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

Transcription

1 Commun. Theor. Phys. 64 (2015) Vol. 64, No. 2, August 1, 2015 Role of Surface Energy Coefficients and Temperature in the Fusion Reactions Induced by Weakly Bound Projectiles R. Gharaei 1, and O.N. Ghodsi 2 1 Department of Physics, Sciences Faculty, Hakim Sabzevari University, P.O. Box 397, Sabzevar, Iran 2 Department of Physics, Sciences Faculty, University of Mazandaran, P.O. Box , Babolsar, Iran (Received January 23, 2015; revised manuscript received March 16, 2015) Abstract 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 6 Li, 7 Li and 9 Be. 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. PACS numbers: Pj, Jj, Eq Key words: weakly bound projectile, proximity formalism, surface energy coefficient, temperature effects 1 Introduction The fusion process of a weakly bound projectile with medium and heavy targets is one of the interesting subjects of nuclear physics which has attracted much attention in recent decades. [1 9] Since the breakup threshold energy of weakly bound nuclei is very low, the fusion reaction induced by such projectiles can be followed through two channels: one is complete fusion (CF) channel and the other is incomplete fusion (ICF) channel. Under these conditions, one can define the total fusion cross section as the sum of CF and ICF cross sections, namely σ TF = σ CF + σ ICF. Since measurement of the CF and ICF cross sections separately is quite laborious, the corresponding experimental data of these channels were reported for a limited number of fusion systems such as 9 Be+ 186 W, [10] 9 Be+ 144 Sm [11] and 9 Be+ 208 Pb. [12] From the theoretical viewpoint, various studies which are performed for fusion reactions with weakly bound projectiles can be generally divided into static and dynamic approaches. It is shown that various concepts such as the shape of interaction potential, the coupled-channels (CC) calculations, the breakup and transfer channels affect the theoretical cross sections in these approaches. [9,13 15] It is well-known that the calculations overestimate the experimental CF cross sections at energies above the Coulomb barrier. In fact, the calculated cross sections using a static approach are required to be scaled down by imposing the so-called suppression factor at hight energy regions. As r.gharaei@hsu.ac.ir c 2015 Chinese Physical Society and IOP Publishing Ltd a result of the literature, this phenomenon appears when we compare the predictions of both the single-barrier penetration model and the CC model with data. [16] Moreover, the authors have attributed this phenomenon to the low binding energy of the projectiles which can break up prior to reaching the fusion barrier, see for example Refs. [17 18] It must be noted that the fusion suppression factor is a very sensitive quantity to the applied theoretical model. [6,19] Double folding [20] and Sao Paulo [21 22] potentials are two dominate approaches for parametrization of the nuclear interactions in such colliding systems. [6,13,19,23 24] Recently, the analysis of the CF channel of the reactions with weakly bound nuclei has also been performed using various versions of the proximity potentials. [9,25] In contrast, many efforts have been reported to explore the fusion dynamics of symmetric or asymmetric heavy-ion fusion reactions, for example see Refs. [26 30]. Phookan and Kalita [9] systematically analyzed the fusion reactions induced by loosely bound projectiles such as 9 Be using eight versions of the proximity potentials, namely Prox 77, Prox 88, Bass 73, Bass 77, Bass 80, CW 76, BW 91, and AW 95. The obtained results of that study show that the potentials Bass 80, AW 95, and BW 91 can reproduce the empirical (or experimental) values of the fusion barrier height and position with more accuracy than the other selected versions. It must be noted that

2 186 Communications in Theoretical Physics Vol. 64 the details of the empirical and experimental efforts of our considered fusion systems have been reported in various corresponding references. According to Ref. [9], the standard deviations of these potentials after evaluation of the percentage deviations of the calculated barrier heights against the corresponding empirical values are 1.89, 2.02, and 2.04, respectively. Whereas, for fusion barrier positions their values are 2.05, 2.19, and 1.94, respectively. The role of the target deformation in the interaction potential and fusion cross section is also discussed for reaction 6 Li+ 152 Sm, see Ref. [9] for details. In the present study, we attempt to investigate the influence of the surface energy coefficient γ and temperature T on the interaction potential and CF cross section of the fusion systems including weakly bound projectiles 6 Li, 7 Li, and 9 Be. For this purpose, we use the proximity potentials BW 91 and AW 95 which have the best predictions for fusion dynamics of the selected colliding systems. Moreover, it is assumed that the target and projectile nuclei to be spherical in nature. It must be noted that the effects of the parameter γ and also the temperature T on the potential Bass 80 are ignored due to the following cases. (i) According to Refs. [31 32], one should point out that the formalism of this model is independent of the surface energy coefficient γ. (ii) It can be indicated that the role of the thermal effects of CN in the predictions of the potential Bass 80 is very negligible. The paper is organized as follows. The theoretical models which are applied to calculate the interaction potential are introduced in Sec. 2. The role of the surface energy coefficient and also temperature of CN in the calculated potentials and fusion cross sections of the different considered systems are discussed in Secs. 3 and 4. Section 5 is devoted to the most important conclusions of this work. 2 Determination of the Nucleus-Nucleus Potential When target and projectile nuclei approach each other, the strong competition between the long-range Coulomb repulsive and short-range nuclear attractive forces causes a potential barrier is formed in front of them. Under these conditions, one can define the total interaction potential as the sum of nuclear and Coulomb potentials as follows, V T (r) = V N (r) + V C (r). (1) It is remarkable that the total potential can be dependent on the angular momentum of interacting system using the extra term V l (r) = l(l +1) 2 /2µr 2, where µ is the reduced mass of target and projectile system. The Coulomb interactions of such systems can be exactly determined using the simple form V C (r) = Z 1 Z 2 e 2 /r, which it is assumed both nuclei to be point charges. In contrast, since the nature of nuclear force not yet fully known, it has still been a challenging problem to provide a comprehensive model for description of the nuclear interactions. However, there are several methods to estimate the nuclear part of total potential. [20,33 35] It is remarkable that using each of these methods, one can examine the role of various physical effects, such as the deformation, [36 37] dynamical [38 40] and thermal [41 44] effects, in the total interaction potential of different fusion reactions. Proximity formalism is one of the applied models to describe the interaction of two nuclei during the fusion process. According to the proximity force theorem, when two surfaces approach each other, approximately at distance of 2 3 fm, an extra force will appear in system which is parameterized as the proximity potential. [18] Blocki and collaborators introduced the original version of this formalism. [34] The improvements of various parameters of proximity formalism, such as radius parameter, surface energy coefficient and universal function, have been performed to explain the experimental fusion data with more accuracy. This subject has led to introduce different versions of proximity potentials. [26,45] As earlier stated, in the present study, the potentials AW 95 and BW 91 are employed to determine the nuclear potential. Description of these models has been briefly followed in the Subsecs. 2.1 and Broglia and Winther 1991 Based on the effective two-body force and the knowledge of the nuclear densities of target and projectile nuclei, Broglia and Winther introduced a nuclear potential by taking Woods Saxon (WS) parametrization. [46] This potential can be written as V N (r) = MeV, (2) 1 + exp((r R 0 )/a) with V 0 = 16πγa R 1R 2. (3) R 1 + R 2 In above relations the diffuseness parameter a = 0.63 fm. Moreover, the radius R 0 reads as, R 0 = R 1 + R fm, (4) where the radius parameter R i has the following form, R i = 1.233A 1/3 i 0.98A 1/3 i fm (i = 1, 2). (5) In Eq. (3), the surface energy coefficient γ is defined as, γ = γ 0 (1 k s A 2 s), (6) here, γ 0 (= a 2 /4πr0; 2 where r 0 and a 2 are the nuclear radius constant and the usual liquid drop model surface energy coefficient, respectively) is the surface energy constant and k s is the surface asymmetry constant. It is remarkable that the values of the constants γ 0 and k s in the present version are 0.95 MeV/fm 2 and 1.8, respectively. Moreover, A s = (N Z)/(N + Z) is asymmetry parameter where N and Z denote the neutron and proton numbers of compound system. This version of proximity potential is marked as BW Aage Winther 1995 The radius and diffuseness parameters of the above potential are refined by Winther using an extensive comparison with experimental data for heavy-ion elastic scattering. [47] These adjustment parameters R i and a are [ 1 ] a = V ( (A 1/3 1 + A 1/3 2 )) fm, (7) R i = 1.20A 1/3 i 0.09 fm (i = 1, 2), (8)

3 No. 2 Communications in Theoretical Physics 187 where R 0 = R 1 + R 2. This potential is labeled as AW Results and Discussion In this section, we initially study the behavior of the interaction potential as a function of the separation distance r based on the considered potentials AW 95 and BW 91. As earlier stated, our systematic study consists of 13 fusion reactions induced by the weakly bound projectiles 6 Li, 7 Li, and 9 Be. In Fig. 1, this behavior is displayed for an arbitrary colliding system such as 6 Li Pb. It is clear that the AW 95 model provides a more attractive nuclear potential than the other one; therefore, the Coulomb barrier predicted by this potential is lower than the potential BW 91. Fig. 1 The calculated nuclear and total interaction potentials vs. the internuclear distance r based on the proximity models AW 95 and BW 91 for fusion reaction 6 Li Pb. 3.1 Description of the Effect of Changing the Surface Energy Coefficient γ As earlier stated, the first purpose of this study is to analyze the role of the surface energy coefficient in the barrier characteristics of the fusion reactions with weakly bound projectiles. It is remarkable that the constants γ 0 and k s in definition of Eq. (6) are considered as adjustable parameters. During the last four decades, with advancement both theoretical and experimental viewpoints, various sets of these constants have been introduced. [48 55] The current study is restricted to four versions of the surface energy coefficient γ, namely γ-mn76, γ-mn95, γ- MS00, and γ-pd03 which are described in more detail below. (a) The Version γ-mn76 The first set of the coefficients γ 0 = MeV fm 2 and k s = 1.79 were parameterized by fitting the experimental binding energies. [48] Later on, Möller and Nix [50] introduced the set γ 0 = MeV fm 2 and k s =4.0 based on the nuclear macroscopic energy calculations. This version of surface energy coefficient is labeled as γ-mn76. (b) The Version γ-mn95 Using a least-squares adjustment to the ground-state masses of 1654 nuclei ranging from 16 O to and fission-barrier heights, Möller and collaborators [54] fitted the strength of the parameters γ 0 and k s to the values MeV/fm 2 and 2.345, respectively. This modified version of coefficient γ is marked as γ-mn95. (c) The Version γ-ms00 Based on the droplet model, [56] Myers and Świ atecki obtained the nuclear surface energy coefficient, which depends on the neutron skin t i of the target and projectile nuclei. In this approach, the coefficient γ is defined as γ = 1 [ 4πr (MeV) Q (t2 1 + t 2 ] 2) 2r0 2, (9) where neutron skin t i is given by, t i = 3 ( 2 r JIi (1/12)c 1 Z i A 1/3 ) i 0 (i = 1, 2). (10) Q + (9/4)JA 1/3 i In this equation, J and Q are the nuclear symmetric energy coefficient and the neutron skin stiffness coefficient, which are taken to be MeV and 35.4 MeV, respectively. Moreover, the value of r 0 is 1.14 fm and c 1 = 3e 2 /5r 0 = MeV. This version is labeled as γ-ms00. (d) The Version γ-pd03 The latest set of the parameters γ 0 and k s has been presented by Pomorski and Dudek. [55] In that study, the nuclear liquid-drop model was revisited using a systematic study on the fission-barrier heights and nuclear binding energies of 2766 nuclei with Z, N 8. The obtained results provide the values MeV/fm 2 and for coefficients γ 0 and k s, respectively. We label this version as γ-pd03. It should be noted that the set γ 0 = 0.95 MeV/fm 2 and k s = 1.8 introduced in Subsec. 2.1 is labeled as γ-aw 95 (-BW 91). Table 1 Calculated values of the surface energy coefficient γ, Eq. (6), using various introduced sets of the parameters γ 0 and k s for colliding system 9 Be+ 144 Sm. γ-version γ (MeV/fm 2 ) γ-bw91 (-AW95) γ-pd γ-ms γ-mn γ-mn So far, the role of the surface energy coefficient in the fusion of heavy-ions has been analyzed using a systematic study over 390 reactions including both the symmetric N = Z and asymmetric N Z reactions. In the present study, this effect is initially explored on the interaction potential of fusion reactions involving the weakly bound projectiles using both proximity versions AW 95 and BW 91. Figure 2 presents the behavior of the nuclear and total potentials based on the different sets of the parameters γ 0 and k s for fusion system 9 Be+ 144 Sm which are compared with the mentioned original versions. It is shown that the use of these modified sets reduces the barrier height; whereas they enhance the depth of the pocket in the inner part of the Coulomb barrier, see right panels of Fig. 2. As a further comment of this figure, one can point out that two sets γ-pd03 and γ-mn76 have the least and the most

4 188 Communications in Theoretical Physics Vol. 64 effects on the shape of nuclear and total potentials, respectively. It is shown that the strength of this coefficient enhances from the version γ-aw 95 (-BW 91) to γ-mn76 for fusion reaction 9 Be+ 144 Sm, see Table 1. Fig. 2 Comparison of the nuclear V N and total V T potentials as a function of distance r using the original and the modified versions of the proximity potentials BW 91 and AW 95 for colliding system 9 Be+ 144 Sm. Table 2 The fusion barrier positions R B (in fm) and heights V B (in MeV) based on the potentials BW 91, BW 91 (γ-pd03). The corresponding empirical data of the barrier characteristics are also listed. Different colliding systems are arranged with respect to their increasing values A 1/3 2. BW 91 BW 91 (γ-pd03) Empirical Reaction A 1/3 2 R B V B R B V B R B V B Ref. 6 Li + 64 Zn a [66] 9 Be + 89 Y a a [67] 6 Li Sm a a [68] 9 Be Sn b b [69] 6 Li Sm b b [25] 6 Li Tb a a [70] 9 Be Sm b b [71] 7 Li Tb a a [72] 6 Li Pb a a [73] 6 Li Bi b b [12] 7 Li Bi b b [12] 9 Be Pb b b [12] 9 Be Bi b b [74] Notes: The data labeled with the superscripts a and b are based on the empirical and experimental efforts, respectively. Using the original and the modified versions of the potential BW 91, we calculate the exact values of the barrier characteristics (height of the barrier and its position) for our considered fusion systems. They are listed in Tables 2 and 3. Similar calculations are also performed based on the potential AW 95, see Tables 4 and 5. In order to compare the calculated barrier characteristics with the corresponding experimental data, the standard deviation (SD) is calculated using the following relation, σ = 1 n n i=1 ( X theor. i ) 2 X expt. i, (11)

5 No. 2 Communications in Theoretical Physics 189 where X = R B or V B. Here, n denotes the number of the fusion reactions under consideration. The obtained values of the standard deviation for different proximity potentials are presented in Tables 6 and 7. It is shown that the modified versions AW 95 (γ-ms00) and BW 91 (γ-ms00) provide the best results for theoretical values of the barrier height, see Fig. 3. As one can see from this figure, these proximity models predict almost identical values for SDs of the barrier height in our selected mass range. Moreover, it is shown that the corrective effect of the version γ-ms00 on the potential BW 91 is more than the other selected model. Fig. 3 The standard deviations of the fusion barrier heights based on the selected proximity versions of the surface energy coefficient γ for different fusion reactions. Table 3 The fusion barrier positions R B (in fm) and heights V B (in MeV) based on the potentials BW 91 (γ-ms00), BW 91 (γ-mn95) and BW 91 (γ-mn76). Different colliding systems are arranged with respect to their increasing values A 1/3 2. BW 91 (γ-ms00) BW 91 (γ-mn95) BW 91 (γ-mn76) Reaction A 1/3 2 R B V B R B V B R B V B 6 Li + 64 Zn Be + 89 Y Li Sm Be Sn Li Sm Li Tb Be Sm Li Tb Li Pb Li Bi Li Bi Be Pb Be Bi Table 4 The fusion barrier positions R B (in fm) and heights V B (in MeV) based on the potentials AW 95, AW 95 (γ-pd03) and AW 95 (γ-ms00). Different colliding systems are arranged with respect to their increasing values A 1/3 2. AW 95 AW 95 (γ-pd03) AW 95 (γ-ms00) Reaction A 1/3 2 R B V B R B V B R B V B 6 Li + 64 Zn Be + 89 Y Li Sm Be Sn Li Sm Li Tb Be Sm Li Tb Li Pb Li Bi Li Bi Be Pb Be Bi

6 190 Communications in Theoretical Physics Vol. 64 Table 5 The fusion barrier positions R B (in fm) and heights V B (in MeV) based on the potentials AW 95 (γ-mn95) and AW 95 (γ-mn76). Different colliding systems are arranged with respect to their increasing values A 1/3 2. AW 95 (γ-mn95) AW 95 (γ-mn76) Reaction A 1/3 2 R B V B R B V B 6 Li + 64 Zn Be + 89 Y Li Sm Be Sn Li Sm Li Tb Be Sm Li Tb Li Pb Li Bi Li Bi Be Pb Be Bi Table 6 Calculated standard deviations for barrier characteristics based on the potential BW 91 and also various considered versions for its γ-dependence. Standard deviation Proximity model R B V B BW BW 91 (γ-pd03) BW 91 (γ-ms00) BW 91 (γ-mn95) BW 91 (γ-mn76) Table 7 Calculated standard deviations for barrier characteristics based on the potential AW 95 and also various considered versions for its γ-dependence. Standard deviation Proximity model R B V B AW AW 95 (γ-pd03) AW 95 (γ-ms00) AW 95 (γ-mn95) AW 95 (γ-mn76) Description of the Effect of Changing the Temperature T From the theoretical viewpoint, the thermal effects can be generally explored using the microscopic approaches (such as the M3Y double-folding model [20] and the Skyrme energy density formalism [33] ) or the macroscopic approaches (such as the proximity model). So far, various theoretical studies based on the mentioned approaches have been performed to analyze the role of the temperature-dependence in the fusion dynamics. [29 30,57 60] It is remarkable that for selected proximity models this dependence can be imposed by employing the modified forms of the surface diffuseness parameter a and the effective sharp radius R i as follows, [61 62] a(t ) = a(t = 0)( T 2 ), (12) R i (T ) = R i (T = 0)( T 2 ), (13) where R i (T = 0) and a(t = 0) are defined in Subsecs. 2.1 and 2.2. Here, temperature T is related to the CN excitation energy ECN or the energy of the projectile nucleus in the center-of-mass frame E c.m. via the entrance channel Q in -value, as [63 64] E = E c.m. + Q in = 1 a AT 2 T, (14) with a = 9 or 10 for intermediate mass or superheavy systems, respectively. Table 8 Calculated standard deviations for barrier characteristics based on the potentials BW 91 and AW 95 with (T-D) and without (T-IND) temperature-dependence for all considered fusion systems. Standard deviation Proximity model R B V B BW 91 (T-IND) BW 91 (T-D) AW 95 (T-IND) AW 95 (T-D) According to Eqs. (12) and (13), the increase of the temperature from T = 0 to a certain value T leads to an increase in the strength of the radius and diffuseness parameters. This phenomenon reduces the height of the barrier and also the pocket energy, see Fig. 4 for custom system 9 Be+ 209 Bi. In this figure, we show that the total interaction potential V T as a function of internuclear distance r using the proximity potentials BW 91 (upper panel) and AW 95 (lower panel) with and without the

7 No. 2 Communications in Theoretical Physics 191 thermal effects of CN. It should be noted that the total potentials are calculated at the temperatures correspond to the energies around the empirical Coulomb barriers. The obtained values for the temperature T and also corresponding energy E c.m. are also illustrated in each panel of Fig. 4. Fig. 4 The behavior of the total interaction potential V T as a function of separation distance r based on the proximity potentials BW 91 (upper panel) and AW 95 (lower panel) with (T-D) and without (T-IND) temperature-dependence for fusion system 9 Be+ 209 Bi. The comparison between the calculated standard deviations before and after imposing of the temperature effects of CN is carried out in Table 8. As a result of this table, we find that these effects improve the agreement between the theoretical and the experimental data of the barrier height for our considered mass range. 4 Analysis of the CF Fusion Cross Sections In order to investigate the behavior of the theoretical values of the CF cross sections σ CF versus the center-ofmass energy, we use the one-dimensional barrier penetration model (1D-BPM). According to this approach, the total fusion cross section can be written as the sum of the partial-wave cross sections, σ CF = σ l (E), (15) l=0 where the partial-wave cross sections are given by, σ l (E) = π 2 2µE (2l + 1)T l(e), (16) here, l and µ are the angular momentum quantum number and the reduced mass of the target and projectile system. In above relation, T l (E) is the quantum-mechanical transmission coefficient through the potential barrier for l th partial-wave. Using the WKB approximation, this coefficient can be evaluated numerically as follows, { [ 2µ R2l ( T l (E) = 1 + exp 2 dr V l=0 (r) + 2 l(l + 1) ) 1/2 ]} 1, 2µr 2 E (17) where R 1l and R 2l are the first and the second classical turning points for the l th partial-wave fusion barrier. By assuming that the Coulomb potential is replaced by a parabola, the transmission coefficient T l (E) can be calculated using a simple expression as, ]} { [ 1 T l (E) = 1 + exp 2π ω (E V Bl). (18) It should be noted that the theoretical values of the CF cross section are calculated by running the code CCFULL. [65] In this program, the WS parametrization is adopted for the nuclear potential. The parameters of this form of the potential have been extracted from the references which are listed in Table 2. The obtained results of the previous studies such as Refs. [6, 12] indicate that the calculations based on the 1D-BPM overestimate the measured cross sections of the fusion reactions with weakly bound projectile at abovebarrier energies. Indeed, one can observe suppression of CF cross sections at this energy range. This is investigated for fusion reactions 6 Li+ 209 Bi and 7 Li+ 209 Bi in Fig. 5. In this figure, we display the linear and logarithmic behaviors of the calculated cross sections based on the potentials BW 91 and AW 95. As expected from the literature, the obtained results of our considered models need to be suppressed at energies above the Coulomb barrier. It is remarkable that the disagreements observed at low energies is not unexpected. One can attribute them to the couplings between the relation motion of the colliding nuclei and their intrinsic degrees of freedom which are ignored in the calculations. Since the fusion suppression phenomenon is important at above-barrier energies, our present study is restricted to the high energy regions. On the other hand, the effects of the couplings to bound states on the CF channel of a system weakly bound projectile+target have been previously explored. [12] It is demonstrated that the calculated fusion cross sections based on the 1D-BPM and CC approaches are consistent with each other. In other words, it can be concluded that the fusion suppression is an insensitive quantity to the coupling the bound states at energies above the Coulomb barrier. 2 R 1l

8 192 Communications in Theoretical Physics Vol. 64 Fig. 5 The linear (right panels) and logarithmic (left panels) of the CF cross sections based on the proximity potentials AW 95 and BW 91 for fusion systems 6 Li+ 209 Bi and 7 Li+ 209 Bi. Fig. 6 Energy dependence of the suppression factors based on the different proximity potentials AW 95, AW 95 (γ- MS00), BW 91 and BW 91 (γ-ms00) for three colliding systems 9 Be+ 144 Sm, 6 Li+ 209 Bi and 7 Li+ 209 Bi. The results of the WS potential have also been presented to achieve further understanding. In each panel, the short-dotted lines are correspond to the mean values of the CF suppression.

9 No. 2 Communications in Theoretical Physics Sensitivity of Fusion Suppression to the Surface Energy Coefficient γ In general, the fusion suppression of CF channel can be determined by comparing the calculated and the measured values of cross sections as the ratio σ expt. CF /σtheor. CF. In Fig. 6, the behavior of the mentioned ratio as a function of the center-of-mass energy divided by the empirical barrier height, E c.m. /V emp. B are illustrated for colliding systems 9 Be Sm, 6 Li Bi and 7 Li Bi. The calculations are based on the original and the modified versions of the proximity potentials AW 95 and BW 91, which have been compared with the corresponding results of the WS potential. As one can see from this figure, for modified versions AW 95 (γ-ms00) and BW 91 (γ-ms00), the ratio of the measured CF cross sections to those from the 1D-BPM calculations are consistent with the WS potential. The mean values of the fusion suppression calculated using the WS and proximity potentials at above-barrier energies are listed in Table 9 for each of the mentioned reactions. Table 9 Mean complete fusion suppression at energies above the Coulomb barrier using the original and the modified (γ-ms00) various of the proximity potentials for colliding systems 9 Be+ 144 Sm, 6 Li+ 209 Bi, and 7 Li+ 209 Bi. Mean CF suppression Fusion reaction WS AW 95 AW 95 (γ-ms00) BW 91 BW 91 (γ-ms00) 9 Be Sm 0.87 a Li Bi 0.64 b Li Bi 0.73 c a The parameters of the WS potential are taken from Ref. [71]. b The parameters of the WS potential are taken from Ref. [12]. c The parameters of the WS potential are taken from Ref. [12]. strength of the coefficient γ from γ = MeV/fm 2 (in version γ-aw95) to γ = MeV/fm 2 (in version γ-mn76), the values of the ratio σ expt. CF /σtheor. CF reduce at each bombarding energy. In fact, one can find that the maximum value of the coefficient γ gives the most fusion suppression, as expected. Fig. 7 The effect of changing the surface energy coefficients γ on the obtained values of the ratio σ expt. CF /σtheor. CF based on the potential AW 95 for fusion system 9 Be+ 124 Sn. To achieve further understanding, the behavior of the ratio σ expt. CF /σtheor. CF versus the quantity E c.m. /V emp. B is analyzed using different values of the surface energy coefficient γ for a custom fusion system such as 9 Be Sn, see Fig. 7. It is clearly obvious that by increasing the 4.2 Sensitivity of Fusion Suppression to the Tem perature T of the CN Using the procedure introduced in Sec. 4, one can explore the behavior of the CF cross sections with energy (or temperature) of CN for fusion reactions induced by the weakly bound projectiles. The obtained results of Fig. 8 for fusion systems 9 Be Pb and 6 Li Bi at abovebarrier energies reveal that we can provide a more accurate description for CF suppression using the thermal effects of CN. Table 10 presents the calculated mean values of suppression factor based on the proximity potentials AW 95 and BW 91, which are supplemented with the mentioned effects. The systematic thermal trend of the ratio of the measured and the calculated CF cross sections for mentioned colliding systems is displayed in Fig. 9. It is shown that the increase of the temperature T enhances the mean value of the CF suppression. Table 10 Mean complete fusion suppression at energies above the Coulomb barrier using the potentials AW 95 and BW 91 with (T-D) and without (T-IND) temperature dependence for fusion reactions 9 Be+ 208 Pb and 6 Li+ 209 Bi. Mean CF suppression Fusion reaction WS AW 95 (T-IND) AW 95 (T-D) BW 91 (T-IND) BW 91 (T-D) 9 Be Pb 0.68 a Li Bi 0.64 a a The parameters of the WS potential are taken from Ref. [12].

10 194 Communications in Theoretical Physics Vol. 64 Fig. 8 The ratio of the calculated CF cross sections to the corresponding experimental data as a function of the ration E c.m./v emp. B based on the proximity potentials BW 91 and AW 95 with (T-D) and without (T-IND) temperaturedependence for reactions 9 Be+ 208 Pb and 6 Li+ 209 Bi. Fig. 9 The effect of changing the temperature on the obtained values of the ratio σ expt. CF /σtheor. CF based on the potential BW 91 for fusion system 9 Be+ 208 Pb. 5 Conclusions The current study is devoted to understand the role of the surface energy coefficients as well as the temperature dependence in the total interaction potential and CF cross section of the colliding systems induced by loosely bound projectiles 6 Li, 7 Li, and 9 Be. The calculations of this investigation are performed using the proximity formalism, including the potentials AW 95 and BW 91. The most important conclusions of the present study can be summarized as follows. (i) By increasing the strength of the surface energy coefficients γ, one can improve the theoretical values of the barrier characteristics based on the proximity potentials AW 95 and BW 91 for selected fusion systems. Among various considered versions, it is shown that the modified potentials BW 91 (γ-ms00) and AW 95 (γ-ms00) provide the best results for predicting the heights of the Coulomb barrier, see Tables 6 and 7 or Fig. 3. (ii) Using the temperature-dependent forms of the proximity potentials AW 95 and BW 91, one can reproduce the experimental data of the barrier characteristics with more accuracy than the temperature-independent ones, see Table 8. (iii) The influence of the surface energy coefficient γ and temperature T on the calculated CF cross sections and especially on the fusion suppression factor has been discussed at energies above the Coulomb barrier. The obtained results of the selected fusion systems indicate that the ratios of the measured and the calculated CF cross sections σ expt. CF /σtheor. CF based on the modified potentials BW 91 (γ-ms00) and AW 95 (γ-ms00) have good agreement with those obtained by the WS potential, see Fig. 6. Moreover, we conclude from an inspection of Fig. 8 that the thermal effects improve the predictions of the mentioned ratio for considered proximity potentials. (iv) The analysis of the systematic trend of the CF suppression factors at above-barrier energies using different values of the coefficient γ and temperature T reveals that the calculated CF cross sections need to be more suppressed by increasing the strength of these values, see Figs. 7 and 9.

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