Using the random matrix model, we discuss the effect of couplings to non-collective states on the penetrability of a one dimensional potential barrier. We show that these non-collective excitations hinder the penetrability and thus smear the barrier distribution at energies above the barrier, while they do not affect significantly the penetrability at deep subbarrier energies. The energy dependence of the Q-value distribution obtained with this model is also discussed.
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 ^{28,30}Si+^{58,62,64}Ni, ^{40}Ca+^{90,96}Zr, ^{28}S+^{94,100}Mo, ^{16,18,20,22,24}O+^{58}Ni and ^{28}Si+^{124,126,128,130,132}Sn 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…
We show that dynamical deformation effects play an important role in fusion reactions involving the $^{64}$Ni nucleus, in particular the $^{64}$Ni+$^{132}$Sn system. We calculate fully microscopic interaction potentials and the corresponding subbarrier fusion cross sections.
We invert experimental data for heavy-ion fusion reactions at energies well below the Coulomb barrier in order to directly determine the internucleus potential between the colliding nuclei. In contrast to the previous applications of the inversion formula, we explicitly take into account the effect of channel couplings on fusion reactions, by assuming that fusion cross sections at deep subbarrier energies are governed by the lowest barrier in the barrier distribution. We apply this procedure to the $^{16}$O +$^{144}$Sm and $^{16}$O +$^{208}$Pb reactions, and find that the inverted internucleus…
A recent publication reports that heavy-ion fusion cross sections at extreme subbarrier energies show a continuous change of their logarithmic slope with decreasing energy, resulting in a much steeper excitation function compared with theoretical predictions. We show that the energy dependence of this slope is partly due to the asymmetric shape of the Coulomb barrier, that is its deviation from a harmonic shape. We also point out that the large low-energy slope is consistent with the surprisingly large surface diffusenesses required to fit recent high-precision fusion data.
Recently measured high precision data of fusion excitation function have enabled a detailed study on the effects of nuclear collective excitations on fusion reactions. Using such highly accurate data of the $^{16}$O + $^{144,148}$Sm reactions, we discuss the anharmonic properties of collective phonon excitations in $^{144,148}$Sm nuclei. It is shown that subbarrier fusion reactions are strongly affected by the anharmonic effects and thus offer an alternative method to extract the static quadrupole moments of phonon states in a spherical nucleus.
We propose a novel extension of the standard coupled-channels framework for heavy-ion reactions in order to analyze fusion reactions at deep subbarrier incident energies. This extension simulates a smooth transition between the diabatic two-body and the adiabatic one-body states. To this end, we damp gradually the off-diagonal part of the coupling potential, for which the position of the onset of the damping varies for each eigen channel. We show that this model accounts well for the steep falloff of the fusion cross sections for the $^{16}$O+$^{208}$Pb, $^{64}$Ni+$^{64}$Ni, and $^{58}$Ni+$^{5…
We analyze the recent experimental data of heavy-ion fusion cross sections available up to deep subbarrier energies in order to discuss the threshold incident energy for a deep subbarrier fusion hindrance phenomenon. To this end, we employ a one-dimensional potential model with a Woods-Saxon internuclear potential. Fitting the experimental data in two different energy regions with different Woods-Saxon potentials, we define the threshold energy as an intersect of the two fusion excitation functions. We show that the threshold energies so extracted are in good agreement with the empirical syste…
For a single potential barrier, the barrier penetrability can be inverted based on the WKB approximation to yield the barrier thickness. We apply this method to heavy-ion fusion reactions at energies well below the Coulomb barrier and directly determine the inter-nucleus potential between the colliding nuclei. To this end, we assume that fusion cross sections at deep subbarrier energies are governed by the lowest barrier in the barrier distribution. The inverted inter-nucleus potentials for the $^{16}$O +$^{144}$Sm and $^{16}$O +$^{208}$Pb reactions show that they are much thicker than phenome…
We investigate the effects of nuclear quadrupole and hexadecapole couplings on the average angular momentum in sub-barrier fusion reactions. This quantity could provide a probe for nuclear shapes, distinguishing between prolate vs. oblate quadrupole and positive vs. negative hexadecapole couplings. We describe the data in the O + Sm system and discuss heavier systems where shape effects become more pronounced.
Recent theoretical developments in using the Interacting Boson Model to describe nuclear structure effects in fusion reactions below the Coulomb barrier are reviewed. Methods dealing with linear and all orders coupling between the nuclear excitations and the translational motion are discussed, and the latter is found to lead to a better description of the barrier distribution data. A systematic study of the available data (cross sections, barrier and spin distributions) in rare-earth nuclei is presented.
Subbarrier fusion of the 7Li + 12C reaction is studied using an antisymmetrized molecular dynamics model (AMD) with an after burner, GEMINI. In AMD, 7Li shows an α+ t structure at its ground state and it is significantly deformed. Simulations are made near the Coulomb barrier energies, i.e., E_{cm} = 3 - 8 MeV. The total fusion cross section of the AMD + GEMINI calculations as a function of incident energy is compared to the experimental results and both are in good agreement at E_{cm} > 3 MeV. The cross section for the different residue channels of the AMD + GEMINI at E_{cm} = 5 MeV is also c…
In this short note we use a simple model to describe the dynamical effects of break-up processes in the subbarrier fusion involving weakly bound nuclei. We model two similar cases involving either a neutron or a proton halo nucleus, both schematically coupled to the break-up channels. We find that the decrease of the coulomb barrier in the proton break-up channel leads, ceteris paribus, to a larger enhancement of the subbarier fusion probabilities with respect to the neutron-halo case.
The hindrance phenomenon of heavy-ion fusion cross sections at deep subbarrier energies often accompanies a maximum of an astrophysical $S$ factor at a threshold energy for fusion hindrance. We argue that this phenomenon can naturally be explained when the fusion excitation function is fitted with two potentials, with a larger (smaller) logarithmic slope at energies lower (higher) than the threshold energy. This analysis clearly suggests that the astrophysical $S$ factor provides a convenient tool to analyze the deep subbarrier hindrance phenomenon, even though the $S$ factor may have a strong…
We discuss the effects of double octupole and quadrupole phonon excitations in $^{144}$Sm on fusion reactions between $^{16}$O and $^{144}$Sm at subbarrier energies. The effects of anharmonicities of the vibrational states are taken into account by using the $sdf$-interacting boson model. We compare the results with those in the harmonic limit to show that anharmonicities play an essential role in reproducing the experimental fusion barrier distribution. From the analysis of the high quality fusion data available for this system, we deduce negative static quadrupole moments for both the first …
At energies below the Coulomb barrier, the fusion excitation function for the $^{12}$C+$^{12}$C system shows prominent fine structures, whereas that for the $^{12}$C+$^{13}$C system behaves more smoothly as a function of energy. We demonstrate that these different behaviors can be simultaneously reproduced using an optical potential in which the strength of the imaginary part is proportional to the level density of each compound nucleus. We also discuss the oscillatory behavior of fusion excitation function for these systems observed at energies above the Coulomb barrier from a view point of q…
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 collectiv…
In this paper, I will update the current status of the carbon-carbon fusion research taking into account that after the latest analysis [Beck {\it et al.} Eur. Phys. J. A {\bf 56}, 97 (2020), Letter to the Editor] new important experimental and theoretical results had been published and will discuss how to advance new THM measurements to extract the low-energy astrophysical $S$-factors.
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…
Since the discovery of molecular resonances in $^{12}$C+$^{12}$C in the early sixties a great deal of research work has been undertaken to study $α$-clustering and resonant effects of the fusion process at sub-Coulomb barrier energies. The modified astrophysical $S^{*}$ factors of $^{12}$C+$^{12}$C fusion have been extracted from direct fusion measurements at deep sub-barrier energies near the Gamow window. They were also obtained by the indirect Trojan horse method (THM). A comparison of direct measurements and the THM, which elucidates problems in the analysis of the THM, is discussed in thi…
We discuss heavy-ion fusion reactions of a well-deformed odd-mass nucleus at energies around the Coulomb barrier. To this end, we consider the $^{16}$O+$^{165}$Ho reaction and take into account the effect of deformation of $^{165}$Ho using the orientation average formula. We show that fusion cross sections are sensitive to magnetic substates of the target nucleus, and cross sections for the side collision, which are relevant to a synthesis of superheavy elements, may be enhanced by a factor of around two by aligning the deformed target nuclei. We also discuss the effect of alignment on the fus…
We discuss anharmonicity of the multi-octupole-phonon states in $^{208}$Pb based on a covariant density functional theory, by fully taking into account the interplay between the quadrupole and the octupole degrees of freedom. Our results indicate the existence of a large anharmonicity in the transition strengths, even though the excitation energies are similar to those in the harmonic limit. We also show that the quadrupole-shape fluctuation significantly enhances the fragmentation of the two-octupole-phonon states in $^{208}$Pb. Using those transition strengths as inputs to coupled channels c…
Based on the adiabatic picture for heavy-ion reactions, in which the neck formation in the one-body system is taken into account, we propose a two-step model for fusion cross sections at deep subbarrier energies. This model consists of the capture process in the two-body potential pocket, which is followed by the penetration of the adiabatic one-body potential to reach a compound state after the touching configuration. We describe the former process with the coupled-channels framework, while the latter with the WKB approximation by taking into account the coordinate dependent inertia mass. The…
We overview the current status of theoretical approaches for heavy-ion fusion reactions at subbarrier energies. We particularly discuss theoretical challenges in the coupled-channels approach, that include i) a description of deep subbarrier hindrance of fusion cross sections, ii) the role of nuclear dissipation, iii) fusion of unstable nuclei, and iv) an interplay between fusion and multi-nucleon transfer processes. We also present results of a semi-microscopic approach to heavy-ion fusion reactions, that combines the coupled-channels approach with state-of-the-art microscopic nuclear structu…
Fusion reactions below the Coulomb barrier provide new insights into multidimensional quantum tunneling, nuclear reaction dynamics and nuclear structure. These reactions are also of considerable interest to nuclear astrophysics. In this summary recent developments in the field are reviewed and open questions related to subbarrier fusion are presented.
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 sm…
The vast knowledge of strong influence of quadrupole deformation $β_2$ 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 $β_4$ and $β_6$ deformations. However, such studies have rarely been carried out, especially for deformation of projectile nuclei. In this article, we investigated the role of $β_4$ 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 $β_4$ paramete…
We discuss several open problems of fusion reactions induced by weakly bound nuclei. For this purpose, we solve a one dimensional three-body Hamiltonian with the coupled-channels formalism. We show that the continuum-continuum couplings substantially reduce the total fusion probability at energies above the barrier compared with the no-breakup case, although the fusion probability remains enhanced at subbarrier energies. We then discuss a role of transfer process in fusion of weakly bound nuclei, and point out that removing spurious Pauli forbidden transfer components from the calculation may …
We discuss the effects of multi-phonon excitations on heavy-ion fusion reactions at energies near and below the Coulomb barrier, focusing especially on the role of anharmonicities. We carry out a systematic study of those effects on the excitation function of the fusion cross section and on the fusion barrier distribution, by using the vibrational limit of the interacting boson model. We also analyze the recently measured high-precision data of the $^{16}$O + $^{148}$Sm fusion reaction with this model and discuss the anharmonic properties of the quadrupole as well as the octupole vibrations in…
A FORTRAN77 program is presented that calculates fusion cross sections and mean angular momenta of the compound nucleus under the influence of couplings between the relative motion and several nuclear collective motions. The no-Coriolis approximation is employed to reduce the dimension of coupled-channels equations. The program takes into account the effects of non-linear couplings to all orders, which have been shown to play an important role in heavy-ion fusion reactions at subbarrier energies.
We extend a recent study that explained the steep falloff in the fusion cross section at energies far below the Coulomb barrier for the symmetric dinuclear system 64Ni+64Ni to another symmetric system, 58Ni+58Ni, and the asymmetric system 64Ni+100Mo. In this scheme the very sensitive dependence of the internal part of the nuclear potential on the nuclear equation of state determines a reduction of the classically allowed region for overlapping configurations and consequently a decrease in the fusion cross sections at bombarding energies far below the barrier. Within the coupled-channels method…
Recent high precision experimental data for heavy-ion fusion reactions at subbarrier energies systematically show that a surprisingly large surface diffuseness parameter for a Woods-Saxon potential is required in order to fit the data. We point out that experimental data for quasi-elastic scattering at backward angles also favor a similar large value of surface diffuseness parameter. Consequently, a double folding approach fails to reproduce the experimental excitation function of quasielastic scattering for the $^{16}$O + $^{154}$Sm system at energies around the Coulomb barrier. We also show …
We describe low-lying collective excitations of atomic nuclei with the multi-reference covariant density functional theory, and combine them with coupled-channels calculations for heavy-ion fusion reactions at energies around the Coulomb barrier. To this end, we use the calculated transition strengths among several collective states as inputs to the coupled-channels calculations. This approach provides a natural way to describe anharmonic multi-phonon excitations as well as a deviation of rotational excitations from a simple rigid rotor. We apply this method to subbarrier fusion reactions of $…
We discuss the role of break-up process of a loosely-bound projectile in subbarrier fusion reactions. Coupled-channels calculations are carried out for $^{11}$Be + $^{208}$Pb and $^{4,6}$He + $^{238}$U reactions by discretizing in energy the particle continuum states. Our calculations show that the coupling to the break-up channel has two effects, namely the loss of flux and the dynamical modulation of fusion potential. Their net effects differ depending on the energy region. At energies above the Coulomb barrier, the former effect dominates over the latter and cross sections for complete fusi…
The nuclear fusion is a reaction to form a compound nucleus. It plays an important role in several circumstances in nuclear physics as well as in nuclear astrophysics, such as synthesis of superheavy elements and nucleosynthesis in stars. Here we discuss two recent theoretical developments in heavy-ion fusion reactions at energies around the Coulomb barrier. The first topic is a generalization of the Wong formula for fusion cross sections in a single-channel problem. By introducing an energy dependence to the barrier parameters, we show that the generalized formula leads to results practically…
Friction coefficients for the fusion reaction $^{16}$O+$^{16}$O $\rightarrow$ $^{32}$S are extracted based on both the time-dependent Hartree-Fock and the time-dependent density matrix methods. The latter goes beyond the mean-field approximation by taking into account the effect of two-body correlations, but in practical simulations of fusion reactions we find that the total energy is not conserved. We analyze this problem and propose a solution that allows for a clear quantification of dissipative effects in the dynamics. Compared to mean-field simulations, friction coefficients in the densit…
We perform semi-classical molecular dynamics simulations of screening by bound electrons in low energy nuclear reactions. In our simulations quantum effects corresponding to the Pauli and Heisenberg principle are enforced by constraints. In addition to the well known adiabatic and sudden limits, we propose a new "dissipative limit" which is expected to be important not only at high energies but in the extremely low energy region. The dissipative limit is associated with the chaotic behavior of the electronic motion. It affects also the magnitude of the enhancement factor. We discuss also numer…
We study the effect of a direct Giant Dipole Resonance ($GDR$) excitation in intermediate dinuclear systems with exotic shape and charge distributions formed in charge asymmetric fusion entrance channels. A related enhancement of the $GDR$ gamma yield in the evaporation cascade of the fused nucleus is expected. The dynamical origin of such $GDR$ extra strength will show up in a characteristic anisotropy of the dipole gamma-emission. A fully microscopic analysis of the fusion dynamics is performed with quantitative predictions of the $GDR$ photon yield based on a dynamics- statistics coupling m…
Atomic nuclei are composite systems, and they may be dynamically excited during nuclear reactions. Such excitations are not only relevant to inelastic scattering but they also affect other reaction processes such as elastic scattering and fusion. The coupled-channels approach is a framework which can describe these reaction processes in a unified manner. It expands the total wave function of the system in terms of the ground and excited states of the colliding nuclei, and solves the coupled Shrödinger equations to obtain the $S$-matrix, from which several cross sections can be constructed. Thi…
A novel quantum dynamical model based on the dissipative quantum dynamics of open quantum systems is presented. It allows the treatment of both deep-inelastic processes and quantum tunneling (fusion) within a fully quantum mechanical coupled-channels approach. Model calculations show the transition from pure state (coherent) to mixed state (decoherent and dissipative) dynamics during a near-barrier nuclear collision. Energy dissipation, due to irreversible decay of giant-dipole excitations of the interacting nuclei, results in hindrance of quantum tunneling.
A model of competition between complete fusion and quasifission channels in fusion of two massive nuclei is extended to include the influence of dissipative effects on the dynamics of nuclear fusion. By using the multidimensional Kramers-type stationary solution of the Fokker-Planck equation, the fusion rate through the inner fusion barrier in mass asymmetry is studied. Fusion probabilities in symmetric 90Zr+90Zr, 100Mo+100Mo, 110Pd+110Pd, 136Xe+136Xe, almost symmetric 86Kr+136Xe and 110Pd+136Xe reactions are calculated. An estimation of the fusion probabilities is given for asymmetrical 62Ni+…
We study the effect of screening by bound electrons in low energy nuclear reactions. We use molecular dynamics to simulate the reactions involving many electrons: D+$d$, D+D, $^3$He+$d$, $^3$He+D, $^6$Li+$d$, $^6$Li+D, $^7$Li+$p$, $^7$Li+H. Quantum effects corresponding to the Pauli and Heisenberg principles are enforced by constraints in terms of the phase space occupancy. In addition to the well known adiabatic and sudden limits, we propose a new "dissipative limit" which is expected to be important not only at high energies but in the extremely low energy region. The dissipative limit is as…
Heavy Ion Collisions (HIC) represent a unique tool to probe the in-medium nuclear interaction in regions away from saturation. In this work we present a selection of reaction observables in dissipative collisions particularly sensitive to the isovector part of the interaction, i.e. to the symmetry term of the nuclear Equation of State (EoS). At low energies the behavior of the symmetry energy around saturation influences dissipation and fragment production mechanisms. We will first discuss the recently observed Dynamical Dipole Radiation, due to a collective neutron-proton oscillation during t…
[Background] Reactions with stable beams have demonstrated a strong interplay between nuclear structure and fusion. Exotic beam facilities open new perspectives to understand the impact of neutron skin, large isospin, and weak binding energies on fusion. Microscopic theories of fusion are required to guide future experiments. [Purpose] To investigate new effects of exotic structures and dynamics in near-barrier fusion with exotic nuclei. [Method] Microscopic approaches based on the Hartree-Fock (HF) mean-field theory are used for studying fusion barriers in $^{40-54}$Ca+$^{116}$Sn reactions fo…
Accounting for viscous damping within Fokker-Planck equations led to various improvements in the understanding and analysis of nuclear fission of heavy nuclei. Analytical expressions for the fission time are typically provided by Kramers' theory, which improves on the Bohr-Wheeler estimate by including the time-scale related to many-particle dissipative processes along the deformation coordinate. However, Kramers' formula breaks down for sufficiently high excitation energies where Kramers' assumption of a large barrier no longer holds. In the regime $T>1$ MeV, Kramers' theory should be replace…
Quasi-fission is the primary reaction mechanism that prevents the formation of superheavy elements in heavy-ion fusion experiments. Employing the time-dependent density functional theory approach we study quasi-fission in the systems $^{40,48}$Ca+$^{238}$U. Results show that for $^{48}$Ca projectiles the quasi-fission is substantially reduced in comparison to the $^{40}$Ca case. This partly explains the success of superheavy element formation with $^{48}$Ca beams. For the first time, we also calculate the repartition of excitation energies of the two fragments in a dynamic microscopic theory. …
Energy dissipative processes play a key role in how quantum many-body systems dynamically evolve towards equilibrium. In closed quantum systems, such processes are attributed to the transfer of energy from collective motion to single-particle degrees of freedom; however, the quantum many-body dynamics of this evolutionary process are poorly understood. To explore energy dissipative phenomena and equilibration dynamics in one such system, an experimental investigation of deep-inelastic and fusion-fission outcomes in the $^{58}$Ni+$^{60}$Ni reaction has been carried out. Experimental outcomes ha…
Applying a macroscopic reduction procedure on the improved quantum molecular dynamics (ImQMD) model, the energy dependences of the nucleus-nucleus potential, the friction parameter, and the random force characterizing a one-dimensional Langevin-type description of the heavy-ion fusion process are investigated. Systematic calculations with the ImQMD model show that the fluctuation-dissipation relation found in the symmetric head-on fusion reactions at energies just above the Coulomb barrier fades out when the incident energy increases. It turns out that this dynamical change with increasing inc…
Heavy Ion Collisions (HIC) represent a unique tool to probe the in-medium nuclear interaction in regions away from saturation. In this work we present a selection of reaction observables in dissipative collisions particularly sensitive to the isovector part of the interaction, i.e. to the symmetry term of the nuclear Equation of State (EoS). At low energies the behavior of the symmetry energy around saturation influences dissipation and fragment production mechanisms. We will first discuss the recently observed Dynamical Dipole Radiation, due to a collective neutron-proton oscillation during t…
