EDWSP Model Research Articles

Role of nuclear structure degrees of freedom and energy dependent interaction potential in sub-barrier fusion dynamics

The role of the nuclear structure degrees of freedom associated with the colliding nuclei and the energy dependence in the nucleus–nucleus interaction potential in fusion dynamics of various projectiles (16O, 27Al and 37Cl) with the Ge-isotopes (74,76Ge) has been investigated. Fusion cross-sections obtained through energy dependent interaction potential (EDWSP model) and coupled channel theory show quite similar behavior in near and above barrier energy regions. In coupled channel view, the couplings to inelastic surface excitations associated with the fusing nuclei results in large fusion enhancement at sub-barrier energies. The consideration of spherical shape for targets (74,76Ge-isotopes) along with their relevant internal structure degrees of freedom in coupled channel approach is sufficient to get a good description to observed fusion enhancement of the studied reactions. In contrast, the energy dependent interaction potential modifies the barrier profile of interaction barrier and subsequently reduces the effective fusion barrier between the colliding systems, henceforth, predicts substantially large fusion enhancement at below barrier energies in comparison to the expectations of the one dimensional barrier penetration model. Furthermore, χ2-analysis and the ratio plots σThσExp for outcomes of the EDWSP model suggested that the model calculations provide a consistent fit to fusion data within 10%. For the chosen reactions, only at 6 fusion data points out of 75 fusion data points does the deviation exceed 5% while 69 fusion data points lie within 5% and hence, the EDWSP model reasonably explained the above barrier fusion data within 5% with a probability more than 90%.

Fusion dynamics of stable and weakly bound colliding systems

This work systematically analyzed the fusion dynamics of the projectile-target combinations involving stable and loosely bound systems within the view of the energy-dependent Woods–Saxon potential model (EDWSP model) and the coupled channel approach. The different projectiles are bombarded onto series of Sm-isotopes, which possess the dominance of the different kinds of the nuclear structure degrees of freedom and with the increase of the neutron richness, the Sm-isotopes gradually shift from spherical shape to a statically deformed shape. In the fusion of [Formula: see text] reaction, the impacts of vibrational degrees of freedom of the colliding nuclei are dominant while in the case of [Formula: see text] systems, the rotational states of the deformed target isotopes have a strong impression on the below-barrier fusion data. The heavier target isotopes ([Formula: see text] also exhibit the higher order deformation such as [Formula: see text], [Formula: see text]-deformation parameter in its ground state and couplings to such channels must be incorporated in theoretical calculations in order to achieve close agreement with the sub-barrier fusion data. However, in the case of the loosely bound systems, the projectile breakup channel significantly affects the fusion excitation functions in the domain of the Coulomb barrier. To ensure the role of the projectile breakup channel, the fusion of the different loosely bound projectiles ([Formula: see text] and [Formula: see text] with Sm-isotopes are investigated, wherein the above-barrier fusion data of these reactions are suppressed with reference to the coupled channel calculations. This hindrance is the result of the projectile breakup effects that occur as a consequence of the breakup of the projectile before reaching the fusion barrier due to its low binding energy. However, in the EDWSP model calculations the magnitude of the hindrance of the above-barrier fusion data of [Formula: see text] and [Formula: see text] reactions is reduced by a factor varying from 7% to 13% with respect to a value reported in the literature. In contrast to this, the sub-barrier fusion enhancement of [Formula: see text] and [Formula: see text] reactions is the result of the dominance of the nuclear structure degrees of freedom of the colliding systems.

Role of Barrier Modification and Nuclear Structure Effects in Sub-Barrier Fusion Dynamics of Various Heavy Ion Fusion Reactions

The role of barrier modifications and the relevant nuclear structure effects in the fusion of the $$ {}_8{}^{16}O+{}_{62}{}^{144,148,150,152,154}Sm $$ and $$ {}_3{}^{6,7}Li+{}_{62}{}^{152}Sm $$ systems is analyzed within the context of the energy-dependent Woods-Saxon potential model (EDWSP model) and the coupled channel model. For the $$ {}_8{}^{16}O+{}_{62}{}^{144,148,150,152,154}Sm $$ reactions, where the colliding pairs are stable against breakup, the collective excitations and/or static deformations are sufficient to account for the observed fusion enhancement. In contrast, the model calculations overpredict the complete fusion data at above - barrier energies for the $$ {}_3{}^{6,7}Li+{}_{62}{}^{152}Sm $$ systems, where the importance of projectile breakup effects has been pointed out. Due to the low threshold of the alpha-breakup channel, the weakly bound projectiles $$ \left({}_3{}^{6,7}Li\right) $$ break up into charged fragments before reaching the fusion barrier and consequently the complete fusion cross section is suppressed by 28% (25%) in the $$ {}_3{}^6Li+{}_{62}{}^{152}Sm\;\left({}_3{}^7Li+{}_{62}{}^{152}Sm\right) $$ reaction with respect to predictions of coupled channel calculations. However, the EDWSP model based calculations can minimize the suppression factor by as much as of 13% (8%) in the $$ {}_3{}^6Li+{}_{62}{}^{152}Sm\;\left({}_3{}^7Li+{}_{62}{}^{152}Sm\right) $$ reaction with reference to the predictions made by the coupled channel calculations. Therefore, the complete fusion data of the $$ {}_3{}^6Li+{}_{62}{}^{152}Sm\;\left({}_3{}^7Li+{}_{62}{}^{152}Sm\right) $$ reaction at above - barrier energies is reduced by 15% (17%) with respect to the expectations of the EDWSP model. The extracted suppression factors for the studied reactions are due to the modifications of the barrier profile as a consequence of the energy - dependence in nucleus-nucleus potential, and thus greater barrier modifications occur for more weakly bound system, which in turn, confirms the breakup of projectile in the incoming channel.

Complete and incomplete fusion dynamics of 6,7Li + 159Tb reactions near the Coulomb barrier

The complete fusion (CF) and incomplete fusion (ICF) cross-sections are estimated for 6, 7Li + 159Tb reactions using the energy-dependent Woods-Saxon potential model (EDWSP model) and dynamical cluster-decay model (DCM). The CF data of the 6Li + 159Tb(7Li + 159Tb) reaction at above barrier energies is suppressed with reference to expectations of the EDWSP model by 25% (20%) which is smaller than the reported data by ∼ 9% (6%). This suppression is correlated with the projectile breakup effect. The projectiles 6,7Li are loosely bound systems, which may break up into charged fragments prior to reaching the fusion barrier and subsequently one of the fragment is captured by the target leading to the suppression of fusion data at above barrier energies. The sum of CF and ICF, which is termed as total fusion cross-section (TF), removes the discrepancies between theoretical predictions and the above barrier complete fusion data and hence is adequately explained via the EDWSP model over a wide range of energy spread across the Coulomb barrier. In addition to fusion, the decay mechanism of 6Li + 159Tb reaction is studied within the framework of the dynamical cluster-decay model (DCM). The breakup of the projectile (6Li) in the entrance channel indicates the presence of ICF, which is investigated further using the collective clusterization approach of DCM. The present theoretical analysis suggests that a larger barrier modification is needed to address the fusion data of chosen reactions in the below barrier energy region.

Sub-barrier fusion excitation function data and energy dependent Woods–Saxon potential

This paper analyzed the role of intrinsic degrees of freedom of colliding nuclei in the enhancement of sub-barrier fusion cross-section data of various heavy ion fusion reactions. The influences of inelastic surface vibrations of colliding pairs are found to be dominant and their couplings result in the significantly larger fusion enhancement over the predictions of the one dimensional barrier penetration model at sub-barrier energies. The theoretical calculations are performed by using energy dependent Woods–Saxon potential model (EDWSP model) in conjunction with the one dimensional Wong formula. The effects of dominant intrinsic channels are entertained within framework of the coupled channel calculations obtained by using the code CCFULL. It is quite interesting to note that the energy dependence in Woods–Saxon potential simulates the effects of inelastic surface vibrational states of reactants wherein significantly larger value of diffuseness parameter ranging from a = 0.85fm to a = 0.95fm is required to address the observed fusion excitation function data of the various heavy ion fusion reactions.

Fusion suppression/fusion enhancement around the Coulomb barrier

This work explored the fusion dynamics of 36Li+2759Co, 49Be+50124Sn, 49Be+82208Pb, 2040Ca+50124Sn and 612C+82208Pb reactions by using energy dependent Woods–Saxon potential model (EDWSP model) and coupled channel model. The roles of projectile breakup channel on fusion process are directly reflected from fusion mechanism of 36Li+2759Co, 49Be+50124Sn and 49Be+82208Pb reactions. Whereas, the impacts of collective excitations and/or nucleon transfer channels on fusion process are evident from analysis of 2040Ca+50124Sn and 612C+82208Pb reactions. The total fusion data (sum of complete and incomplete fusion data) of 36Li+2759Co reaction is reasonably recovered by the EDWSP model calculations which in turn indicate that total fusion excitation function data is not suppressed with reference to theoretical predictions. In contrast, for 49Be+50124Sn (49Be+82208Pb) reaction, the above barrier fusion data is suppressed with respect to the expectations of coupled channel approach and single barrier penetration model by 28% (32%). However, the EDWSP model based calculations reduce the magnitude of suppression factor by 8% (12%) for 49Be+50124Sn (49Be+82208Pb) reaction and consequently the fusion data at above barrier energies is inhibited for both reactions by 20% with reference to the EDWSP model calculations. This fusion suppression can be correlated with loosely bound nature of the projectile. On the other hand, the above barrier fusion data of 2040Ca+50124Sn and 612C+82208Pb reactions is not inhibited with respect to the predictions of coupled channel approach and EDWSP method, henceforth, confirms stability of projectiles against breakup effects.

Role of projectile breakup effects and intrinsic degrees of freedom on fusion dynamics**Supported by Dr. D. S. Kothari Post-Doctoral Fellowship Scheme sponsored by University Grants Commission (UGC), New Delhi, India

This article analyzes the fusion dynamics of loosely bound and stable projectiles with Zr-target isotopes within the context of the coupled channel approach and the energy-dependent Woods-Saxon potential model (EDWSP model). In the case of the 28Si + 90Zr reaction, the coupling to the inelastic surface excitations results in an adequate description of the observed fusion dynamics while in case of the 28Si + 94Zr reaction, the coupling to collective surface vibrational states as well as the neutron (multi-neutron) transfer channel is necessary in the coupled channel calculations to reproduce the below-barrier fusion data. However, the EDWSP model calculation provides an accurate explanation of the fusion data of 28Si + 90,94Zr reactions in the domain of the Coulomb barrier. In the fusion of the 6Li + 90Zr reaction, the inclusion of the nuclear structure degrees of freedom recovers the observed sub-barrier fusion enhancement but results in suppression of the above barrier fusion data by 34% with respect to the coupled channel calculations. Using EDWSP model calculations, this suppression factor is reduced by 14% and consequently, the above-barrier fusion data of 6Li + 90Zr reaction is suppressed by 20% with reference to the EDWSP model calculations. Such fusion suppression at above-barrier energies can be correlated with the breakup of the projectile (6Li) before reaching the fusion barrier, as a consequence of low binding energy.

Absence of isotopic dependence of sub-barrier fusion enhancement in the formation of Sn-isotopes via different entrance channels

The roles of the intrinsic degrees of freedom of the colliding pairs, the entrance channel mass asymmetry effects and variation of the N/Z ratio in the formation of the Sn-isotopes via different channels are analyzed within the view of coupled channel approach and the energy dependent Woods-Saxon potential model (EDWSP model). In the fusion of T2248i + N2858,60,64i, T2246i + N2864i, T2250i + N2860i, and F919 + N4193b reactions, which result in the formation of different Sn-isotopes, the effects of inelastic surface excitations and multi-nucleon transfer channels are found to be dominant while the entrance channel mass asymmetry and variation of the N/Z ratio with neutron richness show a weak influence on the sub-barrier fusion dynamics. With the increase of the N/Z ratio, the strong isotopic sub-barrier fusion enhancement is expected for neutron rich target isotope. But such effects are found to be absent in the fusion dynamics of these reactions and consequently display a weak isotopic fusion enhancement at below barrier energies. Both coupled channel approach and the EDWSP model reasonably describe the observed fusion dynamics of chosen reactions. In addition, the EDWSP model based calculations are capable of providing an agreement with the experimental fusion cross-section data within 10%. For six heavy ion fusion reactions, only at 7 fusion data points out of 54 fusion data points does the deviation exceed 5% while 47 fusion data points lie within 5%. Therefore, within the view of the EDWSP model, it is possible to recover the above barrier portion of the fusion cross-section data within 5% with a probability greater than 87%.

Role of Barrier Modification and Inelastic Surface Excitations in Sub-Barrier Fusion of 16 32 S + 40 94 Zr Reaction

The fusion dynamics of 16 S+40 Zr reaction at near and sub-barrier energies is investigated within the context of different theoretical approaches. The various theoretical models like one-dimensional Wong formula, l-summed ex- tended Wong formula, the energy-dependent Woods-Saxon potential model (EDWSP model), and coupled channel for- mulation have been used to address the impacts of nuclear structure degrees of freedom of the colliding pairs. The roles of different Skyrme forces along with Wong formalism are also tested in the analysis of the sub-barrier fusion dynamics of the 16 S+40 Zr reaction. The influence of the low-lying sur- face vibrational states of the collision partners is investigated within the framework of coupled channel calculations per- formed by the code CCFULL. In the present work, it has been observed that the EDWSP model introduces barrier modifica- tion effects somewhat similar to those of the coupled channel approach, as well as those of using different Skyrme forces and hence it reasonably addresses the observed fusion data of 16 S+40 Zr reaction in the close vicinity of the Coulomb barrier.

Role of Projectile Degrees of Freedom in Sub-Barrier Fusion Dynamics

This work theoretically investigates the role of the projectile degrees of freedom on the fusion dynamics of various heavy-ion fusion reactions. The impact of the projectile breakup channel is studied for the fusion mechanism of the 4 9 Be + 39 89 Y, 6 12 C + 39 89 Y, and 16 32,34 S + 39 89 Y reactions within the view of the coupled channel approach and the energy-dependent Woods-Saxon potential model (EDWSP model). The above-barrier fusion cross-section data of the 4 9 Be + 39 89 Y reaction is suppressed by about 20 % with respect to the theoretical predictions of the coupled channel approach and the single barrier penetration model while this suppression factor is reduced to 10 % within the context of the EDWSP model calculations. Such fusion hindrance at above-barrier energies can be understood in terms of the projectile breakup effects that arise due to its low breakup threshold. However, the observed fusion enhancement of the 6 12 C + 39 89 Y and 16 32,34 S + 39 89 Y reactions, wherein the colliding pairs are stable against breakup, is adequately explained by the EDWSP model and the coupled channel approach in the whole range of energy around the Coulomb barrier. This reveals that the energy dependence in the nucleus-nucleus potential governs barrier modification effects (barrier height, barrier position, barrier curvature) in closely similar way as reflected from the coupled channel formulation.

Entrance Channel Mass Asymmetry Effects in Sub-Barrier Fusion Dynamics by Using Energy Dependent Woods–Saxon Potential**Supported by Dr. D.S. Kothari Post-Doctoral Fellowship Scheme sponsored by University Grants Commission (UGC), New Delhi, India

The present article highlights the inconsistency of static Woods–Saxon potential and the applicability of energy dependent Woods–Saxon potential to explore the fusion dynamics of , and reactions leading to formation of different Sn-isotopes via different entrance channels. Theoretical calculations based upon one-dimensional Wong formula obtained by using static Woods–Saxon potential unable to provide proper explanation for sub-barrier fusion enhancement of these projectile-target combinations. However, the predictions of one-dimensional Wong formula based upon energy dependent Woods–Saxon potential model (EDWSP model) accurately describe the observed fusion dynamics of these systems wherein the significantly larger value of diffuseness parameter ranging from a = 0.85 fm to a = 0.97 fm is required to address the experimental data in whole range of energy. Therefore, the energy dependence in nucleus-nucleus potential simulates the influence of the nuclear structure degrees of freedom of the colliding pairs.

Analysis of fusion dynamics of colliding systems involving stable, loosely bound and halo nuclei

This article analyzed the fusion dynamics of stable nuclei, loosely bound nuclei and halo nuclei within the view of coupled channel approach and the energy dependent Woods–Saxon potential model (EDWSP model). The different projectiles ( and are bombarded onto common spherical target isotope ( wherein target isotope exhibits inelastic surface excitations as dominant mode of coupling. The fusion dynamics of reactions, wherein there is no signature of fusion suppression at above barrier energies, are adequately explained by both theoretical approaches in domain of Coulomb barrier. In case of reactions, the couplings of inelastic surface excitations of colliding systems remove the discrepancies between below barrier fusion data and expectations of one-dimensional barrier penetration model. However, at above barrier energies, the fusion excitation function data are suppressed in comparison to the theoretical predictions of both approaches. Interestingly, this suppression of above barrier fusion data is minimized by the present model calculation by a factor of 7% with respect to a value reported in literature. Within the view of the EDWSP model calculations, the above barrier fusion data of reaction are hindered by a factor of 25% and by a factor of 17% for reaction which is smaller than a value as pointed out in literature. The observed sub-barrier fusion dynamics of and reactions are reasonably described by the coupled channel model and the EDWSP model. Furthermore, the applicability of the EDWSP model has been tested for exploration of the fusion of reaction and hence due to barrier modification effects, the EDWSP model successfully explains the observed fusion enhancement of reaction. The present calculations clearly indicate that different theoretical approaches (coupled channel formulation and EDWSP model) induce similar kinds of barrier modification effects in the heavy ion fusion reactions.

Static versus energy-dependent nucleus-nucleus potential for description of sub-barrier fusion dynamics of reactions**Supported by Dr. D. S. Kothari Post-Doctoral Fellowship Scheme sponsored by University Grants Commission (UGC), New Delhi, India

The static and energy-dependent nucleus-nucleus potentials are simultaneously used along with the Wong formula for exploration of fusion dynamics of reactions. The role of internal structure degrees of freedom of colliding pairs, such as inelastic surface vibrations, are examined within the context of coupled channel calculations performed using the code CCFULL. Theoretical calculations based on the static Woods-Saxon potential along with the one-dimensional Wong formula fail to address the fusion data of reactions. Such discrepancies can be removed if one uses couplings to internal structure degrees of freedom of colliding nuclei. However, the energy-dependent Woods-Saxon potential model (EDWSP model) accurately describes the sub-barrier fusion enhancement of reactions. Therefore, in sub-barrier fusion dynamics, energy dependence in the nucleus-nucleus potential governs barrier modification effects in a closely similar way to that of the coupled channel approach.

Effects of intrinsic degrees of freedom in enhancement of sub-barrier fusion excitation function data and energy-dependent one-dimensional barrier penetration model

We have analyzed the role of barrier modification effects (barrier height, barrier position, barrier curvature) introduced due to the energy-dependent Woods–Saxon potential model (EDWSP model) and the coupled channel model on the sub-barrier fusion dynamics of $$ {}_{16}^{32,36} {\text{S}} + {}_{40}^{90,96} {\text{Zr}} $$ reactions. The influence of inelastic surface excitations of colliding pairs and multi-neutron transfer channels is found to be a dominant mode of couplings. The coupling of relative motion of colliding nuclei to these dominant intrinsic degrees of freedom leads to a substantially large fusion enhancement at below-barrier energies over the expectations of one-dimensional barrier penetration model. The coupled channel calculations based upon static Woods–Saxon potential must include the internal nuclear structure degrees of freedom of colliding nuclei for complete description of experimental data. On the other hand, theoretical calculations based upon the EDWSP model along with Wong formula provide a complete description of sub-barrier fusion enhancement of various heavy-ion fusion reactions. In EDWSP model calculations, significantly larger values of diffuseness parameter ranging from a = 0.98 fm to a = 0.85 fm are required to address the observed sub-barrier fusion enhancement of $$ {}_{16}^{32,36} {\text{S}} + {}_{40}^{90,96} {\text{Zr}} $$ reactions. Furthermore, within the context of EDWSP model, it is possible to achieve an agreement with the experimental fusion cross-sectional data within 10 %. For four heavy-ion fusion reactions, only at 4 fusion data points out of 90 fusion data points deviates exceeding 5 %, while 86 fusion data points lie within 5 % and hence the EDWSP model is able to account the above-barrier portion of the fusion cross-sectional data within 5 % with a probability greater than 90 %.

Isotopic dependence of fusion enhancement of various heavy ion systems using energy dependent Woods–Saxon potential

In the present work, the fusion of symmetric and asymmetric projectile–target combinations are deeply analyzed within the framework of energy dependent Woods–Saxon potential model (EDWSP model) in conjunction with one dimensional Wong formula and the coupled channel code CCFULL. The neutron transfer channels and the inelastic surface excitations of collision partners are dominating mode of couplings and the coupling of relative motion of colliding nuclei to such relevant internal degrees of freedom produces a significant fusion enhancement at sub-barrier energies. It is quite interesting that the effects of dominant intrinsic degrees of freedom such as multi-phonon vibrational states, neutron transfer channels and proton transfer channels can be simulated by introducing the energy dependence in the nucleus–nucleus potential (EDWSP model). In the EDWSP model calculations, a wide range of diffuseness parameter ranging from a = 0.85 fm to a = 0.97 fm , which is much larger than a value ( a = 0.65 fm ) extracted from the elastic scattering data, is needed to reproduce sub-barrier fusion data. However, such diffuseness anomaly, which might be an artifact of some dynamical effects, has been resolved by trajectory fluctuation dissipation (TFD) model wherein the resulting nucleus–nucleus potential possesses normal diffuseness parameter.