Sub-barrier Heavy-ion Fusion Research Articles

The slopes of sub-barrier heavy-ion fusion excitation functions shed light on the dynamics of quantum tunnelling

Quantum tunnelling plays a crucial role in heavy-ion fusion reactions at sub-barrier energies, especially in the context of nuclear physics and astrophysics. The nuclear structure of the colliding nuclei and nucleon transfer processes represent intrinsic degrees of freedom. They are coupled to the relative ion motion and, in general, increase the probability of tunnelling. The influence of couplings to nucleon transfer channels relatively to inelastic excitations, on heavy-ion fusion cross sections, is one of the still open problems in this field. We present a new analysis of several systems, based on the combined observation of the energy-weighted excitation functions Eσ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$E\\sigma $$\\end{document} in relation to their first energy derivatives d(Eσ)/dE\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d(E\\sigma )/dE$$\\end{document}. The relation between d(Eσ)/dE\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d(E\\sigma )/dE$$\\end{document} and Eσ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$E\\sigma $$\\end{document} removes the basic differences due to the varying Coulomb barrier when comparing different systems. We show that, depending on the nuclear structure and/or the presence of strong transfer channels, this representation reveals characteristic features below the barrier. The possible presence of cross section oscillations makes this analysis less clear for light- or medium-light systems.

Influence of the Z = 50 shell closure on heavy-ion sub-barrier fusion

Influence of the Z = 50 shell closure on heavy-ion sub-barrier fusion

Heavy-ion fusion reactions of astrophysical interest

The hindrance in fusion of heavy-ion reactions crops up in the region of extreme sub-barrier energies. This phenomenon can be effectively analyzed using a simple diffused barrier formula derived assuming a Gaussian distribution of fusion barrier heights. Folding the Gaussian barrier distribution with the classical expression for the fusion cross section for a fixed barrier, the fusion cross section is obtained. The energy dependence of the fusion cross section provides good description to the existing data on sub-barrier heavy-ion fusion for lighter systems of astrophysical interest. Using this simple formula, an analysis has been presented from $^{16}$O + $^{18}$O to $^{12}$C + $^{198}$Pt, all of which were measured down to $<$ 10 $\mu$b. The agreement of the present analysis with the measured values is better than those calculated even from the sophisticated coupled channels calculations. The relatively smooth variation of the three parameters of this formula implies that it may be exploited to estimate the excitation function or to extrapolate cross sections for pairs of interacting nuclei which are yet to be measured. Possible extensions of the present methodology and its limitations have also been discussed.

Survey of deep sub-barrier heavy-ion fusion hindrance phenomenon for positive and negative Q-value systems using the proximity-type potential

A systematic survey of the accurate measurements of heavy-ion fusion cross sections at extreme sub-barrier energies is performed using the coupled-channels (CC) theory that is based on the proximity formalism. This work theoretically explores the role of the surface energy coefficient and energy-dependent nucleus-nucleus proximity potential in the mechanism of the fusion hindrance of 14 typical colliding systems with negative -values, including 11B+197Au, 12C+198Pt, 16O+208Pb, 28Si+94Mo, 48Ca+96Zr, 28Si+64Ni, 58Ni+58Ni, 60Ni+89Y, 12C+204Pb, 36S+64Ni, 36S+90Zr, 40Ca+90Zr, 40Ca+40Ca, and 48Ca+48Ca, as well as five typical colliding systems with positive -values, including 12C+30Si, 24Mg+30Si, 28Si+30Si, 36S+48Ca, and 40Ca+48Ca. It is shown that the outcomes based on the proximity potential along with the above-mentioned physical effects achieve reasonable agreement with the experimentally observed data of the fusion cross sections , astrophysical factors, and logarithmic derivatives in the energy region far below the Coulomb barrier. A discussion is also presented on the performance of the present theoretical approach in reproducing the experimental fusion barrier distributions for different colliding systems.

Potential roots of the deep subbarrier heavy-ion fusion hindrance phenomenon within the sudden approximation approach

We analyse the origin of the unexpected deep sub-barrier heavy-ion fusion hindrance in 64Ni+100Mo and 28Si+64Ni recations. Our analysis is based on the improved coupled-channels approach, implemented by means of the finite element method. With the aid of the Woods-Saxon potential the experimental cross sections and the S-factors of these reactions are remarkably well reproduced. We found that the account on the non-diagonal matrix elements of the coupling matrix, traditionally neglected in the conventional coupled-channels approaches in setting the left boundary conditions inside the potential pocket, and its minimal value are crucially important for the interpretation experimental data. Within our approach we found a good agreement with the experimental data for the S-factor of the fusion reaction 12C+12C, which has no a pronounced maximum for this system.

Fusion hindrance in light and heavy systems

The phenomenon of hindrance in sub-barrier heavy-ion fusion has been confirmed by several experimentalevidences and it is now recognised as a general phenomenon of heavy-ion fusion process. In many cases the signature of fusion hindrance lies in the trend of the logarithmic slope of the excitation function and of the S factor at low energies. The comparison with stadard Coupled-Channels calculations is a more quantitative evidence for its existence. In many medium-heavy systems the hindrance effect has been recognised with different features depending on the various couplings to the inelastic and transfer channels. Different theoretical appro ches have been proposed to explain the hindrance but the underlying physics is still a matter of debate. Hindrance is observed in light systems, independent of the sign of the fusion Q-value, with different features. In the case of the 12C + 30Si system the effect is smallbut it is clearly observed. Near-by cases show evidence for systematic behaviours. A very recent experiment has concerned the lighter case 12C + 24Mg where hindrance shows up clearly, because a maximum of the S factor appears already at a relatively high cross section σ=1.6mb. The consequences for the dynamics of stellar evolution have to be clarified by further experimental and theoretical work.

Role of hexadecapole deformation of projectile Si28 in heavy-ion fusion reactions near the Coulomb barrier

The vast knowledge of strong influence of quadrupole deformation $\beta_2$ of colliding nuclei on heavy-ion subbarrier fusion reactions inspires a desire to quest the sensitivity of fusion dynamics to higher order deformations, such as $\beta_4$ and $\beta_6$ deformations. However, such studies have rarely been carried out, especially for deformation of projectile nuclei. In this article, we investigated the role of $\beta_4$ of the projectile nucleus in fusion of the $^{28}$Si + $^{92}$Zr system. We demonstrated that the fusion barrier distribution is sensitive to the sign and the value of the $\beta_4$ parameter of the projectile, $^{28}$Si, and confirmed that the $^{28}$Si nucleus has a large positive $\beta_4$. This study opens an indirect way to estimate deformation parameters of radioactive nuclei using fusion reactions, which is otherwise difficult due to experimental constraints.

Recent experimental results in sub- and near-barrier heavy-ion fusion reactions

Recent advances obtained in the field of near and sub-barrier heavy-ion fusion reactions are reviewed. Emphasis is given to the results obtained in the last decade, and focus will be mainly on the experimental work performed concerning the influence of transfer channels on fusion cross sections and the hindrance phenomenon far below the barrier. Indeed, early data of sub-barrier fusion taught us that cross sections may strongly depend on the low-energy collective modes of the colliding nuclei, and, possibly, on couplings to transfer channels. The coupled-channels (CC) model has been quite successful in the interpretation of the experimental evidences. Fusion barrier distributions often yield the fingerprint of the relevant coupled channels. Recent results obtained by using radioactive beams are reported. At deep sub-barrier energies, the slope of the excitation function in a semi-logarithmic plot keeps increasing in many cases and standard CC calculations over-predict the cross sections. This was named a hindrance phenomenon, and its physical origin is still a matter of debate. Recent theoretical developments suggest that this effect, at least partially, may be a consequence of the Pauli exclusion principle. The hindrance may have far-reaching consequences in astrophysics where fusion of light systems determines stellar evolution during the carbon and oxygen burning stages, and yields important information for exotic reactions that take place in the inner crust of accreting neutron stars.

Present status of coupled-channels calculations for heavy-ion subbarrier fusion reactions

The coupled-channels method has been a standard tool in analyzing heavy-ion fusion reactions at energies around the Coulomb barrier. We investigate three simplifications usually adopted in the coupled-channels calculations. These are i) the exclusion of non-collective excitations, ii) the assumption of coordinate independent coupling strengths, and iii) the harmonic oscillator approximation for multi-phonon excitations. In connection to the last point, we propose a novel microscopic method based on the beyond-mean-field approach in order to take into account the anharmonic effects of collective vibrations.

Study of fusion Q-value rule in sub-barrier fusion of heavy ions**Supported by National Nature Science Foundation of China (11475013, 11035007, 11175011), State Key Laboratory of Software Development Environment (SKLSDE-2014ZX-08), Fundamental Research Funds for the Central Universities and the Key Laboratory of High Precision Nuclear Spectroscopy, Institute of Modern Physics, Chinese Academy of Sciences

A vast body of fusion data has been analyzed for different projectiles and target nuclei. It is indicated that the sub-barrier fusion depends on the fusion Q-value. In terms of a recently introduced fusion Q-value rule and an energy scaling reduction procedure, the experimental fusion excitation functions are reduced and compared with each other. It is found that the reduced fusion excitations of selected fusion systems show a similar trend. The fusion data for massive nuclei are in agreement with the Q-value rule. In the fusion process, the Q contribution should be considered. Within this approach, the sub-barrier fusion cross sections of most fusion systems can be predicted without involving any structure effects of colliding nuclei. Instances of disagreement are presented in a few fusion systems. The use of the energy scaling as a criterion of possible experimental data inconsistency is discussed. More precise experimental fusion data need to be measured.

Subbarrier Fusion Reactions and Many-Particle Quantum Tunneling

Low energy heavy-ion fusion reactions are governed by quantum tunneling through the Coulomb barrier formed by a strong cancellation of the repulsive Coulomb force with the attractive nuclear interaction between the colliding nuclei. Extensive experimental as well as theoretical studies have revealed that fusion reactions are strongly influenced by couplings of the relative motion of the colliding nuclei to several nuclear intrinsic motions. Heavy-ion subbarrier fusion reactions thus provide a good opportunity to address a general problem on quantum tunneling in the presence of couplings, which has been a popular subject in the past decades in many branches of physics and chemistry. Here we review theoretical aspects of heavy-ion subbarrier fusion reactions from the view point of quantum tunneling in systems with many degrees of freedom. Particular emphases are put on the coupled-channels approach to fusion reactions, and the barrier distribution representation for multi-channel penetrability. We also discuss an application of the barrier distribution method to elucidation of the mechanism of dissociative adsorption of H$_2$ melecules in surface science.

Jensen inequalities for tunneling probabilities in complex systems

The Jensen theorem is used to derive inequalities for semiclassical tunneling probabilities for systems involving several degrees of freedom. These Jensen inequalities are used to discuss several aspects of sub-barrier heavy-ion fusion reactions. The inequality hinges on general convexity properties of the tunneling coefficient calculated with the classical action in the classically forbidden region.

Transmission ion chamber: Design and application

A Transmission Ion Chamber (TIC) was constructed for use at the focal plane of the Fragment Mass Analyzer (FMA). The motivation for the construction of the TIC was to allow energy loss information to be added to the focal plane identification data. Three experiments immediately benefited form this information. First, an astrophysics experiment studying 69Br via the β decay of 69Kr would benefit greatly by adding a TIC to the existing implantation station. Second the study of extreme sub-barrier heavy-ion fusion evaporation reaction could be made more sensitive. Third, a study was performed using a PPAC–TIC–DSSD configuration to separate 101Sn from the other evaporation products of highly excited 108Te.

Subbarrier heavy ion fusion enhanced by nucleon transfer

We discuss a model for the description of subbarrier fusion of heavy ions which takes into account the coupling to the low-energy surface vibrational states and to the few-nucleon transfer with arbitrary reaction Q -value. The fusion reactions 28,30Si+58,62,64Ni, 40Ca+90,96Zr, 28S+94,100Mo, 58Ni+64Ni, 16,18,20,22,24O+58Ni and 28Si+124,126,128,130,132Sn are analyzed in detail. The model describes rather well the experimental fusion cross section and mean angular momentum for reactions between nuclei near the β-stability line. It is shown that these quantities are significantly enhanced by few-nucleon transfer with large positive Q-value. A shape independent parameterization of the heavy ion potential at distances smaller then the touching point is proposed.

Effects of finite width of excited states on heavy-ion sub-barrier fusion reactions

We discuss the effects of coupling of the relative motion to nuclear collective excitations which have a finite lifetime on heavy-ion fusion reactions at energies near and below the Coulomb barrier. Both spreading and escape widths are explicitly taken into account in the exit doorway model. The coupled-channels equations are numerically solved to show that the finite resonance width always hinders fusion cross sections at subbarrier energies irrespective of the relative importance between the spreading and the escape widths. We also show that the structure of fusion barrier distribution is smeared due to the spreading of the strength of the doorway state.

Low-energy surface vibrations and neutron transfer initiating heavy-ion sub-barrier fusion

This paper reviews recent experimental results on heavy-ion sub-barrier fusion, which have been obtained with the beams of the XTU Tandem accelerator of LNL and a simple set-up using an electrostatic beam deflector. Various systems have been studied, i.e. , and more recently . In most cases, the sub-barrier fusion cross section enhancements can be attributed to inelastic channel couplings which also produce discrete structures in the representations of fusion barrier distributions obtained as the second derivatives of the excitation functions. From combined analyses of the evaporation residue cross sections and of the fusion barrier distributions, important effects of low-energy surface vibrations up to the two- and possibly three-phonon level have been put in evidence. In two cases (40Ca + 96Zr and 32S + 110Pd) the effect of transfer couplings on cross sections and barrier distributions is rather clear, although the two barrier distributions have very different shapes. Further comparisons are made with the results of older experiments on heavy-ion systems with various nuclear structure situations, and preliminary data are shown on the sub-barrier fusion of 40Ca +116,124Sn, where interesting connections may be deduced with the multi-nucleon transfer reactions already measured for the heavier target.

Coupled-Channel Effects on Heavy-Ion Sub-barrier Fusion Within the Doorway Expansion Method

Coupled-channel effects on the heavy-ion sub-barrier fusion cross section σF are investigated with the doorway expansion method. The case of two coupled channels is considered in detail. It is found that the two-doorway approximation, which leads to a simple result for σF, is in excellent agreement with exact coupled-channel calculations. The contributions from the elastic and inelastic channels to the enhanced fusion cross section are discussed.

Influence of resonant channels on subbarrier heavy-ion fusion.

The subbarrier fusion of heavy ions is discussed when one of the nuclei is excited to a resonant state. The effect of the width of the resonant state on the barrier penetration is calculated within a schematic model. It is concluded that the width could either enhance or hinder the fusion probability, depending on the relative importance of the spreading to escape parts of the width. Application of the theory to the fusion of $^{11}\mathrm{Li}$ with $^{208}\mathrm{Pb}$ at near-barrier energies is made. The resulting fusion cross section calculated with coupling to the soft giant dipole state in $^{11}\mathrm{Li}$ was found to be more than an order of magnitude smaller, in the barrier region, and larger, at subbarrier energies, than the uncoupled one.

Absence of anomalous entrance channel effects in sub-barrier heavy ion fusion.

Fusion cross sections and gamma ray multiplicities have been measured for the $^{28}\mathrm{Si}$${+}^{142}$Ce, $^{32}\mathrm{S}$${+}^{138}$Ba, and $^{48}\mathrm{Ti}$${+}^{122}$Sn entrance channels leading to the $^{170}\mathrm{Hf}$ compound nucleus. The mean spins deduced from the gamma multiplicities do not show any anomalous behavior as observed in some other measurements.

The α Transfer Effect on Sub-barrier Fusion of Heavy Ions

The α transfer effect on sub-barrier fusion is calculated for 16O+148Sm system by using the linear combination of nuclear orbits theory. The coupling between elastic scattering channel and α transfer reaction channel causes the enhancement of the fusion cross section at sub-barrier energies.