Double folding model calculation applied to fusion reactions

The interaction potential between a spherical and a deformed nucleus is calculated within the double-folding model for deformed nuclei.We solve the double folding potential numerically by using the truncated multipole expansion method.The shape,separation and orientation dependence of the interaction potential,fusion cross section and barrier distribution of the system 16O+154Sm are investigated by considering the quadrupole and hexadecapole deformations of 154Sm.It is shown that the height and the position of the barrier depend strongly on the deformation and the orientation angles of the deformed nucleus.These are quite important quantities for heavy-ion fusion reactions,and hence produce great effects on the fusion cross section and barrier distribution.     

Calculation of Interaction Potentials between Spherical and Deformed Nuclei

The interaction potential for spherical-deformed reaction partners is calculated. The shape, separation and orientation dependence of the interaction potential and fusion cross section of the system ^32S＋^154Sm are investigated within the double-folding model of the deformed nuclei. The effective nucleon-nucleon interaction is taken to be the M3Y-Reid potential. The density is considered for three terms of the expansion using the truncated multipole expansion method, which is a deformed Fermi shape With quadrupole and hexadecapole for the density distribution of ^154Sm. It is found for the interaction potential that the height and the position of barrier strongly depend on the deformations, the orientation angle of the deformed nucleus, and hence produce great effects on fusion cross section. The integrated fusion cross section is in good agreement with the experimental data.     

Alpha-induced reaction cross-section for Sm,U, Np targets: influence of hexadecapole deformation and deformed surface diffuseness

Alpha-induced reactions on~(154)Sm,~(233,235,236,238)U, and ~(237)Np deformed nuclei are studied theoretically.The effects of hexadecapole deformation, deformed surface diffuseness parameter, and orientation on barrier height and position, fusion cross-section at any angle, and fusion cross-section have been investigated. Both hexadecapole deformation and deformed surface diffuseness can affect barrier characteristics and enhance fusion cross-section. Good agreement between experimental data and theoretical calculations with quadrupole and hexadecapole deformation and deformed surface diffuseness were observed for the ~4He+~(154)Sm,~(235)U,~(237)Np reactions.     

重离子碰撞中熔合位垒的研究

熔合位垒的研究对熔合反应以及超重核合成有重要的意义。在改进的同位旋相关的量子分子动力学(ImIQMD)模型框架下,提取了熔合反应体系40Ca+40Ca,48Ca+208Pb,48Ca+204Pb和16O+154Sm的熔合位垒。研究了壳修正能对熔合位垒的影响、动力学位垒的能量依赖性、同位旋效应以及形变核的方向效应。计算发现壳修正能降低了熔合反应的位垒。在研究动力学位垒的能量依赖性时,发现位垒高度和位垒半径表现出相反的能量依赖行为。在动力学反应中,当两个核距离接近时,缺中子体系的库仑势同样表现出一定的能量依赖性。对于丰中子体系,由于中子屏蔽作用,库仑势基本没有能量依赖性。     

The sub-barrier fusion of40Ar with144,148,154Sm

The cross sections for production of evaporation residues (σ _er) and for fusion-fission (σ _ff) have been measured for⁴⁰Ar+^{144, 148, 154}Sm at sub-barrier energies by observation of x-ray emission from radioactive products and by direct,ΔE?E identification of fission fragments, respectively. These isotopes span the transition region from spherical (¹⁴⁴Sm) to strongly deformed (¹⁵⁴Sm) equilibrium shapes. The cross section for fusion,σ _fus=σ _er+σ _ff, is found to vary markedly at low energies with the isotope number and, hence, with the quadrupole collectivity of the target. The thresholds for fusion of¹⁴⁸Sm and¹⁴⁴Sm are, respectively, ～3.5 MeV and ～7 MeV (c.m.) higher than for fusion with¹⁵⁴Sm. These differences and the energy dependence of the fusion cross sections are discussed in terms of the effect of nuclear deformation on heavy-ion fusion. A comparative analysis of results for¹⁶O+Sm and⁴⁰Ar+Sm in terms of static deformation indicates thatσ _fus for the Ar+Sm system at very low energies is enhanced relative to the prediction for a one-dimensional barrier based on a fit toσ _fus for¹⁶O+Sm. This may be an indication that additional degrees of freedom (such as formation of a neck or fragment elongation) may be important for fusion with the larger projectile. At energies above the fusion barrier, values ofσ _fus for^{144, 148}Sm are nearly equal, but are significantly smaller than for¹⁵⁴Sm. This is in contrast to the results of previous experiments with¹⁶O projectiles in whichσ _fus (¹⁶O+¹⁴⁸Sm) andσ _fus (¹⁶O+¹⁵⁴Sm) were nearly equal above the barrier. These differences, observed for^{144, 148}Sm and¹⁵⁴Sm at energies above the barrier may reflect a new mechanism which is not encompassed by a static theory.     

Interaction and fusion of deformed nuclei

It is shown that the height of the barrier and its position, as well the depth of the capture well, are highly sensitive to the relative orientation of colliding strongly deformed nuclei. It is found that the fusion/capture cross sections and the nucleus-nucleus potential for heavy nuclear systems depend greatly on the magnitude and sign of the quadrupole deformation of nuclear surfaces. In order to describe correctly the cross section for the capture of heavy strongly deformed nuclei, it is necessary to perform averaging over all three angles that describe their relative orientation. Allowance for a hexadecapole deformation leads to a significant increase in the capture cross section for very heavy nucleus-nucleus systems.     

Systematic study of deformation effects on fusion cross-sections using various proximity potentials

The influence of static quadrupole and hexadecapole (positive & negative) deformation of targets are studied using eleven different versions of nuclear potentials. The height and position of the interaction barrier for the reactions induced by spherical projectile (¹⁶O) on the deformed targets such as ¹⁶⁶Er, ¹⁵⁴Sm and ¹⁷⁶Yb have been estimated. It is found that the nucleus-nucleus potential strongly depends on the value of the deformation parameters and orientation of the target. The experimental fusion cross-section of the reactions ¹⁶O + ¹⁷⁶Yb, ¹⁶O +¹⁶⁶Er and ¹⁶O +¹⁵⁴Sm are investigated by applyingWong’s formula using various parameterizations of the proximity potential as well as an assessment of the results of a multi-dimensional barrier penetration model (BPM). The fusion cross-sections by Prox 77, Prox 88, Prox 00, Prox 00DP, Denisov DP, Bass 80, CW 76 and AW 95 potentials are found to be better than the rest in comparison to experimental data.     

Coulomb reorientation in near-barrier fusion of deformed+spherical systems in classical dynamical approach

A classical rigid-body dynamics model which takes into account all the translational and the rotational degrees of freedom is developed to study Coulomb reorientation of deformed nuclei in heavy-ion collisions. Various aspects of the collision dynamics in the case of near-barrier fusion of ²⁴Mg + ²⁰⁸Pb system due to the Coulomb reorientation are studied; the dependence of the extent of reorientation of the symmetry axis of the deformed nucleus, isotropy of the initial orientations, barrier parameters, and rotational excitation energy are discussed in detail. It is found that the barrier parameters not only depend on the initial orientations of the deformed nucleus but also on the collision energy; with maximum reorientation effect at near- and below-barrier energies. Even small amount of the rotational excitation energy gained by the deformed nucleus at large separation distances is crucial in determining the conditions at the barrier. Study of ¹⁵⁴Sm + ¹⁶O and ²³⁸U + ¹⁶O systems involving heavier deformed nuclei shows that the extent of reorientation also depends on the moment of inertia of the deformed nucleus.     

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.     

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.     

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.     

Alignment effects in subbarrier fusion reactions

Subbarrier fusion reactions in which one constitutent is deformed and aligned are considered in the framework of a coupled-channel model with incoming-wave boundary conditions. The effect of alignment on the fusion cross section is estimated to be 40% for aligned ¹⁶⁵Ho. The semiclassical analog to the coupled-channel equations is developed, yielding an expression for the cross section in terms of a weighted average over all orientations of the deformed nucleus.     

Investigation of the role of the projectile-target orientation angles on the evaporation residue production

The measured yield of evaporation residues in reactions with massive nuclei have been well reproduced by using the partial fusion and quasifission cross sections obtained in the dinuclear-system model. The influence of the orientation angles of the projectile- and target-nucleus symmetry axes relative to the beam direction on the production of the evaporation residues is investigated for the ⁴⁸Ca + ¹⁵⁴Sm reaction as a function of the beam energy. At the low beam energies only the orientation angles close to αP = 30° (projectile) and αP = 0°–15° (target) can contribute to the formation of evaporation residues. At large beam energies (about E _c.m. = 140–180 MeV) the collisions at all values of orientation angles αP and α _T of reactants can contribute to the evaporation residue cross section which ranges between 10–100 mb, while at E _c.m. > 185 MeV the evaporation residue cross section ranges between 0.1–1 mb because the fission barrier for the compound nucleus decreases by increasing its excitation energy and angular momentum.     

Near-barrier transfer in16O +144, 154Sm

The production of nitrogen and carbon fragments in the reactions induced by¹⁶O on¹⁴⁴Sm and¹⁵⁴Sm targets was studied at two bombarding energies (65 MeV and 72 MeV) around the interaction barrier. The measured angular distributions exhibit the characteristic behaviour of direct transfer reactions, and theQ-value spectra are typically centered at the values predicted by kinematic selectivity. The comparison between fusion and charged-particle transfer cross sections shows that whereas fusion is the most important process at the highest bombarding energy, transfer reactions become comparable or even dominant (as in the¹⁴⁴Sm case) at near-barrier energies. The different transfer probabilities observed for the two systems are analyzed in terms of nuclear deformation.     

INFLUENCE OF SHELL STRUCTURE ON DEEP INELASTIC COLLISIONS

A kinematically complete study of the symmetric systems ¹⁵⁴Sm+¹⁵⁴Sm and ¹⁴⁴Sm+¹⁴⁴Sm has been performed at energies 30% in excess of the interaction barrier. They have been chosen because of their different internal structure: ¹⁴⁴Sm has a closed N=82 neutron shell and hence a spherical ground state confighration; ¹⁵⁴Sm with ten neutrons outside this shell is strongly deformed. Over the whole range of kinetic energy loss the variances of the measured mass distributions were found to be similar in both reactions, whereas the variances of the element distributions are considerably larger at small energy losses in the ¹⁴⁴Sm system. Based on the shell-corrected potential energy surface these observations are attributed to the closed N=82 neutron shell which, for ¹⁴⁴Sm, hinders the neutron exchange and leads to a preferential transfer of protons.     

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.     

Quadrupole moments and nucleon excitation strengths in even-A Sm isotopes

We present a coherent coupled-channel analysis of 7 MeV neutron and 16 MeV proton elastic and inelastic scattering from ^{148, 152, 154}Sm. The optical potential and nuclear deformation parameters are determined so as to fit not only these elastic and inelastic scattering data but also the low-energy neutron scattering properties and the total cross sections over a wide energy range. This analysis provides evidence of the same excitation strengths for both projectiles in the case of ^{152, 154}Sm, and of a smaller excitation strength for the proton than for the neutron in case of ¹⁴⁸Sm. Moreover the quadrupole moments of these deformed optical potentials are in good agreement with those extracted from Coulomb excitation measurements and from nuclear matter distribution calculations.     

The interaction and fusion of arbitrarily oriented deformed nuclei

The fusion/capture cross sections between various deformed nuclei are calculated. It is shown that quadrupole and hexadecapole deformations of heavy nuclei are important for evaluating the barrier height, capture well depth, and fusion/capture cross sections of heavy nuclei. It is found that calculations of the capture cross section of two heavy deformed nuclei must be performed with averaging over all possible mutual orientations of the colliding nuclei.     

Shape-enhanced fusion: increasing the reactivity for some advanced fusion fuels

We show that an order of magnitude enhancement of the fusion cross section is possible for some advanced fusion fuels involving deformed nuclei providing the reacting nuclei fuse in a directed orientation. This process is a manifestation of Coulomb barrier reduction and we identify some candidate advanced fusion fuels which could exploit this phenomenon including p-¹¹B, p-⁷Li and p-¹⁰B.