Entrance Channel Mass Asymmetry Effects in Sub-Barrier Fusion Dynamics by Using Energy

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1 Home Search Collections Journals About Contact us My IOPscience Entrance Channel Mass Asymmetry Effects in Sub-Barrier Fusion Dynamics by Using Energy Dependent Woods Saxon Potential This content has been downloaded from IOPscience. Please scroll down to see the full text Commun. Theor. Phys ( View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: This content was downloaded on 08/01/2016 at 05:43 Please note that terms and conditions apply.

2 Commun. Theor. Phys. 64 (2015) Vol. 64, No. 6, December 1, 2015 Entrance Channel Mass Asymmetry Effects in Sub-Barrier Fusion Dynamics by Using Energy Dependent Woods Saxon Potential Manjeet Singh Gautam School of Physics and Material Science, Thapar University, Patiala (Punjab) , India (Received January 6, 2015; revised manuscript received July 28, 2015) Abstract 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 22Ti+ 58,60, Ni, 22Ti Ni, 50 22Ti Ni, and 9 F Nb 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 onedimensional 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. PACS numbers: Pj, Ev, Eq Key words: depth and diffuseness of Woods Saxon potential, heavy-ion sub-barrier fusion reactions, coupled channel equations, diffuseness anomaly 1 Introduction A lot of information about the nuclear structure and nuclear interaction between collision particles has been collected from the study of sub-barrier fusion reactions. In this regard, many theoretical and experimental investigations have been done in the last few decades but many questions in this field remain unanswered and hence the dynamics of sub-barrier fusion reactions is still far from good understanding. Fusion process, which occurs as a consequence of barrier penetration effects, involves interactions between the colliding nuclei in the presence of nuclear structure degrees of freedom and the collision partners come close together with a sufficient kinetic energy resulting in the formation of a compound nucleus either by overcoming or by a quantum mechanical tunneling through the classical forbidden region. It is well recognized fact that the heavy ion fusion reactions at energies near and below the Coulomb barrier are strongly affected by the couplings to intrinsic structure degrees of freedom of fusing nuclei and hence the large enhancement of sub-barrier fusion excitation functions with respect to the predictions of one-dimensional barrier penetration model has been observed in many projectile-target combinations. [1 4] The experimental and theoretical investigations have recommended that such fusion enhancement can be understood in terms of coupling of relative motion to internal structure degrees of freedom of reactants such as permanent deformation (deformed nuclei), vibration of nuclear surface (spherical nuclei), rotations, neck formation and (nucleon) multi-nucleon transfer reactions. A correlation between permanent deformation and inelastic surface vibrations of colliding pairs with subbarrier fusion enhancement has been well established by many theoretical and experimental studies. [1 7] However, the impacts of nucleon (multi-nucleon) transfer channel on sub-barrier fusion enhancement are not fully explored and trigger further investigations to understand the interplay of multi-nucleon transfer channel in fusion dynamics. In transfer mechanism, a complex rearrangement of nucleons between the colliding pairs occurs in the neck region and consequently it is very diffcult to handle nucleon (multi-nucleon) transfer channels in theoretical description. Although, some semi-classical approaches have been successfully used to single out the roles of neutron transfer on fusion process but the understanding of basic mechanism involved in transfer process is challenging and still remain an open question. [8 13] The nucleus-nucleus potential is a key for understanding of the different aspects of reaction dynamics and for complete understanding of various aspects of nuclear interactions between heavy ions; the knowledge of accurate picture of nucleus-nucleus potential is essentially required. The nuclear phenomena like elastic scattering and inelastic scattering are quite sensitive to the surface region of nuclear potential whereas the fusion reactions strongly depend upon the inner part of nuclear potential. [14] The Supported by Dr. D.S. Kothari Post-Doctoral Fellowship Scheme sponsored by University Grants Commission (UGC), New Delhi, India gautammanjeet@gmail.com c 2015 Chinese Physical Society and IOP Publishing Ltd

3 No. 6 Communications in Theoretical Physics 711 Coulomb and centrifugal terms are well understood because they consist of simple expression. However, due to large ambiguities in the optimum choice of nuclear potential, the correct picture of different nuclear phenomena is still unexplored. In this connection, large set of parameterizations of nuclear potential are widely used to clarify the nature of nuclear interactions and the role of nuclear structure of interacting nuclei in compound nucleus formation reactions. [15 23] The standard energy independent Woods Saxon potential is one of them which has been generally used to preview the nature of heavy ion collisions. The different coupled channel codes such as CCFUS, [24] CCDEF, [25] CCMOD, [26] FRESCO [27] and CCFULL [28] has been employed the static Woods Saxon potential to entertain influences of nuclear structure of reacting nuclei on the fusion dynamics. In literature, it has been well populated that the significantly larger values of diffuseness parameter ranging from a = 0.75 fm to a = 1.5 fm are essentially required to address the sub-barrier fusion data. It is quite interesting to note that a much smaller value of diffuseness parameter (a = 0.65 fm) is generally used for description of elastic scattering data. The cause of this diffuseness anomaly is still far from clear understanding and its solution requires more intensive studies on theoretical as well as on the experimental front. The diffuseness anomaly, which might be an artifact of various kinds of static and dynamical physical effects, reflects the inconsistency of static Woods Saxon potential for simultaneous reconcile of the elastic scattering data and fusion data. [29 33] In order to resolve diffuseness anomaly, the previous work undertook several attempts by using energy dependent nucleus-nucleus potential [34 45] wherein it introduces similar kinds of static and dynamical physical effects that are reflected from the channel coupling effects. The energy dependence in the real Woods Saxon potential is introduced in such a way that it becomes much more attractive in the domain of the Coulomb barrier and consequently predicts larger sub-barrier fusion cross-section with reference to energy independent one-dimensional barrier penetration model. Very recently, the energy dependent Woods Saxon potential model (EDWSP model) was successfully used to entertain the influences of nuclear structure degrees of freedom of the colliding pairs like inelastic surface vibrations and multi-neutron transfer channels in the fusion dynamics of various heavy ion reactions. This work is motivated to track the limitations of static Woods Saxon potential and applicability of energy dependent Woods Saxon potential by analyzing the fusion dynamics of 22Ti + 58,60, Ni, 28Ni, 50 28Ni, and 41 Nb reactions.[12 13] In addition, the effects of entrance channel mass asymmetry effects, which is expected to have strong influence on the magnitude of sub-barrier fusion enhancement, leading to different Sn-isotopes are also evident from the present paper. It is expected that the magnitude of enhancement of sub-barrier fusion crosssection data increases with increase of entrance channel mass asymmetry. The role of low lying inelastic surface excitations of colliding pairs are included in the coupled channel calculations, which are obtained by using coupled channel code CCFULL. [28] The EDWSP model along with one-dimensional Wong formula [46] accurately reproduces the fusion dynamics of all fusing systems under consideration and hence directly reveals the limitations of static Woods Saxon potential for analysis of sub-barrier fusion dynamics. It is quite interesting to note that the EDWSP model has an effect that is closely similar to that of static Woods Saxon potential with abnormally large diffuseness parameter and consequently with that of M3Y+ repulsion potential. Ghodsi et al. [47] have shown that the static Woods-Saxon potential with abnormally large diffuseness parameter and M3Y+ repulsion potential predict closely similar behavior of heavy ion fusion dynamics. This conclusion was further supported by the work of Esbensen et al. [ 49] and Stefanini et al. [10,50] The details of the theoretical formalism adopted for this article are discussed in Sec. 2. The results of theoretical calculation are discussed in detail in Sec. 3 while the summary of the work is presented in Sec Theoretical Formalism 2.1 One-Dimensional Wong Formula The partial wave fusion cross-section is defined by the following expression σ F = π k 2 (2l + 1)Tl F. (1) l=0 Based upon parabolic approximation of the effective interaction potential between heavy ions, Hill and Wheeler proposed an expression for tunneling probability (Tl F) which is given by the following expression. [51] Tl HW 1 = 1 + exp[(2π/ ω l )(V l E )]. (2) This parabolic approximation was further simplified by Wong using the following assumptions for barrier position, barrier curvature and barrier height. [46] R l = R l=0 = R B, ω l = ω l=0 = ω, [ V l = V B + 2 2µRB 2 l + 2] 1 2. Using these assumptions and Eq. (2) into Eq. (1), the fusion cross-section can be written as σ F = π (2l + 1) k 2 [1 + exp(2π/ ω)(v l E)]. (3) l=0 Wong assumes that the infinite number of partial waves contribute to the fusion process, therefore one can change the summation over l into integral with respect to l in

4 712 Communications in Theoretical Physics Vol. 64 Eq. (3) and by solving the integral one can get the following expression of Wong formula. [46] σ F = ω [ ( R2 B 2π ) ] 2E ln 1 + exp ω (E V B). (4) 2.2 Energy Dependent Woods Saxon Potential Model (EDWSP Model) Very recently, the EDWSP model was successfully used to explore the dynamics of sub-barrier fusion crosssection. In theoretical calculations, the EDWSP model is used along with one-dimensional Wong formula [34 45] V 0 = [ A 2/3 P + A2/3 where I P = (N P Z P )/A P and I T = (N T Z T )/A T are the isospin asymmetry of colliding systems. This parameterization of potential depth has been deduced by reproducing the fusion excitation function data of large number of fusion reactions ranging from Z P Z T = 84 to Z P Z T = [34 45] In heavy ion collisions, the various static and dynamical physical effects occur in the surface region of colliding nuclei or in the tail region of nuclear potential, which in turn modify the parameters of nuclear potential. In the present model, the fluctuation in the surface energy of colliding nuclei that strongly depends upon the collective motion of all the nucleons inside the nucleus and variation of densities of collision partners in neck region are accommodated through the first term in a(e) = 0.85 while static Woods Saxon potential has been employed in the coupled channel calculations. Therefore, the shape of static Woods Saxon potential is defined as V N (r) = T (A P + A T ) 2/3] [ (1 + I P + I T ) A1/3 [ V exp((r R 0 )/a), (5) with R 0 = r 0 (A 1/3 P + A 1/3 T ). The quantity V 0 is depth and a is diffuseness parameter of Woods Saxon potential. In EDWSP model, the depth of real part of Woods Saxon potential is given by the following expression P A1/3 T A 1/3 P + A1/3 T ] MeV, (6) the square bracket of Eq. (6). Such static and dynamical physical effects bring the requirement of abnormally large diffuseness parameter ranging from a = 0.75 fm to a = 1.5 fm for accounting the fusion excitation function data. The second term inside the square bracket of Eq. (6) is directly proportional to isospin asymmetry effects of colliding nuclei, which is different for different isotopes of a particular element. The isotopic effects of reactants entered in the nucleus-nucleus potential through this term. In the essence of the importance of variation of diffuseness parameter of Woods Saxon potential, the energy dependence in Woods Saxon potential is taken via its diffuseness parameter. Therefore, the energy dependent diffuseness parameter is defined as (A 1/3 P + A 1/3 T )(1 + exp( E/VB )) r 0 ] fm. (7) In fusion dynamics, the variation of surface energy, N/Z ratio, variation of densities in neck region, dissipation of kinetic energy of relative motion to internal structure of collision partners or other static and dynamical physical effects, which brings the modification in the value of diffuseness parameter of static Woods Saxon potential, are accurately accommodated in the EDWSP model. In EDWSP model calculations; such kinds of static and dynamical physical effects are entering through energy dependent diffuseness parameter (see Eq. (7)). In heavy ion fusion reactions, the EDWSP model provides a wide range of diffuseness parameter depending upon the value of r 0 and bombarding energy of reactants. The free parameter r 0 is adjusted in order to vary the value of diffuseness parameter required to address the observed fusion dynamics of fusing system under consideration. To obtain this expression, the fusion excitation function data have been reproduced for wide range of projectile-target combinations by varying the diffuseness of standard Woods Saxon potential. The different values of diffuseness are needed to fit the data in different energy regions and by using the sigmoidal fitting; the above expression has been deduced. [34 45] It will be shown later that the theoretical calculations based upon static Woods Saxon potential (CCFULL calculations) must incorporate the couplings to internal nuclear structure degrees of freedom such as inelastic surface excitations of colliding pairs, rotational states of deformed nuclei and multi-nucleon transfer channels or other static and dynamical physical effects to reproduce the sub-barrier fusion data. However, the energy dependence in Woods Saxon potential produces similar kinds of channel coupling effects that arise due to coupling of relative motion of reactants to their intrinsic degrees of freedom and consequently adequately explain the observed fusion dynamics of various heavy ion fusion reactions. The underlying reason of the equivalency of these two different physical mechanisms (coupled channel approach and the EDWSP model) is that in principle these approaches produce similar kinds of barrier modification effects (barrier height, barrier position, barrier curvature) in heavy ion fusion dynamics.

5 No. 6 Communications in Theoretical Physics Coupled Channel Model In this section, the details of coupled channel model are presented. The influences of internal structure degrees of freedom of colliding nuclei are entertained within the context of the coupled channel calculations. Although, it is very difficult to include the impacts of all intrinsic degrees of freedom of collision partners but it is fruitful to include the relevant channels. [28,52 53] Therefore, the set of coupled channel equation can be written as [ 2 d 2 J (J + 1) 2 + 2µ dr2 2µ r 2 + V N (r) + Z PZ T e 2 r + ε n E cm ]ψ n (r) + V nm (r) ψ m (r) = 0, (8) m where, r is the radial coordinate for the relative motion between fusing nuclei. µ is defined as the reduced mass of the projectile-target system. The quantities E cm and ε n represent the bombarding energy in the centre of mass frame and the excitation energy of the n th channel respectively. The V nm is the matrix elements of the coupling Hamiltonian, which in the collective model consists of Coulomb and nuclear components. For coupled channel calculations, one can use the code CCFULL [28] wherein the coupled channel equations are numerically solved by using the certain basic approximations. The no-coriolis or rotating frame approximation has been used to reduce the number of the coupled channel equations. [28,52 53] If there is no transfer of the angular momentum from the relative motion of colliding nuclei to their intrinsic motion, the total orbital angular momentum quantum number L can be replaced by the total angular momentum quantum number J. The ingoing wave boundary conditions (IWBC), which are well applicable for the heavy ion reactions, are used to obtain the numerical solution of the coupled channel equations. According to IWBC, there are only incoming waves at the minimum position of the Coulomb pocket inside the barrier and there are only outgoing waves at infinity for all channels except the entrance channel. [28] The code CCFULL [28] makes the use of static Woods Saxon potential (see Eq. (5)) to account the effects of nuclear structure degrees of freedom. Thus, by including all the relevant channels, the fusion cross-section becomes σ F (E) = σ J (E) = π k 2 (2J + 1)P J (E), (9) J 0 J where, P J (E) is the total transmission coefficient corresponding to the angular momentum J. 3 Results and Discussion The present work analyzed the limitations of static Woods Saxon potential in conjunction with onedimensional Wong formula by considering the fusion of 22 58,60, Ti + 28Ni, 22Ti + 28Ni, 22Ti + 28 Ni, and 9 F Nb reactions. [12 13] In addition, effects of entrance channel mass asymmetry of colliding nuclei are directly reflected from the fusion dynamics of these systems. Theoretical calculations have been performed by using static Woods Saxon potential and the energy dependent Woods Saxon potential along with one-dimensional Wong formula. The effects of low lying surface vibrational states are included in the coupled channel calculations which are performed by using the code CCFULL. [28] The values of the deformation parameters and corresponding excitation energies of low lying 2 + and 3 vibrational states as used in coupled channel calculations for all fusing nuclei are listed in Table1. The barrier height corresponding to different projectile-target system is used in the energy dependent Woods Saxon potential model while the other parameters like barrier position and barrier curvature as listed in Table 2 are used in the Wong formula for evaluation of the fusion excitation functions. The potential parameters such as range, depth and diffuseness parameters as used in the EDWSP model calculations for various combinations of colliding nuclei are listed in Table 3. Table 1 The deformation parameter (β λ ) and the energy (E λ ) of the quadrupole and octupole vibrational states of colliding nuclei. Nucleus β 2 E 2 /MeV β 3 E 3 /MeV Reference 9 F [13] 46 22Ti [13] 22Ti [12] 50 22Ti [13] 58 28Ni [12] 60 28Ni [13] 64 28Ni [13] 93 41Nb [13] The limitations of static Woods Saxon potential and the validity of energy dependent Woods-Saxon potential on fusion process have been discussed for the fusion dynamics of Ni, 22Ti + 28Ni, 22Ti + 28Ni, and 41 Nb reactions.[12 13] Theoretical calculations have been performed by using static Woods Saxon potential and the energy dependent Woods Saxon potential model (EDWSP model) in conjunction with one-dimensional Wong formula. For these fusing systems, theoretical calculations based upon static Woods Saxon potential along with one-dimensional Wong formula systematically fail to account the observed experimental data. In above barrier energy regions, there is close agreement between theoretical predictions and fusion data while the deviations between theoretical calculations and fusion data increases as bombarding energy droops below the energy of the Coulomb barrier. However, energy dependent Woods Saxon potential model (EDWSP model) provides the complete description of experimental data of these reactions Ti + 58,60,64 28

6 714 Communications in Theoretical Physics Vol. 64 in whole range of energy. In EDWSP model, the energy dependent diffuseness parameter produces a spectrum of barriers of varying heights. The barriers whose heights are smaller than that of single Coulomb barrier are responsible for transmitting of maximum fraction of flux from elastic channel to fusion channel and hence can be manifested for a substantially large fusion cross-section at sub-barrier energies over the predictions of energy independent one-dimensional Wong formula as evident from Figs. 1 and 2. Furthermore, in EDWSP model calculations, at below barrier energies, a = 0.97 fm is the largest value of diffuseness parameter which results in the lowest fusion barrier and the passage through this lowest barrier is much more probable which in turn results in the large sub-barrier fusion enhancement. As incident energy increases, the value of diffuseness parameter decreases and the height of corresponding fusion barrier increases. The barriers whose heights are smaller than that of uncoupled nominal barrier produce substantially larger fusion enhancement at below barrier energies. At above barrier energies, wherein fusion cross-section is almost insensitive to the various channel coupling effects (internal structure of colliding nuclei), the value of diffuseness parameter gets saturated to its minimum value (a = 0.85 fm). In EDWSP model calculations, significantly large value of diffuseness parameter ranging from a = 0.85 fm to a = 0.97 fm is necessarily required to account the below barrier fusion data. This reflects that the energy dependence in Woods Saxon potential model (EDWSP model) has an effect that is closely similar to that of static Woods Saxon potential with large diffuseness. It is very important to note that in EDWSP model, the effects of variation of diffuseness parameter is effectively equivalent to increase of capture radii of colliding nuclei. Such increase of capture radii of fusing nuclei suggested that the fusion process starts at much larger inter-nuclear separation between the collision partners. [34 45] Analogous static and dynamical physical effects that are produced due to coupling between elastic channel and internal degrees of freedom of reacting nuclei are evident from the energy dependent nucleus-nucleus potential. Therefore, the coupled channel calculations and the EDWSP model calculations mirror the similar behavior of heavy ion fusion reactions. For all fusing systems, no coupling calculations based on static Woods Saxon potential obtained by using the code CCFULL are substantially smaller than the fusion data (see Figs. 3 and 4) which clearly suggests that the coupling to internal degrees of freedom must be incorporated to explain sub-barrier fusion enhancement of 22 58,60,64 46 Ti + 28Ni, 22 systems. Ti Ni, Ti + 28 Ni, and 9 F Nb Table 2 The values of V B0, R B, and ω used in the EDWSP model [34 45] calculations for various heavy ion systems. System V B0 /MeV R B /fm ω/mev Reference 22Ti Ni [28] 28Ni [28] 28Ni [28] 46 28Ni [28] 50 28Ni [28] 41Nb [28] Table 3 Range, depth, diffuseness of Woods Saxon potential used in the EDWSP model calculations for various heavy ion systems. [34 45] System r 0 /fm V 0 /MeV a Present Energy Range 22Ti Ni to to Ni to to Ni to to Ni to to Ni to to Nb to to 60 ( ) fm MeV In general, when a common projectile is bombarded onto different isotopes of a given element, it results in the substantially larger fusion enhancement for heaviest isotope. Such fusion enhancement can be correlated with the increase of deformation and the existence of larger probability of neutron transfer channels. To single out the impacts of the nucleon (multi-nucleon) transfer channel from the influence of collective excitations in the sub-barrier fusion enhancement, the projectile-target combinations are chosen in such a way that the target isotope has a closed shell or closed sub-shell so that neutron transfer channels exists with positive Q-value. Furthermore, the strong collectivity and the possibility of neutron transfer channels are directly linked with the neutron richness of target isotope and thus couplings to multi-nucleon transfer channels is the only factor that results in the strong isotopic fusion enhancement at sub-barrier energies. In this sense, the fusion dynamics of 58,60,64 22Ti + 28Ni reactions attracts researchers to explore their fusion dynamics. [12] Due to increase of isotopic mass of target nuclei, a strong sub-barrier fusion enhancement is expected for heavier target isotope but authors claim a mild isotopic dependence of sub-barrier fusion enhancement of 22Ti+ 58,60,64 28 Ni

7 No. 6 Communications in Theoretical Physics 715 reactions rather than strong isotopic sub-barrier fusion enhancement. [12] To clarify these facts, the present article explores the fusion dynamics of 22Ti + 58,60,64 28Ni reactions within the view of the EDWSP model and the coupled channel formulation. In case of these reactions, the collective excitations of fusing nuclei play a crucial role in the sub-barrier fusion dynamics and hence the inclusion of such dominant channel results in the substantially large sub-barrier fusion enhancement over the predictions of one-dimensional barrier penetration model. The inclusion of inelastic surface vibrations such as one phonon 2 + and 3 vibrational states along with their mutual couplings in either of colliding pairs fails to account the below barrier experimental data in a reasonable way but the above barrier fusion data are adequately addressed by such coupling scheme. It is worth noting here that the difference between the couplings to one phonon 2 + and 3 vibrational states along with their mutual couplings in target and one phonon 2 + and 3 vibrational states along with their mutual couplings in projectile decreases as one shift from lighter target isotope ( 58 28Ni) to heavier target isotope ( 64 28Ni) as inferred from Fig. 3. This particular effect can be correlated with the nuclear structure effects of target isotopes. In Ni-isotopes, quadrupole deformation and the corresponding excitation energies varies as one moves from lighter target isotope ( 58 28Ni) to heavier target isotope ( Ni).[12] Therefore, the coupling strengths of qudrupole deformation as well as octupole deformation remain almost same while their corresponding excitation energy decreases and consequently results in larger sub-barrier fusion enhancement associated with the heavier target isotope. In addition, the entrance channel mass asymmetry of 22 Ti+58,60,64 28Ni systems increases weakly with increase of neutron richness of target isotopes. For instance, the entrance channel mass asymmetry (η) for 22 Ti+58 28Ni system is η = (A P A T )/(A P + A T ) = 0.094, for 60 22Ti + 28 Ni system is η = (A P A T )/(A P + A T ) = and for 64 22Ti + 28 Ni system is η = (A P A T )/(A P + A T ) = The weak target isotopic dependence of sub-barrier fusion enhancement of 22 Ti+58,60,64 28Ni systems can be understood in terms of inelastic surface excitations of colliding nuclei as well as the increasing entrance channel mass asymmetry effects. Fig. 1 The fusion excitation functions of 22Ti + 58,60,64 28Ni systems obtained by using static Woods Saxon potential model and the energy dependent Woods Saxon potential model (EDWSP model) along with Wong formula. [34 45] The results are compared with available experimental data (*) taken from Ref. [12]. Fig. 2 Same as Fig. 1 but for 46 28Ni (a), 50 28Ni (b) and 9F Nb (c) systems. The results are compared with available experimental data (*) taken from Ref. [13].

8 716 Communications in Theoretical Physics Vol. 64 Fig. 3 The fusion excitation function of 28Ni systems obtained by using the EDWSP model along with Wong formula [34 45] and the coupled channel code CCFULL. [28] The results are compared with the available experimental data (*) taken from Ref. [12]. 22Ti+ 58,60,64 Fig. 4 Same as Fig. 3 but for 46 22Ti Ni (a), 50 22Ti Ni (b) and 9F + 41Nb 93 (c) systems and the results are compared with the available experimental data (*) taken from Ref. [13]. Generally, in sub-barrier fusion dynamics, the inelastic surface excitations contribute dominantly but in some projectile-target combinations in addition to collective surface vibrations, the inclusion of nucleon transfer channels is necessarily required to achieve the close agreement between the theoretical calculations and the fusion data. The significance of the multi-nucleon transfer channels with the positive ground state Q-value is directly reflected from the analysis of fusion dynamics of Ti Ni, Ti + 28 Ni and 41Nb reactions. The deformation parameters and their corresponding excitation energies in the odd nuclei as used in coupled channel model are taken from the average values of neighboring even-even nuclei. [54 55] The coupled channel calculations based upon inelastic surface excitations of the colliding pairs like single phonon 2 + and 3 vibrational states along with their mutual couplings in either projectile or target produce substantially larger sub-barrier fusion enhancement in reference to the no coupling calculations but fails to recover the observed fusion dynamics and hence point towards the additions of more intrinsic channels in the coupled channel calculations. The coupling to one phonon 2 + and 3 vibrational states along with their mutual couplings in reactants quantitatively improves the results and bring theoretical calculations nearer to the experimental data but still unable to provide the complete description of the below barrier experimental data as inferred from Fig. 4. Such discrepancies between theoretical calculations and fusion data can be correlated with the presence of neutron transfer channel and entrance channel mass asymmetry effects. For fusion of 46 28Ni and 50 28Ni systems, the coupled channel calculations reasonably explain the sub-barrier fusion enhancement while the small difference between theoretical calculations and below barrier fusion data can be overcome by including the couplings to nucleon transfer channels. [13] It is worth noting here that the significantly larger sub-barrier fusion enhancement of Ti + 28Ni system relative to 22Ti + 28Ni system are due to the larger probability of two neutron pickup channel with positive ground state Q-value (for 46 28Ni system, Q-value is MeV while for 50 28Ni system, it is equal to 6.21 MeV). For 50 28Ni reaction, all the nucleon transfer channels have the negative Q-value except for two proton pickup channel ( Q = MeV for two proton pickup channels) which has negligible influence on the fusion process at below barrier energy. [13] For 41Nb system, the deviation of the coupled channel calculations and the sub-barrier fusion data is quite large and such additional fusion enhancement can be correlated with the possibility of the one proton pick up channel with

9 No. 6 Communications in Theoretical Physics 717 large positive ground state Q-values (Q = MeV). [13] The other underlying reason for large sub-barrier fusion enhancement is the entrance channel mass asymmetry effects which strongly influence the fusion dynamics of the colliding nuclei. Specifically, the size of the neck formed between the collisions partners that lead to the formation of the compound nucleus via different entrance channels strongly depends upon the entrance channel mass asymmetry. As discussed in Refs. [35 56], the fusion inhibition factor decreases with the increase in the entrance channel mass asymmetry and hence the large magnitude of fusion cross-section data was observed for larger mass asymmetric projectile-target combinations. Therefore, the magnitude of sub-barrier fusion enhancement increases with increase of entrance channel mass asymmetry. For instance, for Ti + 28Ni system is η = (A P A T )/(A P + A T ) = 0.090, for Ti + 28Ni system is η = (A P A T )/A P + A T = and for 41Nb system is η = (A P A T )/(A P + A T ) = Furthermore, the fusion of Ti + 28Ni system and 22Ti + 28 Ni system leads to the formation of same compound nucleus Sn. Similarly, the fusion of 22Ti + 28Ni system and 41Nb system leads to the formation of same compound nucleus Sn. All these projectile-target combinations results in the formation of different isotope of Snnucleus and hence the larger magnitude sub-barrier fusion enhancement is found for fusing system, which has larger entrance channel mass asymmetry. It is worth mention to here that the authors of Ref. [13], compared the experimental data in reduced scale and showed that besides smaller entrance channel mass asymmetry of 28Ni reaction, the sub-barrier fusion enhancement of 22 Ti Ni reaction is slighter larger than that of 9F Nb reaction. In literature, it is well established that inclusion of inelastic excitations of projectile (target) or permanent deformations of projectile (target) or nucleon (multinucleon) transfer results in a distribution of barriers of varying heights instead of single Coulomb barrier. This distribution of barriers is the mirror image of the type of coupling responsible for the sub-barrier fusion enhancement. On the other hand, the energy dependence in nucleus-nucleus potential also adequately explains the observed fusion dynamics of various heavy ion fusion reactions in closely similar way to that of coupled channel calculations. This clearly suggested that channel coupling effects and the EDWSP model introduce similar kinds of barrier modification effects (barrier height, barrier position, barrier curvature) in heavy ion fusion reactions. Ghodsi et al. [47] have shown that the M3Y+repulsion and static Woods-Saxon potential with large diffuseness parameter accurately reproduce the fusion dynamics of 12 6 C Zr, 16 8 O Zr, Si Zr, and 17Cl + 40 Zr systems. This clearly reveals that M3Y+repulsion and static Woods Saxon potential with large diffuseness parameter mirrors similar features of heavy ion fusion reactions and hence the effects of M3Y+repulsion potential can be accurately entertained by static Woods-Saxon potential with abnormally large diffuseness parameter ranging from a = 0.75 fm to a = 1.5 fm. The similarity between M3Y+repulsion potential and static Woods-Saxon potential with large diffuseness parameter is also supported by the work of Esbensen et al. [ 49] and Stefanini et al. [10,50] For heavy ion fusion reactions, the present work suggested that the EDWSP model has an effect that is closely similar to that of static Woods Saxon potential with large diffuseness parameter and consequently from M3Y+repulsion potential. Furthermore, the number of questions with regard the role of nuclear structure degrees of freedom of colliding pairs is still open. Whether the various static and dynamical physical effects that are induced due to inelastic surface vibrations, nucleon transfer channel and other static and dynamical physical effects represent a true picture of the relevant channels in the fusion enhancement or simply mirror the limitations of energy independent Woods Saxon potential parameters is still not clear. Furthermore, the energy dependence of Woods Saxon potential is true picture of the nuclear potential or mocks up other static and dynamical physical effects is still not clear and hence more intensive studies are required to answer all these questions. 4 Conclusions The present work addresses validity of static Woods Saxon potential and energy dependent Woods Saxon potential for exploration of fusion dynamics of 22 Ti + 58,60, Ni, 22Ti + 28Ni, 22Ti + 28 Ni, and 41Nb reactions. The theoretical calculations based upon the coupled channel model and the EDWSP model result in the closely similar behavior of the fusion dynamics of 28 Ni, 22Ti+ 58,60, Ni, 50 28Ni, and 9F Nb reactions. However, the predictions of static Woods Saxon potential in conjunction with one dimensional Wong formula fails to provide the complete description of fusion dynamics these fusion reactions while the EDWSP model calculation adequately reproduces the experimental data of these fusing systems in whole range of energy and consequently simulates the various channel coupling effects. Furthermore, the EDWSP model calculations require a significantly large values of diffuseness parameter ranging from a = 0.85 fm to a = 0.97 fm, which is much larger than a value deduced from the elastic scattering data (a = 0.65 fm). Therefore, the EDWSP model has an effect that is closely similar to that of static Woods Saxon potential with abnormally large diffuseness parameter ranging from a = 0.75 fm to a = 1.5 fm.

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