Barrier Penetration Model Research Articles

Channel coupling effects on the fusion excitation functions forSi28+Zr90,94in sub- and near-barrier regions

Fusion excitation functions and angular distributions of evaporation residues (ERs) have been measured for $^{28}\mathrm{Si}+^{90,94}\mathrm{Zr}$ systems around the Coulomb barrier using the recoil mass spectrometer, Heavy Ion Reaction Analyzer (HIRA). For both systems, the experimental fusion cross sections are strongly enhanced compared to the predictions of the one-dimensional barrier penetration model (1-d BPM) below the barrier. Coupled channels formalism has been employed to theoretically explain the observed sub-barrier fusion cross section enhancement. The enhancement could be explained by considering the coupling of the low-lying inelastic states of the projectile and target in the $^{28}\mathrm{Si}+^{90}\mathrm{Zr}$ system. In the sub-barrier region, the measured fusion cross sections for $^{28}\mathrm{Si}+^{94}\mathrm{Zr}$ turned out to be about an order of magnitude higher than the ones for the $^{28}\mathrm{Si}+^{90}\mathrm{Zr}$ system, which could not be explained by coupling to inelastic states alone. This observation indicates the importance of multinucleon transfer reaction channels with positive $Q$ values in the sub-barrier fusion cross section enhancement, because $^{90,94}\mathrm{Zr}$ are believed to have similar collective strengths. This implies that no strong isotopic dependence of fusion cross sections is expected as far as the couplings to collective inelastic states are concerned. In addition, the role of projectile and multiphonon couplings in the enhancement has been explored.

Fusion of deformed nuclei:C12+C12

The $S$ factor and the fusion cross-section values for the $^{12}\mathrm{C}+^{12}\mathrm{C}$ reaction are evaluated in the framework of a simple barrier-penetration model, which takes into account quadrupole and hexadecapole surface deformations of $^{12}\mathrm{C}$. The calculated values agree well with corresponding experimental data at off-resonances energies. It is shown that the strong oblate quadrupole deformation of $^{12}\mathrm{C}$ induces the increasing of both the $S$ factor and the fusion cross-section values at subbarrier energies. The $S$ factor and the fusion cross section for the $^{12}\mathrm{C}+^{12}\mathrm{C}$ reaction are slightly reduced at low energy due to the influence of the hexadecapole deformation.

Suppression of complete fusion in theLi6+Sm144reaction

Complete fusion excitation function for the $^{6}\mathrm{Li}+^{144}\mathrm{Sm}$ reaction has been measured at near barrier energies by the activation technique. Coupled-channel calculations show an enhancement in fusion cross section at energies below the barrier compared to the one-dimensional barrier penetration model calculation, but they overpredict it in the entire energy range compared to the experimental data. Reduced fusion cross sections for the present system at energies normalized to the Coulomb barrier were also found to be systematically lower than those with strongly bound projectiles forming a similar compound nucleus. These two observations conclusively show that the complete fusion cross section, at above barrier energies, is suppressed by $~32%$ in the $^{6}\mathrm{Li}+^{144}\mathrm{Sm}$ reaction. Reanalyses of existing fusion data for $^{7}\mathrm{Li}+^{165}\mathrm{Ho}$ and $^{7}\mathrm{Li}+^{159}\mathrm{Tb}$ also show a suppression compared to those with strongly bound projectiles, which contradicts earlier conclusions. The fusion suppression factor seems to exhibit a systematic behavior with respect to the breakup threshold of the projectile and the atomic number of the target nucleus.

Heavy-ion fusion cross sections and barrier distributions for 12C, 16O, 28Si and 35Cl on 92Zr

The fusion excitation functions and barrier distributions for the fusion of 12C, 16O, 28Si and 35Cl on 92Zr have been calculated using one-dimensional barrier penetration model, taking scattering potential as the sum of Coulomb and proximity potential. Calculated results are compared with experimental data and CC calculations using code CCFULL. At and above the barrier the computed cross sections match well with the experimental data, whereas below the barrier, calculations with nuclear surface tension coefficient improved by Reisdorf in the proximity potential give good fit. Reduced reaction cross sections for the systems have also been described.

Sub-barrier fusion excitation for the system7Li + 28Si

The sub-barrier fusion excitation functions are measured for the first time for the system ${}^{7}\mathrm{Li}+ {}^{28}\mathrm{Si}$ by the characteristic \ensuremath{\gamma}-ray method in the energy range ${E}_{\mathrm{lab}}=7\text{\ensuremath{-}}11.5$ MeV. The results show an enhancement, below the barrier, by about a factor of two when compared with the one-dimensional barrier penetration (1D BPM) model. Introduction of coupling with the rotational ${2}^{+}$ state (1.779 MeV) of the target improves the fit somewhat, but still an enhancement of about 25--40% remains.

Fusion of neutron-rich oxygen isotopes in the crust of accreting neutron stars

Fusion reactions in the crust of an accreting neutron star are an important source of heat, and the depth at which these reactions occur is important for determining the temperature profile of the star. Fusion reactions depend strongly on the nuclear charge $Z$. Nuclei with $Z\ensuremath{\leqslant}6$ can fuse at low densities in a liquid ocean. However, nuclei with $Z=8$ or 10 may not burn until higher densities where the crust is solid and electron capture has made the nuclei neutron rich. We calculate the $S$ factor for fusion reactions of neutron rich nuclei including $^{24}\mathrm{O}+^{24}\mathrm{O}$ and $^{28}\mathrm{Ne}+^{28}\mathrm{Ne}$. We use a simple barrier penetration model. The $S$ factor could be further enhanced by dynamical effects involving the neutron rich skin. This possible enhancement in $S$ should be studied in the laboratory with neutron rich radioactive beams. We model the structure of the crust with molecular dynamics simulations. We find that the crust of accreting neutron stars may contain micro-crystals or regions of phase separation. Nevertheless, the screening factors that we determine for the enhancement of the rate of thermonuclear reactions are insensitive to these features. Finally, we calculate the rate of thermonuclear $^{24}\mathrm{O}+^{24}\mathrm{O}$ fusion and find that $^{24}\mathrm{O}$ should burn at densities near ${10}^{11}$ g/cm${}^{3}$. The energy released from this and similar reactions may be important for the temperature profile of the star.

Detailed determination of the nuclear fusion radius by a simultaneous optical model calculation of elastic scattering and fusion cross sections in reactions involving weakly bound projectiles

Within the optical model for direct reactions, simultaneous calculations of elastic scattering, complete fusion, and total reaction cross sections for energies around the Coulomb barrier are presented for reactions involving the weakly bound projectile $^{9}\mathrm{Be}$ on $^{64}\mathrm{Zn}$. Volume (${W}_{F}$) and surface (${W}_{\mathrm{DR}}$) Woods-Saxon optical potentials are used such that the former is responsible only for complete fusion reactions while the latter for all direct reactions plus incomplete fusion. Simultaneous fits can be obtained with several sets of potential parameters, but if we impose the condition that the strength of ${W}_{F}$ is smaller than the strength of ${W}_{\mathrm{DR}}$ at the tail region of the potential (this condition is discussed in detail), then values are required for ${r}_{F}$ and ${r}_{\mathrm{DR}}$ of around $1.6$ and 1.7--1.9 fm, respectively. These values are much larger than those frequently used in barrier penetration model calculations. Through the energy dependence of the real and imaginary parts of the polarization potentials, we show that the usual threshold anomaly does not show up for this system, but instead there is evidence of the presence of a breakup threshold anomaly.

Experimental investigation of fusion ofLi7+Si28above the Coulomb barrier

Excitation functions for the above-barrier fusion cross sections are measured for the first time for the $^{7}\mathrm{Li}+^{28}\mathrm{Si}$ system by two methods---the characteristic \ensuremath{\gamma}-ray method and the evaporation \ensuremath{\alpha} measurement method---in the energy range ${E}_{\mathrm{lab}}=11.5\text{\ensuremath{-}}26$ MeV. Experimental results are consistent and agree with each other, and the one-dimensional Barrier Penetration Model (BPM) predictions describe the data well up to twice the Coulomb barrier, but they overestimate the data by about 15--20% at higher energies.

New γ-ray measurements forC12+C12sub-Coulomb fusion: Toward data unification

Fusion excitation functions are measured for the {alpha},p, and n evaporation channels in the fusion of the {sup 12}C+{sup 12}C system at center-of-mass energies between 4.42 and 6.48 MeV, with energy steps of 75 keV. The {gamma}-ray technique is used with a new absolute normalization method which is independent of charge collection, allowing one to monitor at the same time the carbon buildup at the target. Discrepancies between previous works are discussed and a simple unification is proposed. Barrier penetration model (BPM) calculations are consistent with a belly-to-belly orientation of the oblate deformed {sup 12}C nuclei at the touching point. As in previous works, much structure is seen in the excitation function which is nearly consistent with the positions of resonances reported in the literature for this system. Using BPM predictions for the background excitation function and doing a simultaneous fit of the relevant Breit-Wigner terms in the measured energy region produced a modified set of resonance parameters. The extrapolation of the astrophysical S factor to lower energies is discussed on the basis of the unified data.

Isomeric cross section ratios and angular momentum distribution in sub-barrier fusion ofC12withY89

The isomeric cross section of $^{99}\mathrm{Rh}$ has been measured in the $^{12}\mathrm{C}+^{89}\mathrm{Y}$ reaction at sub-barrier energies using recoil catcher and $\ensuremath{\gamma}$-ray spectroscopy. The experimental results were compared with CASCADE calculations. The average angular momentum of the CN was obtained from the measured isomeric cross section ratios using GROGI2 and compared with the predictions of a one-dimensional barrier penetration model and a coupled-channel code. The average angular momenta were found to decrease with decreasing beam energy until the fusion barrier, below which it remained constant.

Comparison between heavy-ion reaction and fusion processes for hundreds of systems

A large number of heavy-ion fusion and reaction cross section data are compared in different regions of mass and energy. The totally parameter-free São Paulo potential is used as basis for comparison within the context of the barrier penetration model. The systematics indicates that, in average, the fusion process represents about 74% of the total reaction cross section. The regions of mass and energy where this result was obtained is discussed in the text.

Coulomb and nuclear potentials between deformed nuclei applied to the fusion process

We obtained theoretical fusion cross sections based on a new and accurate procedure to calculate the Coulomb and nuclear interactions between deformed nuclei. The calculations were performed within the context of the barrier penetration model with zero point motion for the 64Ni + 64Ni system.

Pycnonuclear reaction rates between neutron-rich nuclei

We present pycnonuclear reaction rate calculations for systems involving neutron-rich oxygen, neon and magnesium isotopes. The results are obtained taking into account the screening potential effects and the internuclear many-body correlations. In this work, the S-factor predictions are derived from the one-dimensional barrier penetration model.

Breakup and fusion ofLi6andHe6withPb208

The effect of $^{6}\mathrm{Li}$ and $^{6}\mathrm{He}$ breakup on the fusion cross section of these nuclei with $^{208}\mathrm{Pb}$ is investigated by means of continuum-discretized coupled-channels (CDCC) calculations. For $^{6}\mathrm{Li}$ the calculations describe reasonably well the experimental data for elastic scattering, $^{6}\mathrm{Li}\ensuremath{\rightarrow}\ensuremath{\alpha}+d$ breakup and the absorption cross section given by the sum of the $^{6}\mathrm{Li}$ fusion and the $\ensuremath{\alpha}$ production cross section not attributed to breakup. The effect of $^{6}\mathrm{Li}$ breakup on the calculated absorption cross section is found to depend strongly on the imaginary part of the diagonal bare potential. A combination of the CDCC technique and the barrier penetration model generates results close to the measured fusion cross section. For $^{6}\mathrm{He}$ the calculated absorption cross section is much larger than the measured $^{6}\mathrm{He}+^{209}\mathrm{Bi}$ complete fusion cross section values. However, it is found to be relatively independent of the form of the imaginary part of the bare potential. The complete fusion cross section is again found to be reasonably well described by the CDCC∕BPM combination.

Elastic scattering and fusion ofBe9+Pb208: Density function dependence of the double folding renormalization

Elastic scattering angular distributions of the weakly bound $^{9}\mathrm{Be}$ projectile from $^{208}\mathrm{Pb}$ have been measured for 11 beam energies ranging from 38 to $75\phantom{\rule{0.3em}{0ex}}\text{MeV}$. Analysis was carried out with the optical model, employing a double folded microscopic real potential with four different $^{9}\mathrm{Be}$ density functions. The surface strengths of the interaction potentials were consistent and all exhibited a threshold anomaly. However, the renormalization factor required to obtain an optimum fit to the data varied significantly depending upon which $^{9}\mathrm{Be}$ density function was used, cautioning that conclusions about the effect of breakup should not be based solely upon the value of this factor. Fusion cross sections predicted from these potentials were extracted using a barrier penetration model. Comparison with recently published experimental data suggests that the flux removed from complete fusion by breakup is balanced by the flux redirected into the partial fusion channel.

Impacts of the Cutoff of Coupled-Channels Effects on Heavy-Ion Fusion Reactions at Extreme Sub-Barrier Energies

Based on the one dimensional barrier penetration model with semiclassical WKB method, the dynamic polarization potentials in heavy-ion reactions are extracted from the fusion cross sections. By choosing the proper parameters, the real polarization potential can be calculated from the imaginary part according to the dispersion relation in the framework of the linear-schematic model. The imaginary potential comes from the coupled-channels effects and shows a rapid cuttoff at energies far below the Coulomb barrier. Due to the absence of the coupled-channels effects, the fusion excitation function fall off more quickly than the usual expectation. Moreover, the pure potential in fusion reactions is proposed in hopes to reproduce the fusion cross sections at all energies in the full coupled-channels calculation.

A Consistent Description of the Heavy-Ion Fusion and Elastic Scattering Processes Using a Nonlocal Model

In earlier works, we have developed a nonlocal model for the nuclear interaction, which has been successfully applied in the description of the heavy-ion elastic scattering. In the present work, we have used this interaction to study the fusion process in the context of the barrier penetration model.

Adiabatic heavy-ion fusion potentials for fusion at deep sub-barrier energies

The fusion cross sections from well above barrier to extreme sub-barrier energies have been analysed using the energy (E) and angular momentum (L) dependent barrier penetration model ({\small{ELDBPM}}). From this analysis, the adiabatic limits of fusion barriers have been determined for a wide range of heavy ion systems. The empirical prescription of Wilzynska and Wilzynski has been used with modified radius parameter and surface tension coefficient values consistent with the parameterization of the nuclear masses. The adiabatic fusion barriers calculated from this prescription are in good agreement with the adiabatic barriers deduced from {\small{ELDBPM}} fits to fusion data. The nuclear potential diffuseness is larger at adiabatic limit, resulting in a lower $\hbar\omega$ leading to increase of logarithmic slope observed at energies well below the barrier. The effective fusion barrier radius and curvature values are anomalously smaller than the predictions of known empirical prescriptions. A detailed comparison of the systematics of fusion barrier with and without L-dependence has been presented.

Angular momentum and cross sections for fusion with weakly bound nuclei: breakup, a coherent effect.

Results for the cross section and average angular momentum for complete fusion at energies around the Coulomb barrier are presented for 7Li with 165Ho. Comparison of the cross sections with a one-dimensional barrier penetration model, using a potential consistent with the measured elastic scattering, showed a reduction above the barrier and an enhancement below it. An increase in the measured average angular momentum, <l>, above the barrier and its consistency with that obtained from the fusion excitation function for weakly bound nuclei, is reported. These results together with a reanalysis of existing data conclusively demonstrate that the effect of breakup on fusion is coherent, like coupling to any nonelastic channel.

Fusion calculations for the6,7Li+16Osystems

Recently reported measurements of fusion for the ${}^{6,7}\mathrm{Li}{+}^{16}\mathrm{O},{}^{13}\mathrm{C},$ and ${}^{12}\mathrm{C}$ systems using the characteristic $\ensuremath{\gamma}$-ray yield method were significantly larger than previous measurements for the same systems using the evaporation residue technique. The question of which set of measurements more accurately represents fusion and, indeed, the precise definition of what constitutes fusion is an important one, considering the recent interest in the question of whether fusion is inhibited or enhanced for weakly bound exotic nuclei such as ${}^{6}\mathrm{He}$ and ${}^{11}\mathrm{Li}.$ We present here fusion cross sections derived from continuum discretized coupled channels (CDCC) calculations using the barrier penetration model (BPM) that agree very well with the characteristic $\ensuremath{\gamma}$-ray method results.