as the “experimental” ones and we altered them by multiplication by small random numbers. This is intended to simulate the error bars, of which we want to examine the influence on the fusion barriers. We find that in spite of the rough predictions yielded by the second derivative method, this task requires data with a precision difficult to reach. Furthermore, a careful check of the predictions of this method for coupled channels calculations shows that, due to the errors bars, this approach adds spurious results. Received: 6 January 1999 / Revised version: 30 March 1999     

Analytic description of the fusion and fission processes through compact quasi-molecular shapes

Recent studies have shown that the characteristics of the entrance and exit channels through compact quasi-molecular shapes are compatible with the experimental data on fusion, fission and cluster radioactivity when the deformation energy is determined within a generalized liquid drop model. Analytic expressions allowing to calculate rapidly the main characteristics of this deformation path through necked shapes with quasi-spherical ends are presented now; namely formulas for the fusion and fission barrier heights, the fusion barrier radius, the symmetric fission barriers and the proximity energy.     

Analysis of the fusion hindrance in mass-symmetric heavy ion reactions

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 hi...     

Effects of the entrance channel mass asymmetry in fusion reactions

The symmetric and asymmetric fusion reaction systems forming the same compound nuclei 26Al,30Si,38Ar and 170Hf are investigated with the frame of improved isospin dependent quantum molecular dynamics model.The entrance channel mass asymmetry dependence of compound nucleus formation is found by analyzing the shell correction energies,the Coulomb barriers and the fusion cross sections.The calculated fusion cross sections agree quantitatively with the experimental data.The results indicate that compound nucleus formation is favorable for the systems with larger mass asymmetry because of the smaller Coulomb contribution to the fusion barrier.     

Zipper mechanism of nanotube fusion: theory and experiment

We propose a new microscopic mechanism to explain the unusually fast fusion process of carbon nanotubes. We identify the detailed pathway for two adjacent (5,5) nanotubes to gradually merge into a (10,10) tube, and characterize the transition states. The propagation of the fused region is energetically favorable and proceeds in a morphology reminiscent of a Y junction via a zipper mechanism, involving only Stone-Wales bond rearrangements with low activation barriers. The zipper mechanism of fusion is supported by a time series of high-resolution transmission electron microscopy observations.     

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.     

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.     

核对称轴不同相对取向对熔合动力学的影响

计算了核对称轴不同相对取向时的熔合位垒.基于双核模型观念，考虑了熔合与准裂变的竞争，通过数值法求解主方程，计算了⁷⁶Ge+²⁰⁸Pb，⁴⁸Ca+²⁴⁴Pu核对称轴不同相对取向对熔合概率的影响，探索了最有利于超重元素合成的弹靶相对取向.取向不同时，对熔合反应的影响较大，计算结果表明弹靶碰撞为腰对腰时，更有利于发生熔合反应. 关键词： 超重元素 熔合概率 变形核 方向角度     

Subbarrier fusion of spherical nuclei of intermediate atomic number

A two-humped model was proposed for explaining the suppression of the cross sections for the deep-subbarrier fusion of spherical nuclei that have intermediate atomic numbers. It was shown that the suppression of these cross sections was due to the interference between the incident and reflected fluxes between the two barriers. The results of fitting the cross sections calculated within the two-humped model revealed the invariability of the distance between the vertices of the two barriers. As the charges of colliding nuclei increase, the inner barrier becomes higher than the outer one, and this determines the competition with cross sections for deep-inelastic-transfer reactions.     

Neutron excess in analytical expressions for fusion barrier parameters

The neutron excess dependence of heavy-ion fusion barriers is investigated, guided by predictions of different ion-ion potentials. We develop phenomenological expressions for the fusion barrier radii and heights, involving both the entrance channel mass asymmetry and neutron excess of the projectile and target. Compared to commonly used formulas, the developed expressions reproduce theoretical barrier parameters with a higher accuracy. Furthermore, they provide a means to assess the importance of the neutron excess degree of freedom implied by each potential.Received: 1 April 2003, Revised: 31 January 2004, Published online: 10 August 2004PACS: 24.10.-i Nuclear reaction models and methods - 25.70.-z Low and intermediate energy heavy-ion reactions - 25.70.Jj Fusion and fusion-fission reactions - 25.60.Pj Fusion reactions