Proximity Formalism Research Articles

Systematic study of the fusion hindrance phenomenon using the proximity potential approach: signature of the energy-dependent effects of the surface energy coefficient

In the present study, we have systematically studied the role of surface energy coefficient as well as temperature dependence in the fusion hindrance phenomenon of the heavy-ion reactions using the proximity potential formalism. To this end, we have performed the calculations of the interaction potential using the original proximity potential 1977 (Prox. 77) and the fusion cross sections are calculated based on the coupled-channels (CC) approach. The considered fusion systems are including the heavy-ion reactions 11B+197Au, 12C+198Pt, 16O+208Pb, 28Si + 64Ni, 28Si + 94Mo, 58Ni+58Ni, 32S+89Y,34S+89Y, 12C+204Pb and 36S + 64Ni with conditions of Q <0 and charge product of the participant nuclei 392≤ Z1Z2 ≤784. Our preliminary calculations show the Prox. 77 model predicts the theoretical values ​​of the fusion cross-sections less than the corresponding experimental data especially in the energy regions below the fusion barrier. However, the imposing of the mentioned physical effects increases the calculated values of the fusion cross sections and thus improves their agreement with the experimental cross sections for the selected reactions. In addition, by considering the energy dependence on the surface energy constant γ0 of the proximity formalism at the low energy region we are able to reproduce well the fusion cross sections, the astrophysical factor S(E) as well as the logarithmic derivative L(E) in these regions.‎

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.

Systematic study on $$\alpha $$-decay half-lives: a new dependency of effective sharp radius on $$\alpha $$-decay energy

Within the framework of the proximity formalism, we present a systematic study to analyze the effects of the $$\alpha $$ -decay energy through the effective sharp radius parameter on the $$\alpha $$ -decay half-lives of 227 nuclei in the range $$61 \le Z \le 99$$ . Wentzel-Kramers-Brillouin (WKB) calculations with the proximity potential Zhang 2013 are carried out to obtain the theoretical values of the $$\alpha $$ -decay half-lives. In this work, we introduce a new $$Q_{\alpha }$$ -dependent (QD) form of the effective sharp radius which significantly reduces the standard deviation of estimated half-lives using the Zhang 2013 model in comparison with the corresponding experimental data in our selected mass range. We evaluate the validity of this simple formula using the Geiger-Nuttall (G-N) plots and semi-empirical formulas. The modified form of the Zhang 2013 model is also found to work well in $$\alpha $$ -decay studies of superheavy nuclei (SHN) with $$Z=117-120$$ . Our results reveal that the calculated half-lives for the use of new proposed form of the effective sharp radius in the proximity potential can reproduce the closed-shell effects at neutron magic number $$N=126$$ and $$N=184$$ .

Role of nuclear surface tension coefficients in the isotopic dependence of fusion dynamics for neutron-rich colliding nuclei: New parametrization of fusion barriers

A theoretical approach is developed to study systematically the influence of the nuclear surface tension coefficient, γ, of the proximity formalism on the isotopic behavior of the fusion dynamics over a wide range of neutron-rich colliding systems with condition of 1 ≤ N/Z < 1.6 for their compound nuclei. Here six different versions of the surface energy coefficient (γ-PD03, γ-MS67, γ-MS66, γ-MN81-I, γ-MN81-II and γ-MN76) in the calculation of the nuclear potential based on the original proximity potential 1977are employed. It is found that the fusion barrier characteristics and fusion cross sections follow a linear isotopic dependence which is sensitive to the change in the nuclear surface tension between the reacting nuclei. The maximum sensitivity of the data to this technical parameter is obtained for the use of the γ-PD03 version. Further, it is shown that the sensitivity of the isotopic behavior of the fusion cross sections to the coefficient γ increases by decreasing the incident energy. After the calculation of fusion barrier heights and positions using the different versions of proximity potentials, we have searched for their parametrization. Firstly, we present a pocket formula for fusion barriers which is directly dependent on both the coefficient γ and the N/Z ratio of the isotopic systems. Our analysis for the studied isotopic systems reveals that the agreement with the empirical data is improved when we consider the dependence of the analytical parameterized formulas on this coefficient. Further, the fusion cross sections obtained by the present pocket formula are in good agreement with the corresponding experimental data. A comparison with other well known parameterized forms in the context of isotopic studies is also made.

A new formalism for the study of the surface tension coefficient of α-nuclei systems using the density-dependent nucleon-nucleon interactions

This study investigates the nuclear surface tension coefficient, γ, of the proximity formalism by using the microscopic double-folding (DF) model with the realistic density-dependent (DD) nucleon-nucleon interaction of the effective M3Y forces type (including DDM3Y1, CDM3Y4 and BDM3Y1) for the ground-state to ground-state α transition of 230 parent nuclei with Z = 61-99. In fact, the present work can be considered as an expansion of the previous study which has been performed by Gharaei and Mohammadi using CDM3Y6 version in 2019. Within the propsed approach, we have tried to present a new approach for the calculation of the surface energy coefficient, γ, in alpha-decay by integrating the proximity and DF potential models. In addition, we present a new dependency of the surface energy coefficient, γ, on the asymmetry parameter, , of the considered α-nuclei systems by fitting all of the calculated values. The obtained results suggest a new formalism for the coefficient γ that is dependent directly on the selection of the interaction type. We also test the validity of the suggested formula. To this aim, by using the the obtained formula of the coefficient γ in the original version of the proximity potentials, we calculate the theoretical values of the alpha-decay half-lives for different nuclei in the framework of the WKB approximation. The calculated results are compared with the corresponding experimental data and those obtained from the original proximity potential 1977. It is shown that the modified forms of the proximity potential model, labeled as Prox. New (DDM3Y1), Prox. New (CDM3Y4) and Prox. New (BDM3Y1), provide better descriptions of the experimental α-decay half-lives than the proximity potential 1977 (Prox. 77). Further, the best results are obtained using the Prox. New (CDM3Y4) potential model for our selected mass range. Using the modified forms of the proximity potential, we examined the closed-shell effects in nuclei and the validity of the Geiger-Nuttall law. Additionally, the results of the Prox. New (CDM3Y4) potential model are compared with the various empirical formulas for alpha decay half-lives. Ultimately, the prediction of alpha decay half lives is made for superheavy nuclei with Z=117-120.

Study of the surface energy coefficient used in nuclear proximity potential of the $ \alpha$-nuclei systems from density-dependent nucleon-nucleon interactions

An attempt has been made to study systematically the nuclear surface tension coefficient, $ \gamma$ , of the proximity formalism by using the microscopic double-folding (DF) model with the realistic density-dependent (DD) nucleon-nucleon interaction for the ground-state-to-ground-state $ \alpha$ -decays of 235 parent nuclei with $ Z=61\mbox{--}99$ . In the calculations of DF model, the nucleon-nucleon (NN) interactions are formulated using the effective M3Y force of the CDM3Y6 based on the G -matrix elements of the Paris interaction. We present a new dependence of the surface energy coefficient $ \gamma$ on the asymmetry parameter $ A_s$ of the $ \alpha$ -nuclei systems by fitting all of the calculated values. Using the presently obtained formula of the coefficient $ \gamma$ in the proximity potential we calculate the theoretical values of the alpha-decay half-lives for different nuclei in the framework of the Wentzel-Kramers-Brillouin (WKB) approximation. Results of our calculations are compared to the values of the experimental data and proximity potential 1977. Good agreements are found. The quality of the proximity potential accompanied by the analytical formula of the surface energy coefficient is tested for fusion reactions between $ \alpha$ -particles and heavy nuclei 40 Ca, 48 Ti, 51V , 59Co , 63Cu , 93Nb , 154Sm and 162Dy The evaluated $ \alpha$ -capture cross sections agreed well with the corresponding experimental data. In the present study, the role of the proton and neutron magic numbers of daughter nuclei and also the Geiger-Nuttall plots of $ \log_{10} (T_{1/2})$versus$ Q_{\alpha}^{-1/2}$ for various isotopes of heavy parent nuclei have been studied.

A systematic study on the influence of nuclear surface tension and temperature upon the parameterization of the fusion dynamics

For the first time the influence of the surface energy coefficient $\gamma$ and temperature T on the parameterization of the fusion barriers is systematically analyzed within the framework of the proximity formalism, namely proximity 1977, proximity 1988 and proximity 2010 models. A total of 114 fusion reactions with the condition $39 \leq Z_{1} Z_{2} \leq 1520$ for the charge product of their participant nuclei have been studied. We present $\gamma$ -dependent and T -dependent pocket formulas which reproduce the theoretical and empirical data of the fusion barrier height and position for our considered reactions with good accuracy. It is shown that the quality of the $\gamma$ -dependent formula enhances by increasing the strength of the surface energy coefficient. Moreover, the obtained results confirm that imposing the thermal effects improves the agreement between the parameterized and empirical data of the barrier characteristics.

Parameterization of fusion barriers for light-projectiles-induced reactions using the proximity approach

In this article we propose a pocket formula for fusion barriers calculated by three versions of the proximity formalism, namely AW 95, Bass 80 and Prox. 2010 potentials, for fusion reactions involving the collisions of the proton and helium projectiles with different targets in mass ranges $ 51\leq A_{T} \leq 130$ and $ 40\leq A_{T} \leq 233$ , respectively. For the first type of the colliding systems, it is shown that the proposed pocket formulas are able to predict the actual values of $ R_{B}$ and $ V_{B}$ within accuracies of ±0.4% and ±0.45% , respectively. Moreover, for the second type of the selected reactions, these accuracies are obtained ±0.24% and ±0.36% , respectively. In this study, the ability of the present pocket formulas is also demonstrated to predict the exact values of the fusion cross sections for our selected mass ranges. A comparison with the results of the previous pocket formulas reveals that our parameterized forms are more successful to reproduce the empirical data of the barrier height and position in the proton- and helium-induced reactions.

Temperature-dependent potential in cluster-decay process

Role of the thermal effects of the parent nucleus in the Coulomb barrier and the half-life of 28 cluster-decays is systematically analyzed within the framework of the proximity formalism, namely proximity potential 2010. The WKB approximation is used to determine the penetration probability of the emitted cluster. It is shown that the height and width of the Coulomb barrier in the temperature-dependent proximity potential are less than its temperature-independent version. Moreover, this investigation reveals that the calculated values of half-life for selected cluster-decays are in better agreement with the experimental data when the mentioned effects are imposed on the proximity approach. A discussion is also presented about the predictions of the present thermal approach for cluster-decay half-lives of the super-heavy-elements.

Role of Surface Energy Coefficients and Temperature in the Fusion Reactions Induced by Weakly Bound Projectiles

A systematic study is provided to analyze the behaviors of the interaction potential and complete fusion cross section which are influenced by the effects of the surface energy coefficients γ and temperature T. Our framework is restricted to the proximity formalism for fusion reactions induced by weakly bound projectiles 6Li, 7Li and 9Be. The different surface energy coefficients (γ-MN76, γ-MN95, γ-MS00 and γ-PD03) are used to study the role of the parameter γ in the proximity potentials AW 95 and BW 91. Comparison of the theoretical and the experimental values of the barrier characteristics (barrier heights and its positions) indicates that the modified versions AW 95 (γ-MS00) and BW 91 (γ-MS00) give the least deviations for fusion barrier heights. Moreover, it is shown that the temperature-dependence improves the calculated barrier heights based on the potentials AW 95 and BW 91. In the present study, the analysis of the mentioned effects on the complete fusion cross sections has been also discussed for the systems of interest. The obtained results reveal that the above-modified versions provide a more accurate description for behavior of the complete fusion cross sections than the original potentials at above-barrier energies. It is demonstrated that the increase of the temperature T enhances the complete fusion suppression at this energy range.

Systematic study of the isotopic behavior of the fusion cross section at energies near and below the fusion barrier using the proximity formalism

The isotopic behavior of the fusion cross sections and barrier curvature at energies near and below the barrier are explored using the proximity formalism. For this purpose, various versions of the proximity potentials are used to calculate the nuclear part of the interaction potential. Our study has been restricted to the isotopic systems which obey the condition of $0.5\ensuremath{\le}N/Z\ensuremath{\le}1.6$ for their compound nuclei. The obtained results show that the barrier curvature and the fusion cross section follow a similar behavior at near- and below-barrier energies. In fact, these quantities decrease nonlinearly (second order) with the addition of a neutron. In the present study, the energy dependence of the fusion cross sections is also discussed for the considered isotopic systems.

Analysis of heavy-ion fusion reactions at extreme sub-barrier energies using the proximity formalism

The recent measured values of the fusion excitation functions of the heavy-ion colliding systems ${}^{28}$Si${+}^{100}$Mo, ${}^{58}$Ni${+}^{54}$Fe, and ${}^{64}$Ni${+}^{64}$Ni are investigated using the original version of the proximity formalism. The fusion cross sections are calculated based on the coupled-channels approach, including couplings to the low-lying ${2}^{+}$ and ${3}^{\ensuremath{-}}$ states in both target and projectile nuclei. The comparison between the calculated and the measured values of the fusion excitation functions indicates that the potential Prox.77 needs to be modified considerably at sub-barrier energies. In the present study, the role of the surface energy coefficient $\ensuremath{\gamma}$ and also the temperature $T$ of the compound nucleus in nuclear potential and fusion cross section has been explored for our colliding systems. Moreover, the mutual and the multiphonon excitations of the lowest ${2}^{+}$ and ${3}^{\ensuremath{-}}$ states are considered in the coupled-channels calculations. It is demonstrated that the potential Prox.77 with these corrective effects can reproduce the experimental data of the fusion cross section, the $S$ factor and the logarithmic derivative for fusion reactions ${}^{28}$Si${+}^{100}$Mo, ${}^{58}$Ni${+}^{54}$Fe, and ${}^{64}$Ni${+}^{64}$Ni with good accuracy especially at below-barrier energies.

ROLE OF SURFACE ENERGY COEFFICIENT AND TEMPERATURE OF COMPOUND NUCLEUS IN THE α-DECAY PROCESS

The proximity formalism is used to study the influence of the surface energy coefficient γ and temperature T of compound nucleus on the half-life of the α-decay process. For this purpose, we perform a systematic study based on the original version of the proximity model. The obtained results of the present investigation reveal that the half-life of the α-decay process reduces by increasing the strength of the parameter γ. Whereas, temperature-dependence of the compound nucleus increases the calculated values of the α-decay half-life.

Studying the Effects of Compound Nucleus Energy on Coefficient of Surface Tension in Fusion Reactions Using Proximity Potential Formalism

In this investigation, the energy effects of compound nucleus are studied using formalism of proximity potential on the surface tension coefficient (γ). For this purpose, this coefficient in both main and final versions of proximity formalism, which are: Prox.77 and Prox.2010, is corrected based on systematic studying on 60 of fusion reactions. The correction in γ is caused to improve agreeing fusion cross sections with empirical amounts of them specifically in sub-barrier energies. This correction has significant effect on increasing the theoretical amounts of cross sections in the range of Coulomb sub-barrier energies. The produced results show improvement of agreement on theoretical amounts of Coulomb barrier height with corresponding laboratory data in the various collision systems. In addition, the obtained results show that the MP.2010 potential is able to produce laboratory data more accurate rather than other introduced potentials.

Role of the saturation properties of hot nuclear matter in the proximity formalism

Temperature dependence of the surface thickness parameter $b$ in the original version of proximity formalism is investigated by including the modification effects of equation of state (EOS) at finite temperature on nuclear potential. Here, this modification is performed using the evaluation of the increasing of total energy in the overlapping region by the EOS extracted from the extended Thomas-Fermi approach for asymmetric nuclear matter at finite temperature. This EOS is also supplemented with the saturation effects of nuclear matter density. Our results show a different temperature dependence for surface thickness $b$ of the excited compound nucleus, which is formed during the fusion process. The original version of Proximity 77 with this new form of $b(T)$ provides a reasonable result for theoretical values of the barrier characteristics and fusion cross section. Our obtained results are also compared with other theoretical approaches.

The Influence of the Dependence of Surface Energy Coefficient to Temperature in the Proximity Model

We introduce a new surface energy coefficient in proximity formalism, which is dependent on temperature, and apply it to a systematic study of barrier height and position. This proximity model can effectively predict the barrier heights and positions, as well as the fusion cross sections, over a wide range of incident energies, especially in light-heavy nuclei interaction.

Systematic study of the isotopic dependence of fusion dynamics for neutron- and proton-rich nuclei using a proximity formalism

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 well as nuclear $V_N$ and Coulomb $V_C$ potentials (at $R=R_B$) as a function of ($N/Z-1$) quantity are non-linear (second-order) whereas the fusion cross sections follow a linear-dependence.

FUSION CROSS-SECTIONS FOR 35Cl+92Zr REACTION CALCULATED IN PROXIMITY AND DOUBLE-FOLDING MODELS

In this paper, the proximity formalism which has been recently generalized to calculate the nuclear potential in the fusion reaction of deformed nuclei, employed to study the 35 Cl +92 Zr system. We also employ different versions of surface energy coefficients and discuss the effect of surface energy coefficients in the proximity potential. Comparison between results of double-folding (DF) and proximity model (PM) are made.

Probing sub-barrier fusion and extra-push by measuring fermium evaporation residues in different heavy ion reactions

The production of fermium isotopes was attempted by complete fusion of different targets and projectiles spanning a wide range of effective entrance channel fissilities below and above the predicted threshold valuex eff thr ≃0.7. For the most asymmetric systems where fusion is expected to occur without dynamical hindrance we investigate to what extent the expected amount of sub-barrier fusion contributes to the production of fermium evaporation residues. For increasingly symmetric systems the experimental fusion barriers are found to exceed the fusion barriers predicted by the proximity formalism. The barrier heights are discussed in the framework of both the extra-push model and the surface friction model.

Ho165+Fe56reaction atElab=462MeV

The system $^{165}\mathrm{Ho}$ + $^{56}\mathrm{Fe}$ was studied at a bombarding energy of ${E}_{\mathrm{lab}}=462$ MeV. The experimental $Z$ distributions measured as a function of angle and energy are discussed. The integrated fragment $Z$ distribution of the light reaction fragment is fairly broad and centered close to the charge of the projectile. The kinetic energy distribution extends from the quasielastic region down to energies smaller than the Coulomb energy of touching spherical nuclei. The angular distribution is peaked forward of the quarter-point angle and rises at small angles, which is interpreted as due to orbiting. Correlations of observed fragment $Z$ distributions with energy loss give an indication of the progress of the reaction. Charge distributions show an average drift towards larger asymmetry and an increase in width with increasing energy damping. Results are presented taking account of the Coulomb barrier for two choices of the scission radius. With the aid of a phenomenological description, the initial angular momenta ${l}_{i}$ and interaction times are deduced. Interaction times are compared to other characteristic times for the reaction as a function of $l$. Results from a classical dynamical model using a proximity formalism and one-body transport are shown for comparison to experimental observables and deduced quantities. The influence of a neck degree of freedom on the calculated trajectories is studied in some detail.NUCLEAR REACTIONS $^{165}\mathrm{Ho}$($^{56}\mathrm{Fe}$,$x$), ${E}_{\mathrm{lab}}=462$ MeV; measured $\ensuremath{\sigma}(\ensuremath{\theta},E,Z)$; damped reaction; fusion-fusion; deduced angular momenta, interaction times; comparison to dynamical calculations.