Fusion Barrier Heights Research Articles

Study of Fusion Dynamics Using Skyrme Energy Density Formalism with Different Surface Corrections

Within the framework of Skyrme energy density formalism, we investigate the role of surface corrections on the fusion of colliding nuclei. The coefficient of surface correction is varied between 1/36 and 4/36, and its impact is studied on about 180 reactions. The detailed investigations indicate a linear relationship between the fusion barrier heights and strength of the surface corrections. Our analysis of the fusion barriers advocate the strength of surface correction of 1/36.

Analytical parametrization of fusion barriers using proximity potentials

Using the three versions of proximity potentials, namely proximity 1977, proximity 1988, and proximity 2000, we present a pocket formula for fusion barrier heights and positions. This was achieved by analyzing as many as 400 reactions with mass between 15 and 296. Our parametrized formula can reproduced the exact barrier heights and positions within an accuracy of $\pm1%$. A comparison with the experimental data is also in good agreement.

Comparison of different proximity potentials for asymmetric colliding nuclei

Using the different versions of phenomenological proximity potential as well as other parametrizations within the proximity concept, we perform a detailed comparative study of fusion barriers for asymmetric colliding nuclei with asymmetry parameter as high as 0.23. In all, 12 different proximity potentials are robust against the experimental data of 60 reactions. Our detailed study reveals that the surface energy coefficient as well as radius of the colliding nuclei depend significantly on the asymmetry parameter. All models are able to explain the fusion barrier heights within $\pm10%$ on the average. The potentials due to Bass 80, AW 95, and Denisov DP explain nicely the fusion cross sections at above- as well as below-barrier energies.

THE EFFECT OF SURFACE NUCLEON DENSITY ON THE TOTAL POTENTIAL OF DI-NUCLEAR SYSTEMS

The effect of the density of the surface nucleons on the calculation of the total potential in reactions where the participant nuclei are on the stability line and at least one of the participant nuclei is rich in neutrons has been studied. The results obtained by the analysis of the 16 O +209 Bi , 16 O +144 Nd , and 16 O +144 Sm reactions using the Hartree–Fock calculations of the neutron and the proton densities show a decrease of about 0.67, 0.53, and 0.45 MeV of the fusion barrier height in comparison to the corresponding values obtained when the nuclear density is taken to be proportional to the nuclear charge. It is shown that this decrease is due to the variation of the ratio of the density of neutrons to that of the protons in outer regions of the nuclear densities of 209 Bi , 144 Nd , and 144 Sm nuclei.

132Sn+96Zrreaction: A study of fusion enhancement/hindrance

Capture-fission cross sections were measured for the collision of the massive nucleus {sup 132}Sn with {sup 96}Zr at center-of-mass energies ranging from 192.8 to 249.6 MeV in an attempt to study fusion enhancement and hindrance in this reaction involving very neutron-rich nuclei. Coincident fission fragments were detected using silicon detectors. Using angle and energy conditions, deep inelastic scattering events were separated from fission events. Coupled-channels calculations can describe the data if the surface diffuseness parameter, a, is allowed to be 1.10 fm instead of the customary 0.6 fm. The measured capture-fission cross sections agree moderately well with model calculations using the dinuclear system model. If we use this model to predict fusion barrier heights for these reactions, we find the predicted fusion hindrance, as represented by the extra push energy, is greater for the more neutron-rich system, lessening the advantage of the lower interaction barriers with neutron-rich projectiles.

Capture cross sections for the near symmetricSn124+Zr96reaction

Capture-fission cross sections were measured for the near symmetric reaction between the massive nuclei {sup 124}Sn and {sup 96}Zr for center of mass energies from 195 to 265 MeV. Coincident fission fragments were detected and separated from elastic and deep inelastic scattering products by angle/energy/mass conditions. The measured capture cross sections agree quite well with calculations using the dinuclear system (DNS) model. The DNS model also predicts the fusion cross section for this reaction with a fusion barrier height of 208.0 MeV. The deduced extra push energy, corresponding to this barrier height, differs from that deduced from evaporation residue measurements.

Barrier for cold-fusion production of superheavy elements

We estimate the fusion-barrier height ${B}_{\mathrm{fu}}^{\text{(two-body)}}$ for approaching ions in cold-fusion reactions in a model where the projectile deformation and quadrupole zero-point vibrational energy are taken into account. This barrier height is defined as the barrier energy at the target and projectile separation distance where an original oblate deformation of projectile and/or target caused by a repulsive Coulomb force turns into a large prolate deformation caused by the attractive nuclear force as the target and projectile come closer. The instability develops before touching because the attractive short-range nuclear force overcomes the repulsive Coulomb force and the shape-stabilizing effect of shell structure. The shell structure of the doubly magic $^{208}\mathrm{Pb}$ target is sufficiently strong that its shape remains very close to spherical in all cases studied here. The fusion potential for approaching ions in the two-body channel is calculated in the macroscopic-microscopic model with the quadrupole vibrational zero-point energy obtained in the WKB approximation. We compare our results with data from 10 experimental cold-fusion reactions and with the Bass barriers. Differences and similarities between the Yukawa-plus-exponential model and the Bass model are discussed. We also calculate five-dimensional potential-energy surfaces for the single compound system and show that well-established fission and fusion valleys are both present. For heavy systems, ${B}_{\mathrm{fu}}^{\text{(two-body)}}$ becomes lower than the fission barrier just beyond the ground state of the compound system. In the vicinity of this transition, the optimum collision energy for formation of evaporation residues can be expected to depend in a delicate fashion on the interplay among ${B}_{\mathrm{fu}}^{\text{(two-body)}}$, the fusion valley, the fission barrier of the compound system, and the one- and two-neutron separation energies ${S}_{1n}$ and ${S}_{2n}$. We discuss these issues in detail and calculate fission-barrier heights. Except for reactions in which the projectile is doubly magic or near doubly magic, the calculated quantities are consistent with the observed optimal energies for evaporation-residue formation.

Isotopic dependence of fusion probabilities for neutron-deficient and -rich colliding nuclei

Isotopic dependence of the fusion dynamics is studied by analyzing the collision of a large number of isotopes of Ca and Ni with 0.6 \( \leqslant\)N/Z\( \leqslant\) 2. This study, which results from the Skyrme energy density formalism, reveals that the addition of neutrons favors fusion of reacting partners, whereas the reverse happens with the removal of neutrons. The fusion barrier heights and positions follow a non-linear second-order dependence on (\( {\frac{{{N}}}{{{Z}}}}\)-1 ), whereas fusion cross-sections can be parameterized by a straight line.-1

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.

Analytic expressions for the proximity energy, the fusion process and the [formula omitted] emission

The entrance and exit channels through quasimolecular shapes are compatible with the experimental data on fusion, light nucleus and α emissions when the proximity energy is taken into account. Analytic expressions allowing to determine rapidly this proximity energy are presented as well as formulas for the fusion barrier heights and radii and for the α emission barriers. Predictions for half-lives of exotic α emissions are proposed.

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.

Isotopic dependence of fusion cross-sections – linear relationships

A systematic study of isotopic dependence of fusion cross-section is carried out by adding neutrons gradually to N=Z colliding nuclei. We find that fusion barrier position increases and height decreases, both linearly with the increase of N/Z ratio of the compound system. The increase in barrier position is larger compared to decrease in barrier height. In terms of these linear relationships, a parameterized form of fusion cross-sections is given for the neutron-rich colliding nuclei. The fusion cross-sections are also enhanced linearly with the N/Z ratio, and this enhancement is larger for lower incident centre-of-mass energies and independent of the choice of reaction partners. Experimental data and other theoretical studies are called for to verify these results.

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.

Collisions of deformed nuclei: A path to the far side of the superheavy island

A detailed understanding of complete fusion cross sections in heavy-ion collisions requires a consideration of the effects of the deformation of the projectile and target. Our aim here is to show that deformation and orientation of the colliding nuclei have a very significant effect on the fusion-barrier height and on the compactness of the touching configuration. To facilitate discussions of fusion configurations of deformed nuclei, we develop a classification scheme and introduce a notation convention for these configurations. We discuss particular deformations and orientations that lead to compact touching configurations and to fusion-barrier heights that correspond to fairly low excitation energies of the compound systems. Such configurations should be the most favorable for producing superheavy elements. We analyze a few projectile-target combinations whose deformations allow favorable entrance-channel configurations and whose proton and neutron numbers lead to compound systems in a part of the superheavy region where α half-lives are calculated to be observable, that is, longer than 1 μs.

Fusion barrier lowering induced by nuclear deformations

Ellipsoidal deformations have been introduced in the entrance channel of asymmetric heavy-ion reactions within the liquid-drop model and including the nuclear proximity energy. The corresponding maximal lowering of the fusion barrier heights is about 10 MeV and takes place for Z1Z2 approximately=750. When Z1Z2<or=650-750 and with regard to the two-sphere configuration, the barrier is more external and its width is reduced. At the top, a transition to prolate shapes occurs due to the increase of the proximity force effects. For Z1Z2>or=650-750, the barrier occurs at smaller inter-nuclear distances and corresponds to the transition from two-body shapes to one-body compact and creviced configurations. Before the neck formation, the heaviest colliding ion has a prolate shape while the lightest one is oblate. The sub-barrier fusion enhancement might be explained by the release of the deformation degree of freedom, the reaction time allowing some adiabaticity.

Predictions of the macroscopic model of nuclei: Barriers to fusion and to light fragment emission.

Fission and fusion barriers have been calculated for decay or formation of the composite nuclei $^{149}\mathrm{Tb}^{\mathrm{*}}$ and $^{194}\mathrm{Hg}^{\mathrm{*}}$ via the Yukawa-plus-exponential-finite-range macroscopic model. The predicted properties of these barriers are used for comparisons to measured total kinetic energies for the emission of light fragments (2\ensuremath{\le}${Z}_{L}$\ensuremath{\le}9) and to empirical systematics of s-wave fusion barrier heights. The fusion barriers are consistent with the model. Values of total kinetic energies were calculated with the assumption of no energy loss into excitation beyond the saddle point. For each light fragment, the observed values of total kinetic energies are smaller than those calculated. The deviation increases with ${Z}_{L}$ and is correlated with the nuclear shape at the corresponding conditional saddle point. It scales with the neck thickness and the fragment deformation, as does the error involved in our approximation. This suggests (at least for the higher ${Z}_{L}$ values) that this macroscopic fission model might apply for light-fragment emission. Dynamical effects must, however, be included.

Fusion barriers for heavy-ion systems

Analytical expressions for the fusion barrier height and radius have been derived from a four-parameter empirical fusion cross section formula for heavy ions. The fusion barrier parameters calculated, using these expressions, are in good agreement with the literature values.NUCLEAR REACTIONS Fusion cross section excitation functions, fusion barrier parameters.

Fusion ofNi58,60,62with 113-170 MeVCa40ions

Results of measurements of the fusion excitation functions of $^{58,60,62}\mathrm{Ni}$ targets with 113-170 MeV (lab) $^{40}\mathrm{Ca}$ ions are reported. These results are used to extract fusion barrier heights and radii by several different methods. Subbarrier fusion cross sections are analyzed in terms of static deformations. Possible importance of dynamic deformation is discussed. Conclusions as to the role of static deformation are similar to those from the $^{35}\mathrm{Cl}$ + $^{58,60,62}\mathrm{Ni}$ fusion system; a large part of the subbarrier behavior can be attributed to this cause.NUCLEAR REACTIONS $^{52,58}\mathrm{Ni}$ ($^{40}\mathrm{Ca}$, fusion), ${E}_{\mathrm{Ca}}=113\ensuremath{-}170$ MeV lab, measured evaporation residue excitation functions.

Diffuse surface nuclear and Coulomb potential energies for heavy-ion reactions

We calculate the nuclear and Coulomb potential energies for heavy-ion reactions using a modfied form of the liquid-drop model of the nucleus. Effects of the finite range of the nuclear force, of the surface diffuseness and intrinsic deformation of spheroidal interacting ions are included explicitly. In order to evaluate the potential energies before the density distributions of the two nuclei overlap appreciably, multipole expansion formulae are derived, from which the dependence of the interaction energy upon the relative orientation of the ions can be studied in detail. After the densities begin to overlap, an adiabatic approximation for the nuclear and Coulomb energies is achieved by prescribing a continuous deformation of the density distributions, as proposed by Nix and Sierk. Special attention is paid to the choice of the radius and diffuseness parameters of the ions. Furthermore, the constants of the model are selected from a study of fusion radii and barrier heights, fission saddle-point energies, nuclear energy at the strong absorption radius, and fusion cross sections for light ion systems at energies below the Coulomb barrier. In particular, we examine the fusion of two oblate 12C nuclei and show that the corresponding cross section below the Coulomb barrier depends strongly on their relative orientation. The fusion of two spherical 16O nuclei is also discussed.

Fusion and interaction barrier parameters and critical angular momenta fromCl35-induced reactions

Cross sections for evaporation residue formation and fission following complete fusion of /sup 35/Cl with /sup 27/Al, /sup 48/Ti, /sup 54/,/sup 56/Fe, /sup 58/,/sup 60/,/sup 62/,/sup 64/Ni, /sup 90/Zr, and /sup 116/,/sup 124/Sn have been measured with counter telescopes for laboratory projectile energies between 70 and 170 MeV. Elastic scattering of /sup 35/Cl + /sup 58/,/sup 62/Ni has been measured near the interaction barrier. The excitation functions for complete fusion are analyzed with a semiclassical model, and fusion barrier heights and radii are extracted. These data are discussed in terms of the nuclear density overlap and the nuclear potential contributions to the fusion barrier. The results are compared with the predictions of several heavy ion potential models and with parameters for the interaction deduced from elastic scattering data. For the systems with masses A < 100 about 70% of the total reaction cross section appears as complete fusion. This fraction decreases with increasing mass of the compound system. Evaporation residue and fission excitation functions have been analyzed by the Bohr-Wheeler model using the rotating liquid drop model of Cohen, Plasil, and Swiatecki. To first order these calculations give a satisfactory description of both sets of results. (AIP)