The double folding (DF) approach is one of the widely used methods for finding nucleus-nucleus interaction potential. In the present work, the influence of the nuclear matter density on the DF potential and on the Coulomb barrier parameters is studied systematically for collisions of spherical nuclei. The value of the parameter $B_Z=Z_P Z_T/(A_P^{1/3}+A_T^{1/3})$ (estimating the Coulomb barrier height) varies in these calculations from 10 MeV up to 150 MeV. The novel feature of this study is that the nuclear densities came from the Relativistic Mean Field approach (RMF). For the nucleon-nucleon effective interaction, the M3Y forces with the finite range exchange term and density dependence are employed. The above barrier fusion cross sections are calculated within the framework of the trajectory model with surface friction. Results are compared with the previous study in which the nuclear density came from the Hartree-Fock (HF) calculations and with the high precision experimental data. This comparison demonstrates that i) agreement between the theoretical and experimental cross sections obtained with RMF and HF densities is of the same quality and ii) the values of the only adjustable parameter (friction strength) obtained with RMF and HF densities strongly correlate.
A detailed study fusion of neutron-rich colliding nuclei is performed using various isospin dependent potentials. For present study, Three different series namely, Ne-Ne, Ca-Ca, and Zr-Zr are taken into account and N/Z ratio.A monotonous increase (decrease) in the fusion barrier positions (heights) using a unified second order nonlinear parametrization in the normalized fusion barrier positions and heights with (N/Z-1) is presented. These predications are in good agreement with the available theoretical as well as experimental results.
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…
By using 14 different versions and parametrizations of a proximity potential and two new versions of the potential proposed in this paper, we perform a comparative study of fusion barriers by studying 26 symmetric reactions. The mass asymmetry $η_{A}$ = $(\frac{A_{2}-A_{1}}{A_{2}+A_{1}}),$ however, is very large. Our detailed investigation reveals that most of the proximity potentials reproduce experimental data within $\pm$8% on the average. A comparison of fusion cross sections indicates that Bass 80, AW 95, and Denisov DP potentials have a better edge than other potentials. We also propose …
For the first time, the precise data on the above barrier fusion (capture) cross-sections for the reactions involving spherical colliding nuclei are quantitatively analyzed using the relativistic mean-field effective interaction. The parameter sets NL1, NL2, NL3, and HS are employed. The analysis is performed within the framework of the fluctuation-dissipation model with surface friction based on the double-folding approach for the nucleus-nucleus potential. The effective interactions, as well as the resulting potentials, are confronted with the ones obtained using the M3Y NN forces. Of the fo…
[Background] Significant enhancement of sub-barrier fusion cross sections owing to neutron rearrangement with positive $Q$-values were found for many combinations of colliding nuclei. However several experimental results on fusion reactions were reported recently in which such enhancement has not been observed in spite of a possibility for neutron rearrangement with positive $Q$-values. [Purpose] We aim to clarify much better the mechanism of neutron rearrangement in sub-barrier fusion reactions to find the other requirements (beside positive $Q$-value) which favour (or prevent) sub-barrier fu…
A potential model to describe the total cross section of nuclear fusion reactions at low energies is proposed. It is shown that within an approach with a simple, single barrier potential a satisfactory description of data is hindered, while with models with more complex interactions, e.g., with two-barrier potentials, a good description of data can be achieved in a large interval of colliding nuclei and center of mass energies. In particular, the two-barrier model allows to describe data at low, sub-barrier energies where data exhibits a steep falloff of the cross section. It is also shown tha…
The roles of nuclear deformation and neutron transfer in sub-barrier capture process are studied within the quantum diffusion approach. The change of the deformations of colliding nuclei with neutron exchange can crucially influence the sub-barrier fusion. The comparison of the calculated capture cross section and the measured fusion cross section in various reactions at extreme sub- barrier energies gives us information about the fusion and quasifission.
The breakup of the $^{9}$Be projectile on the $^{209}$Bi target at bombarding energies above and near the Coulomb barrier is studied in the adiabatic two-center shell model approach. The effect of $^{9}$Be$% \to $n$+2α$ breakup channel on complete fusion, elastic and inelastic cross sections is investigated. Results show that the breakup of the projectile $^{9}$Be could be due to a molecular single-particle effect shortly before the colliding nuclei reach the Coulomb barrier.
We present fission-barrier-height calculations for nuclei throught the Periodic Table based on a realistic macroscopic-microscopic model. Compared to other calculations: (1) we use a deformation space of sufficiently high dimension, sampled densely enough to describe the relevant topography of the fission potential, (2) we unambiguously find the physically relevant saddle points in this space, and (3) we formulate our model so that we obtain continuity of the potential energy at the division point between a single system and separated fission fragments or colliding nuclei, allowing us to (4) d…
The shell correction is proposed in the improved isospin dependent quantum molecular dynamics (ImIQMD) model, which plays an important role in heavy-ion fusion reactions near Coulomb barrier. By using the ImIQMD model, the static and dynamical fusion barriers, dynamical barrier distribution in the fusion reactions are analyzed systematically. The fusion and capture excitation functions for a series of reaction systems are calculated and compared with experimental data. It is found that the fusion cross sections for neutron-rich systems increase obviously, and the strong shell effects of two co…
Dissipation and fluctuations of one-body observables in heavy-ion reactions around the Coulomb barrier are investigated with a microscopic stochastic mean-field approach. By projecting the stochastic mean-field dynamics on a suitable collective path, transport coefficients associated with the relative distance between colliding nuclei and a fragment mass are extracted. Although microscopic mean-field approach is know to underestimate the variance of fragment mass distribution, the description of the variance is much improved by the stochastic mean-field method. While fluctuations are consisten…
Fusion reactions of heavy ions are investigated by employing a simple stochastic semi-classical model which includes the coupling between relative motion and low frequency collective surface modes of colliding ions similarly to the quantal coupled-channels description. The quantal effect enters into the calculation through the initial zero-point fluctuations of the surface vibrations. Good agreement with the result of coupled-channels calculations as well as data is obtained for the fusion cross sections of nickel isotopes. The internal excitations in non-fusing events as well as the fusion ti…
The enhancement of the sub-barrier fusion cross sections was explained as the lowering of the dynamical fusion barriers within the framework of the improved isospin-dependent quantum molecular dynamics (ImIQMD) model. The numbers of nucleon transfer in the neck region are appreciably dependent on the incident energies, but strongly on the reaction systems. A comparison of the neck dynamics is performed for the symmetric reactions $^{58}$Ni+$^{58}$Ni and $^{64}$Ni+$^{64}$Ni at energies in the vicinity of the Coulomb barrier. An increase of the ratios of neutron to proton in the neck region at i…
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…
The behaviors of barrier characteristics and fusion cross sections are analyzed by changing neutron over wide range of colliding systems. For this purpose, we have extended our previous study (Eur. Phys. J. A \textbf{48}, 21 (2012), it is devoted to the colliding systems with neutron-rich nuclei) to 125 isotopic systems with condition of $0.5\leq N/Z \leq 1.6$ for their compound nuclei. The AW 95, Bass 80, Denisov DP and Prox. 2010 potentials are used to calculate the nuclear part of interacting potential. The obtained results show that the trend of barrier heights $V_B$ and positions $R_B$ as…
Within the framework of Skyrme energy density formalism, we investigate the role of surface corrections on the fusion of colliding nuclei. For this, the coefficient of surface correction was varied between 1/36 and 4/36, and its impact was studied on about 180 reactions. Our 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.
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 demonstrate the damping of quantum octupole vibrations near the touching point when two colliding nuclei approach each other in the mass-asymmetric $^{208}$Pb + $^{16}$O system, for which the strong fusion hindrance was clearly observed. We, for the first time, apply the random-phase approximation method to the heavy-mass asymmetric di-nuclear system to calculate the transition strength $B$(E3) as a function of the center-of-mass distance. The obtained $B$(E3) strengths are substantially damped near the touching point, because the single-particle wave functions of the two nuclei strongly mi…
We demonstrate that when two colliding nuclei approach each other, their quantum vibrations are damped near the touching point. We show that this damping is responsible for the fusion hindrance phenomena measured in the deep sub-barrier fusion reactions. To show those, we for the first time apply the random-phase-approximation (RPA) method to the two-body $^{16}$O + $^{16}$O and $^{40}$Ca + $^{40}$Ca systems. We calculate the octupole transition strengths for the two nuclei adiabatically approaching each other. The calculated transition strength drastically decreases near the touching point, s…
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…
The conditions of optimum orientations (lowest barrier and largest interaction radius) for deformed colliding nuclei are introduced in "cold" fusion of superheavy nuclei. Also, the role of (octupole and) hexadecupole deformations is studied. We have used the proximity potential and applied our method to Ca-induced reactions.
Experimental results for sub-barrier nuclear fusion reactions show cross section enhancements with respect to bare nuclei which are generally larger than those expected according to electron screening calculations. We point out that energy spread of target or projectile nuclei is a mechanism which generally provides fusion enhancement. We present a general formula for calculating the enhancement factor and we provide quantitative estimate for effects due to thermal motion, vibrations inside atomic, molecular or crystal system, and due to finite beam energy width. All these effects are marginal…
We discuss two recent topics on heavy-ion quasi-elastic scattering at energies around the Coulomb barrier. The first topic is an application of quasi-elastic scattering at deep-subbarrier energies to extracting the surface diffuseness parameter of the nucleus-nucleus potential. The second topic is a coupled-channels analysis for the quasi-elastic barrier distribution for the $^{70}$Zn + $^{208}$Pb reaction. We show that the coupled-channels calculations which include the multi-phonon excitations in the colliding nuclei reproduce reasonably well the experimental excitation function for quasi-el…
The multinucleon transfer reactions near barrier energies has been investigated with a multistep model based on the dinuclear system (DNS) concept, in which the capture of two colliding nuclei, the transfer dynamics and the de-excitation process of primary fragments are described by the analytical formula, the diffusion theory and the statistical model, respectively. The nucleon transfer takes place after forming the DNS and is coupled to the dissipation of relative motion energy and angular momentum by solving a set of microscopically derived master equations within the potential energy surfa…
We perform exact coupled-channels calculations, taking into account properly the effects of Coulomb coupling and the finite excitation energy of collective excitations in the colliding nuclei, for three Fm formation reactions, $^{37}$Cl + $^{209}$Bi, $^{45}$Sc + $^{197}$Au, and $^{59}$Co + $^{181}$Ta. For the $^{37}$Cl + $^{209}$Bi and $^{45}$Sc + $^{197}$Au reactions, those calculations well reproduce the experimental total fission cross sections, and a part of the extra-push phenomena can be explained in terms of the Coulomb excitations to multi-phonon states. On the other hand, for the heav…
The Pauli exclusion principle induces a repulsion between composite systems of identical fermions such as colliding atomic nuclei. Our goal is to study how heavy-ion fusion is impacted by this "Pauli repulsion". We propose a new microscopic approach, the density-constrained frozen Hartree-Fock method, to compute the bare potential including the Pauli exclusion principle exactly. Pauli repulsion is shown to be important inside the barrier radius and increases with the charge product of the nuclei. Its main effect is to reduce tunnelling probability. Pauli repulsion is part of the solution to th…
With the quantum diffusion approach the unexpected behavior of fusion cross section, angular momentum, and astrophysical S-factor at sub-barrier energies has been revealed. Out of the region of short-range nuclear interaction and action of friction at turning point the decrease rate of the cross section under the barrier becomes smaller. The calculated results for the reactions with spherical nuclei are in a good agreement with the existing experimental data.
Couplings between relative motion and internal structures are known to affect fusion barriers by dynamically modifying the densities of the colliding nuclei. The effect is expected to be stronger at energies near the barrier top, where changes in density have longer time to develop than at higher energies. Quantitatively, modern TDHF calculations are able to predict realistic fusion thresholds. However, the evolution of the potential barrier with bombarding energy remains to be confronted with the experimental data. The aim is to find signatures of the energy dependence of the barrier by compa…
The influence on fusion of coupling to the breakup process is investigated for reactions where at least one of the colliding nuclei has a sufficiently low binding energy for breakup to become an important process. Elastic scattering, excitation functions for sub-and near-barrier fusion cross sections, and breakup yields are analyzed for $^{6,7}$Li+$^{59}$Co. Continuum-Discretized Coupled-Channels (CDCC) calculations describe well the data at and above the barrier. Elastic scattering with $^{6}$Li (as compared to $^{7}$Li) indicates the significant role of breakup for weakly bound projectiles. …
In the collision of two heavy ions, the strong repulsion coming from the Coulomb field is enough to produce $e^+e^-$ pair(s) from vacuum fluctuations. The energy is provided by the kinetic energy of the ions and the Coulomb interaction at the production point. If, for instance, the electron is located at the center of mass (C.M.) of the two ions moving along the \emph{z}-axis, and the positron is at a distance \emph{x} from the electron, the ions can be accelerated towards each other since the Coulomb barrier is lowered by the presence of the electron. This screening results in an increase in …
Excitations of colliding nuclei during a nuclear reaction considerably affect fusion cross sections at energies around the Coulomb barrier. It has been demonstrated that such channel coupling effects can be represented in terms of a distribution of multiple fusion barriers. We here apply a Bayesian approach to analyze the so called fusion barrier distributions. This method determines simultaneously the barrier parameters and the number of barriers. We particularly investigate the $^{16}$O+$^{144}$Sm and $^{16}$O+$^{154}$Sm systems in order to demonstrate the effectiveness of the method. The pr…
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…
Within the framework of the dinuclear system model, the reaction mechanisms for synthesizing target-like isotopes from Bk to compound nuclei Lv are thoroughly investigated in complete and incomplete fusion reaction of $^{48}$Ca +$^{248}$Cm around Coulomb barrier energies. Production cross-section of $^{292,293}$Lv as a function of excitation energy in fusion-evaporation reactions and target-like isotopic yields in multinucleon transfer reactions are evaluated, in which a statistical approach is used to describe the decay process of excited nuclei. The available experimental data can be reprodu…
The potential energy of a deformed nucleus has been determined within a Generalized Liquid Drop Model taking into account the proximity energy, the microscopic corrections and compact and necked shapes. Multiple-humped potential barriers appear. A third minimum and third maximum exist in specific exit channels where one fragment is close to a magic spherical nucleus while the other one varies from oblate to prolate shapes. The heights of the fission barriers and half-lives of actinides are in agreement with the experimental results.
Based on a single particle model which describes the time evolution of the wave function during tunneling across a one dimensional potential barrier we study the proton decay of $^{208}$Pb from excited states with non-vanishing angular momentum $\ell$. Several quantities of interest in this process like the decay rate $λ$, the period of oscillation $T_{osc}$, the transient time $t_{tr}$, the tunneling time $t_{tun}$ and the average value of the proton packet position $ r_{av} $ are computed and compared with the WKB results.
Systematic calculations of the Coulomb barrier parameters for collisions of spherical nuclei are performed within the framework of the double folding approach. The value of the parameter $B_Z=Z_PZ_T/(A^{1/3}_P+(A^{1/3}_P)$ (which estimates the Coulomb barrier height) varies in these calculations from 10 MeV up to 150 MeV. The nuclear densities came from the Hartree-Fock calculations which reproduce the experimental charge densities with good accuracy. For the nucleon-nucleon effective interaction two analytical approximations known in the literature are used: the M3Y and Migdal forces. The cal…
With the quantum diffusion approach the behavior of capture cross sections and mean-square angular momenta of captured systems are revealed in the reactions with deformed and spherical nuclei at sub-barrier energies. With decreasing bombarding energy under the barrier the external turning point of the nucleus-nucleus potential leaves the region of short-range nuclear interaction and action of friction. Because of this change of the regime of interaction, an unexpected enhancement of the capture cross section is found at bombarding energies far below the Coulomb barrier. This effect is shown it…
A systematic study of fission-barrier dependence on excitation energy has been performed using the self-consistent finite-temperature Hartree-Fock+BCS (FT-HF+BCS) formalism with the SkM* Skyrme energy density functional. The calculations have been carried out for even-even superheavy nuclei with Z ranging between 110 and 124. For an accurate description of fission pathways, the effects of triaxial and reflection-asymmetric degrees of freedom have been fully incorporated. Our survey demonstrates that the dependence of isentropic fission barriers on excitation energy changes rapidly with particl…
The influence on the fusion process of coupling to collective degrees of freedom has been explored. The significant enhancement of the fusion cross section at sub-barrier energies was compared to predictions of one-dimensional barrier penetration models. This was understood in terms of the dynamical processes arising from strong couplings to collective inelastic excitations of the target and projectile. However, in the case of reactions where at least one of the colliding nuclei has a sufficiently low binding energy, for breakup to become an important process, conflicting model predictions and…
The semi-microscopic potential between heavy nuclei is evaluated for various colliding ions in the approach of frozen densities in the framework of the extended Thomas-Fermi approximation with hbar^2 correction terms in the kinetic energy density functional. The proton and neutron densities of each nucleus are obtained in the Hartree-Fock-BCS approximation with SkM* parameter set of the Skyrme force. A simple expression for the nuclear interaction potential between spherical nuclei is presented. It is shown that muon bound with light projectile induces the superheavy elements production in nuc…
Background: Multinucleon transfer reactions at energies around the Coulomb barrier offer a vital opportunity to study rich physics of nuclear structure and dynamics. Despite the continuous development in the field, we have still limited knowledge about how deformation - one of the representative nuclear structures - affects multinucleon transfer reactions. Purpose: To shed light on the effect of deformation in multinucleon transfer processes, we study the $^{16}$O+$^{154}$Sm reaction at $E_{\rm lab}$=85 MeV (around the Coulomb barrier) and 134 MeV (substantially above the Coulomb barrier), whe…
Entrance-channel potentials in nucleus-nucleus collisions, relevant for the synthesis of superheavy elements, are systematically studied within a semi-microscopic approach, where microscopic nuclear densities of the colliding spherical or deformed nuclei are used in semi-classical expressions of the energy-density functional. From experimental data on fusion windows evidence is found that the existence of pockets in the entrance-channel potentials is crucial for fusion. Criteria for the choice of best collision systems for the synthesis of superheavy elements are discussed.
We perform a systematic study on the surface property of nucleus-nucleus potential in heavy-ion reactions using large-angle quasielastic scattering at energies well below the Coulomb barrier. At these energies, the quasielastic scattering can be well described by a single-channel potential model. Exploiting this fact, we point out that systems which involve spherical nuclei require the diffuseness parameter of around 0.60 fm in order to fit the experimental data, while systems with a deformed target between 0.8 fm and 1.1 fm.
The tensor force is implemented into the time-dependent Hartree-Fock (TDHF) theory so that both exotic and stable collision partners, as well as their dynamics in heavy-ion fusion, can be described microscopically. The role of tensor force on fusion dynamics is systematically investigated for $^{40}\mathrm{Ca}+\mathrm{^{40}Ca}$, $^{40}\mathrm{Ca}+\mathrm{^{48}Ca}$, $^{48}\mathrm{Ca}+\mathrm{^{48}Ca}$, $^{48}\mathrm{Ca}+\mathrm{^{56}Ni}$, and $^{56}\mathrm{Ni}+\mathrm{^{56}Ni}$ reactions which vary by the total number of spin-unsaturated magic numbers in target and projectile. A notable effect …
We use a phenomenological nucleus-nucleus optical potential constructed in the light of a potential developed by Ginocchio to study the elastic angular distributions of different nuclear systems near Coulomb barrier. The differential cross section ratios of elastic to Rutherford are studied for systems $^{14}$N+$^{56}$Fe and $^{14}$N+$^{90}$Zr for several colliding energies. Our theoretical predictions are well competitive with the experimental data, and the significantly small imaginary part of the potential explains the elastic scattering cross section and threshold anomaly in both systems.
Elliptic flow ($v_2$) fluctuations in central heavy-ion collisions are direct probes of the fluctuating geometry of the quark-gluon plasma, and, as such, are strongly sensitive to any deviation from spherical symmetry in the shape of the colliding nuclei. We investigate the consequences of nuclear deformation for $v_2$ fluctuations, and we assess whether current models of medium geometry are able to predict and capture such effects. Assuming linear hydrodynamic response between $v_2$ and the eccentricity of the medium, $\varepsilon_2$, we perform accurate comparisons between model calculations…
The effect of the neutron and proton numbers ratio in colliding nuclei at formation of the evaporation residues in the $^{34}$S+$^{208}$Pb and $^{36}$S+$^{206}$Pb reactions
We present results of collisions of $^{90}$Zr+$^{90}$Zr and $^{96}$Zr+$^{96}$Zr obtained within time-dependent density functional theory (TDDFT) extended to superfluid systems, known as time-dependent superfluid local density approximation (TDSLDA). We discuss qualitatively new features occurring in collisions of two superfluid nuclei at energies in the vicinity of the Coulomb barrier. We show that a \textit{solitonic excitation} -- an abrupt pairing phase distortion -- reported previously [P.~Magierski et al., Phys. Rev. Lett. \textbf{119}, 042501 (2017)], increases the barrier for capture ge…
BiFold calculates the density-dependent (DDM3Y$n$, BDM3Y$n$, CDM3Y$n$) or independent double folded potentials between two colliding spherical nuclei. It is written in a Python package form to give the ability to use the potentials directly in a nuclear reaction/structure code. In addition to using Woods-Saxon/Fermi or Gaussian functions, the code also allows for the definition of nuclear matter densities using pre-calculated densities in a data file. The manuscript provides an overview of the double folding model and the use of the code.
