Survey of deep sub-barrier heavy-ion fusion hindrance phenomenon for positive and negative Q-value systems using the proximity-type potential

  • A systematic survey of the accurate measurements of heavy-ion fusion cross sections at extreme sub-barrier energies has been carried out by using the coupled-channels (CC) theory that is based on the proximity formalism. The present work theoretically explores the role of surface energy coefficient and energy-dependent nucleus-nucleus proximity potential in mechanism of the fusion hindrance of 14 typical colliding systems with negative $Q$-values, including 11B+197Au, 12C+198Pt, 16O+208Pb, 28Si+94Mo, 48Ca+96Zr, 28Si+64Ni, 58Ni+58Ni, 60Ni+89Y, 12C+204Pb, 36S+64Ni, 36S+90Zr, 40Ca+90Zr, 40Ca+40Ca and 48Ca+48Ca as well as 5 typical colliding systems with positive $Q$-values, including 12C+30Si, 24Mg+30Si, 28Si+30Si, 36S+48Ca, and 40Ca+48Ca. It is shown that the outcomes based on the proximity potential along with the above-mentioned physical effects are able to achieve reasonable agreement with the experimentally observed data of the fusion cross sections $\sigma_{\rm{fus}}(E)$, astrophysical $S(E)$ factors, and logarithmic derivatives $L(E)$ in the energy region far below the Coulomb barrier. A discussion is also presented about the performance of the present theoretical approach in reproducing the experimental fusion barrier distributions for different colliding systems.
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R. Gharaei, A. Fuji and B. Azadegan. Survey of deep sub-barrier heavy-ion fusion hindrance phenomenon for positive and negative Q-value systems using the proximity-type potential[J]. Chinese Physics C. doi: 10.1088/1674-1137/ac23d3
R. Gharaei, A. Fuji and B. Azadegan. Survey of deep sub-barrier heavy-ion fusion hindrance phenomenon for positive and negative Q-value systems using the proximity-type potential[J]. Chinese Physics C.  doi: 10.1088/1674-1137/ac23d3 shu
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Survey of deep sub-barrier heavy-ion fusion hindrance phenomenon for positive and negative Q-value systems using the proximity-type potential

    Corresponding author: R. Gharaei, r.gharaei@hsu.ac.ir
  • Department of Physics, Sciences Faculty, Hakim Sabzevari University P. O. Box 397, Sabzevar, Khorasan Razavi, Iran

Abstract: A systematic survey of the accurate measurements of heavy-ion fusion cross sections at extreme sub-barrier energies has been carried out by using the coupled-channels (CC) theory that is based on the proximity formalism. The present work theoretically explores the role of surface energy coefficient and energy-dependent nucleus-nucleus proximity potential in mechanism of the fusion hindrance of 14 typical colliding systems with negative $Q$-values, including 11B+197Au, 12C+198Pt, 16O+208Pb, 28Si+94Mo, 48Ca+96Zr, 28Si+64Ni, 58Ni+58Ni, 60Ni+89Y, 12C+204Pb, 36S+64Ni, 36S+90Zr, 40Ca+90Zr, 40Ca+40Ca and 48Ca+48Ca as well as 5 typical colliding systems with positive $Q$-values, including 12C+30Si, 24Mg+30Si, 28Si+30Si, 36S+48Ca, and 40Ca+48Ca. It is shown that the outcomes based on the proximity potential along with the above-mentioned physical effects are able to achieve reasonable agreement with the experimentally observed data of the fusion cross sections $\sigma_{\rm{fus}}(E)$, astrophysical $S(E)$ factors, and logarithmic derivatives $L(E)$ in the energy region far below the Coulomb barrier. A discussion is also presented about the performance of the present theoretical approach in reproducing the experimental fusion barrier distributions for different colliding systems.

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    I.   INTRODUCTION
    • One of the most interesting developing areas of nuclear physics is the analysis of sub-barrier heavy-ion fusion cross sections which offers various opportunities of studying fusion dynamics, see for example Refs. [1-6]. The most notable outcome of such studies is the hindrance phenomenon of heavy-ion fusion at extreme sub-barrier energies. In fact for medium-heavy-mass systems, it is found that the fusion cross sections $ \sigma_{\rm{fus}} $ drop rapidly with decreasing energy, so that the experimental values of $ \sigma_{\rm{fus}} $ are much steeper than the prediction of standard coupled-channels (CC) calculations using a Woods-Saxon (WS) potential [5,7-9,17]. From a practical standpoint, the low-energy fusion hindrance phenomenon in the mentioned mass region can easily be recognized by two important observations. One is the astrophysical $ S(E) $ factor representation and the other is the logarithmic derivative (slope) $ L(E) $ of the fusion excitation functions. For heavy-ion fusion reactions with negative $ Q $-values, the authors found that the astrophysical $ S(E) $ factor must develop a maximum at an energy $ E_s $ which shows the onset of the sub-barrier fusion hindrance [18-20]. This reality can be grasped by means of the definition of the $ S(E) $ factor as follows,

      $ S(E) = E \sigma_{\rm{fus}} \exp[2\pi(\eta-\eta_0)]\; \; {\rm{mb}}\; {\rm{MeV}}, $

      (1)

      where $ \eta = e^2Z_pZ_t/\hbar\upsilon_{\rm{rel}} $ is the Sommerfeld parameter and $ \upsilon_{\rm{rel}} $ is the relative velocity between the target and projectile in the center-off-mass frame. In addition, $ \eta_0 $ is an adjustable parameter. For $ E \rightarrow -Q $ when $ Q < 0 $, in fact, the fusion cross section $ \sigma_{\rm{fus}}(E) $ and thus $ S(E) $ factor must go to 0 which results in a maximum of the energy-dependent behavior of this factor at energy $ E_s>-Q $. Fusion hindrance phenomenon has also been observed in lighter fusion systems with positive $ Q $-values. For a system with $ Q>0 $, there is no restriction on $ S(E) $ when $ E\rightarrow 0 $ and therefore it is not necessary to have an $ S $-factor maximum. However, there are indications of the occurrence of fusion hindrance in systems with positive Q-values. In fact, the results discussed in the previous studies such as Refs. [4,21,22] confirm that the experimental $ S $-factor for fusion systems with positive $ Q $-values develops a maximum at low energy. As mentioned above, another sensitive method for identifying the unexpected steep falloff in low-energy fusion cross sections is provided by analyzing the energy-dependent behavior of the logarithmic derivative representation $ L(E) $ of the experimental fusion cross section, defined as,

      $ L(E) = \frac{d[\ln(\sigma_{\rm{fus}} E)]}{ dE} = \frac{1}{E\sigma_{\rm{fus}}}\frac{d(E\sigma_{\rm{fus}})}{dE}\; \; {\rm{MeV}}^{-1}. $

      (2)

      As a result of the literature, the experimental values of the logarithmic derivative $ L(E) $ must be continued to grow slowly with decreasing the center-of-mass energy for heavy-ion fusion reactions [7-9,23,24]. The onset of the sub-barrier fusion hindrance phenomenon can be identified by an intersection between the $ L(E) $ factor and the constant $ S(E) $ factor function, $ L_{\rm{CS}}(E) = \dfrac{\pi\eta}{E} $.

      From the theoretical point of view, the sudden approach proposed by Mişicu and Esbensen as well as the adiabatic model proposed by the Ichikawa and co-workers were very successful in reproducing the hindrance phenomenon in several cases [7,9]. In 2019, Cheng and Xu [25] employed 13 different phenomenological nuclear potentials (including KNS model and 12 widely used proximity potentials) to explore systematically the influence of adiabatic process on the fusion hindrance phenomenon observed in 16 typical fusion reactions. The authors found that by considering the damping of the CC effect, the results of fusion cross section, astrophysical $ S(E) $ factor, and logarithmic derivative $ L(E) $ calculated by the coupled-channels model are improved compared with the results without the damping factors. As a result of the literature, Pauli exclusion principle will generate a repulsion between the two reacting nuclei when the overlap between their densities is significant [10]. Microscopic dynamical methods such as the density-constrained frozen Hartree-Fock (DC-FHF) can be used to investigate the impact of Pauli repulsion on the ion-ion potential [11-13]. It is shown that the nuclear potential inside the barrier is sensitive to the effect of Pauli repulsion. In addition, the authors indicated that the inclusion of this repulsion between composite systems of nucleons within the framework of the DCFHF theory plays an important role in describing the behavior of fusion hindrance at deep sub-barrier energies. The fully microscopic density-constrained time-dependent Hartree-Fock (DC-TDHF) approach also shows that for achieving a good description of the data at energies far below the Coulomb barrier the dynamical effects such as the surface properties due to neck formation, energy dependence of the fusion barriers and mass transfer must be considered to modify the nucleus-nucleus potential in the inner region [14,15]. By inspiring the Pauli-blocking effects of $ \alpha $-cluster decay in radioactive nuclei, very recently a microscopic approach has been introduced to explain the fusion hindrance at colliding energies far below the Coulomb barrier, see for example [16]. This approach is based on the $ \alpha $-cluster structures in n$ \alpha $ nuclei such as 12C, 16O and 24Mg. In order to construct the Pauli blocking potential in the n$ \alpha $-nucleus-induced fusion reactions, one can use a single folding procedure. The authors of Ref. [16] indicated that a shallow pocket is formed in the inner part of the potential barrier when we use the standard M3Y nuclear interaction accompanied by the Pauli blocking potential of two colliding nuclei. Further, it is shown that the fusion hindrance phenomenon in the colliding systems 12C+198Pt, 16O+208Pb, 12C+30Si, 24Mg+30Si, and 28Si+30Si can be described well by introducing this Pauli blocking potential.

      Recently, we proposed a dynamical approach to evaluate the heavy-ion fusion cross sections at deep sub-barrier energies within the framework of energy-dependent nucleus-nucleus potential [26]. It is shown that when we take into account the effect of surface energy coefficients as well as temperature-dependence in the original proximity potential 1977, the theoretical fusion cross sections calculated by using the CC calculations provide a nice description for steep falloff phenomenon at low energies for 28Si+100Mo, 58Ni+54Fe, and 64Ni+64Ni systems. Very recently, we used this idea for exploring the fusion hindrance phenomenon for the two fusion reactions 32,34S+89Y [27]. The obtained results demonstrate clear experimental evidence for the occurrence of a steep falloff in the fusion excitation functions of these reactions. In addition, our calculations reveal that the experimental fusion evaporation cross sections $ \sigma_{\rm{fus}}(E) $, astrophysical $ S(E) $ factors, and logarithmic derivatives $ L(E) $ can be reproduced by using the modified form of the proximity potential well.

      In the present paper, we are interested in generalizing the performance of the proximity formalism as a macroscopic nucleus-nucleus potential in reproducing the sub-barrier fusion cross sections for two category of systems. The first category (category I) includes 14 target and projectile combinations 11B+197Au, 12C+198Pt, 16O+208Pb, 28Si+94Mo, 48Ca+96Zr, 28Si+64Ni, 58Ni+58Ni, 60Ni+89Y, 12C+204Pb, 36S+64Ni, 36S+90Zr, 40Ca+90Zr, 40Ca+40Ca and 48Ca+48Ca with product of the atomic number range $ 392 \leqslant Z_1Z_2 \leqslant 1029 $. We notice that the $ Q $-values for all of the studied systems under category I are negative, ranging from $ Q = -90.5192 $ to $ -1.780 $ MeV. In this category, comparisons are made of the hindrance behavior between the present calculations and those previously performed for colliding systems 32S+89Y (with $ Q = -36.626 $ MeV), 34S+89Y (with $ Q = -36.597 $ MeV), 28Si+100Mo (with $ Q = -39.9194 $ MeV), 58Ni+54Fe (with $ Q = -75.8554 $ MeV), and 64Ni+64Ni (with $ Q = -66.4341 $ MeV). The second category (category II) includes 5 typical fusion reactions 12C+30Si, 24Mg+30Si, 28Si+30Si, 36S+48Ca, and 40Ca+48Ca with $ 84 \leqslant Z_1Z_2 \leqslant 400 $. The $ Q $-values for all of the studied systems under category II are positive, ranging from $ Q = 4.5572 $ to 17.8873 MeV. In this study, we also analyze the influence of nuclear surface tension coefficients and temperature dependence on the concept of barrier distributions $ B(E) = d^2(E\sigma_{\rm{fus}} )/dE^2 $. An effort is also made to present a discussion of the energy-dependent behavior of the surface energy coefficient $ \gamma $ at the energy range which corresponds to the steep fall-off phenomenon. The existence of the heavy-ion reactions with positive and negative fusion $ Q $-values allows us to investigate the effect of the sign of $ Q $-value on the energy-dependent behavior of the $ \gamma $ coefficient. As a result of the literature [28-33], an additional enhancement in the sub-barrier fusion process can be observed due to the coupling of the neutron transfer channels with positive $ Q_{xn} $-values. Nevertheless, the effective $ Q $-values for various neutron transfer channels, involving single- and multi-neutron (up to six neutrons) stripping (or pick-up), are negative for most of the reactions under study in this paper. Hence, we ignore the couplings to the neutron transfer channels in the calculations of the fusion cross sections.

      This article is organized in the following way. Section II gives the relevant details of the theoretical frameworks used to calculate the nuclear potential. The setup and the results are presented in section III. Section IV concludes the present work with a short summary.

    II.   THE NUCLEUS-NUCLEUS PROXIMITY POTENTIAL
    • The nuclear potential $ V_N(r) $ of the original proximity potential 1977 (Prox. 77) [36] is given by

      $ V^{\rm{Prox.77}}_{N}(r) = 4\pi\gamma b\overline{R}\; \Phi\bigg(\frac{s}{b}\bigg)\; \; {\rm{MeV}}. $

      (3)

      Here, $ s $ is the distance between the near surfaces of two reacting nuclei, $ \overline{R} $ is the mean curvature radius and $ \Phi(s/b) $ is the universal function [36-38]. In addition, $ b $ represents the width of the nuclear surface and was taken to be 1 fm. In Eq. (3), the nuclear surface tension coefficient $ \gamma $ can be taken from the work of Myers and Świątecki (Lysekil mass formula) [39]

      $ \gamma = \gamma_0\Biggl[1-k_s\Biggl(\frac{N-Z}{A}\Biggl)^2\Biggl]\; \; {\rm{MeV. fm}}^{-2}, $

      (4)

      where $ N $, $ Z $ and $ A $ represent the neutron, proton and mass numbers of the compound nucleus, respectively. In addition, $ \gamma_0 $ and $ k_s $ are the surface energy constant and surface asymmetry constant. In the original version of the proximity formalism, these constants have been parameterized as $ \gamma_0 = 0.9517 \; {\rm{MeV\cdot fm}}^{-2} $ and $ k_s = 1.7826 $.

    III.   RESULTS AND DISCUSSION

      A.   The fusion cross section

    • The measurements of the excitation function for all reactions studied presently confirm that there are strong indications of fusion hindrance (or steep fall-off) phenomenon at energies well below the Coulomb barrier. It has been previously shown that the experimental fusion cross sections for these reactions drop much faster than predicted by standard CC calculations. Accordingly, in the first step of the calculations, we attempt to analyze the heavy-ion fusion cross sections for the studied reactions under categories I and II within the framework of the Prox. 77. In the previous studies [37,38,40], the authors often used the Wong formula for estimating the theoretical fusion cross sections based on the proximity potential formalism. The theoretical calculations performed with the Wong formula fail to reproduce the experimental data at sub-barrier energies at sub-barrier energies for medium-heavy systems [37,38,40-42]. Therefore, in the present study, we compute the theoretical values of fusion cross section with the help of the standard CC calculations based on the computer code CCFULL [43]. It must be noted that these calculations include couplings to low-lying quadrupole and octupole excitation states in both target and projectile as well as mutual and multi-phonon excitations of these states. The level energies and spectroscopic information for the low-lying surface vibrations of the reacting nuclei are extracted from Refs. [44-47]. The results of the calculations are shown in Fig. 1. We conclude from an inspection of this figure that the Prox. 77 model significantly underestimates the measurements of the fusion-evaporation excitation functions at sub-barrier energies for a number of systems. However, it seems that this potential model does a better job for lighter colliding systems 12C+30Si, 28Si+30Si and 24Mg+30Si. These results can be understood by looking at the radial behavior of the Coulomb plus nuclear potentials used in the CC calculations (see the insert of the figure). In order to gain further insight, we compare in Table I the barrier heights provided by the proximity model calculations with those obtained from the standard WS potentials that used in the previous studies. In the cases that these data are not available, we use the parameterized form for fusion barrier height which has been obtained by analyzing the empirical/experimental fusion barrier heights for more than 200 reactions [48]. It is clear that the original version of the proximity potential produces the fusion barriers that are higher and therefore thicker than the empirical ones.

      Figure 1.  (color online) Comparison of the experimental fusion cross sections with those obtained theoretically using the original proximity potential 1977 for different heavy-ion reactions under categories I (upper panels) and II (lower panels). For further understanding, the fusion barriers resulting from the Prox. 77 model are compared with the WS/Expt ones (horizontal dashed lines)

      Reaction $ V_B^{{\rm{Prox}}. 77} \;{\rm{(MeV)}}$ $ V_B^{{\rm{Prox.}}\; 77\; ({\rm{Mod}}-1)} \;{\rm{(MeV)}}$ $ V_B^{{\rm{WS/Expt}}} \;{\rm{(MeV)}}$ Ref.
      Category I
      11B+197Au 48.92 46.78 46.70 [59]
      12C+198Pt 58.01 56.08 56.20 [59]
      16O+208Pb 79.38 77.49 77.60 [60]
      28Si+94Mo 78.14 76.38 73.82 [48]
      48Ca+96Zr 99.33 96.10 95.40 [61]
      28Si+64Ni 54.74 52.51 48.99 [62]
      58Ni+58Ni 103.49 99.64 100.30 [63]
      60Ni+89Y 135.13 132.93 132.40 [18]
      12C+204Pb 60.73 58.00 56.59 [48]
      36S+64Ni 60.63 58.84 58.50 [64]
      36S+90Zr 82.99 78.87 79.80 [65]
      40Ca+90Zr 103.6 101.99 99.34 [61]
      40Ca+40Ca 57.59 55.43 54.43 [67]
      48Ca+48Ca 53.95 52.06 51.70 [68]
      Category II
      12C+30Si 13.67 13.49 13.59 [3]
      24Mg+28Si 25.97 25.30 24.33 [4]
      28Si+30Si 29.88 29.28 29.00 [69]
      36S+48Ca 44.62 42.44 43.30 [70]
      40Ca+48Ca 55.66 52.17 48.75 [68]

      Table 1.  The fusion barrier heights $ V_B $ (in MeV) using the original and modified forms of the proximity potential for different colliding systems under categories I and II. The values of the Coulomb barrier height calculated using the WS potentials and those obtained by the parameterized form suggested in Ref. [48] are also listed.

      In order to improve the results of the proximity potential, we attempt to explore simultaneously the role of two important physical effects in fusion dynamics. In fact, we discuss the effect of surface energy coefficient $ \gamma $ as well as thermal effects of hot nuclei on the proximity potential and ultimately on the fusion cross sections of the considered systems. As a result of the literature [26,27,40], one can find out that the mentioned physical effects play a crucial role in the sub-barrier enhancement. In order to implement this idea, it is crucial to use a modified version of the surface energy coefficient in the proximity potential as follows,

      $ \gamma(T) = \acute{\gamma_0}\Biggl[1-\acute{k_s}\Biggl(\frac{N-Z}{A}\Biggl)^2\Biggl]\Biggl[1-\frac{T-T_B}{T_B}\Biggl]^{3/2}\; \; {\rm{MeV. fm}}^{-2}. $

      (5)

      As evident from the above equation, we use the revised sets of the constants $ (\acute{\gamma_0},\acute{k_s}) $ for the present calculations to impose the effect of surface energy coefficient on the fusion dynamics. The values of these constants are presented in Table II. In order to implement the thermal effects of hot nuclei, we follow the procedure proposed in Ref. [49] for the temperature dependence of the liquid surface tension. In that study, the authors introduced a modified temperature-dependent surface energy coefficient in the proximity potential based on the equation of state proposed by van der Waals [50-52]. We note that, the temperature $ T $ can be related to the energy of the projectile nucleus in the center-of-mass frame $ E_{\rm{c.m.}} $ via the entrance (incoming) channel $ Q_{\rm{in}} $-value [53-55] as follows,

      Reaction $ \acute{\gamma_0} \;{\rm{(MeV\cdot{\rm{fm}}^{-2})}}$ $ \acute{k_s} $ T(MeV) $ E_s\;{\rm{(MeV)}} $ $ \gamma^{{\rm{Prox}}. 77} \;{\rm{(MeV\cdot{\rm{fm}}^{-2})}}$ $ \gamma^{{\rm{Prox}}. 77 ({\rm{Mod}}-1)}\;{\rm{(MeV\cdot{\rm{fm}}^{-2})}} $
      Category I
      11B+197Au 1.460734 4.0 1.25 40.97 0.89238 1.65446
      12C+198Pt 1.460734 4.0 1.54 48.08 0.89029 1.41245
      16O+208Pb 0.918 0.7546 1.08 70.40 0.88624 1.23989
      28Si+94Mo 1.01734 1.79 1.84 66.50 0.94030 1.25084
      48Ca+96Zr 1.27326 2.5 1.93 88.38 0.90457 1.42037
      28Si+64Ni 1.460734 4.0 2.49 45.31 0.93887 1.54155
      58Ni+58Ni 1.27326 2.5 1.93 93.83 0.94968 1.53975
      60Ni+89Y 1.01734 1.79 1.68 122.8 0.93450 1.33418
      12C+204Pb 1.460734 4.0 1.11 53.68 0.89352 1.68692
      36S+64Ni 1.27326 2.5 2.38 51.95 0.92727 1.34221
      36S+90Zr 1.460734 4.0 1.79 73.70 0.93075 1.81337
      40Ca+90Zr 0.918 0.7546 1.87 91.65 0.94166 1.15089
      40Ca+40Ca 1.27326 2.5 2.39 49.00 0.95170 1.47676
      48Ca+48Ca 1.27326 2.5 2.24 48.00 0.90457 1.40977
      Category II
      12C+30Si 1.01734 1.79 2.70 10.43 0.94785 1.07667
      24Mg+28Si 1.1756 2.2 2.95 20.68 0.94937 1.24903
      28Si+30Si 1.01734 1.79 2.69 23.86 0.94968 1.18083
      36S+48Ca 1.460734 4.0 2.28 36.92 0.91707 1.67610
      40Ca+48Ca 1.460734 4.0 2.03 46.73 0.93767 2.06212

      Table 2.  The values of ($ \acute{\gamma_0},\acute{k_s} $) constants for different colliding systems. The calculated values of the temperature $ T $ (in MeV), threshold energy $ E_s $, surface energy coefficients $ \gamma $ based on the Prox. 77 and Prox. 77 (Mod-1) potentials are also presented.

      $ E_{\rm{c.m.}} + Q_{\rm{in}} = \frac{1}{9}AT^2 - T, $

      (6)

      where $ A $ is the mass number of the compound nucleus. In Eq. (5), temperature $ T_B $ corresponds to the energy of the Coulomb barrier evaluated by Eq. (15) of Ref. [49]. In the present work, the results of the proximity potential 1977 accompanied by the modified form (5) are marked as "Prox. 77 (Mod-1)". The temperature values $ T $ we used for the calculations are listed in the forth column of Table II for different colliding systems with both negative and positive $ Q $-values. We notice that a fixed value of the center-of-mass energy $ E_{\rm{c.m.}} $ that approximately corresponds to the experimental barrier height, $ E_{\rm{c.m.}}\approx V_B^{\rm{Expt.}} $, is used for calculating the temperature value $ T $ in each colliding system.

      Concerning this idea, it must be mentioned that the function of $ V_N(r) $ of the Prox. 77 model, Eq. (3), reveals that the nuclear proximity potential depends directly on the surface energy coefficient. On the other hand, the present modifications allow us to obtain the larger strengths of nuclear surface tension between two colliding nuclei, see Table II for comparison of the coefficients $ \gamma^{\rm{Prox. 77}} $ and $ \gamma^{\rm{Prox.77(Mod-1)}} $. So, it is straightforward to see that, we can obtain more attractive nuclear potentials in comparison with the original proximity potentials 1977 when we use the present modified form (5) instead of Eq. (4) in different colliding systems. The accuracy of the Prox. 77 (Mod-1) model must be carefully verified, so we calculate the Coulomb barrier heights $ V_B $ for different heavy-ion fusion reactions in comparison with the corresponding values caused by the standard WS potentials and experimental/empirical data. The calculated barrier heights are shown in the third column of Table I. From this table, it is clearly seen that the modified form of the proximity potential can reproduce the fusion barrier heights in a better manner than the original version of this formalism. To further understand the role of the modified form (5), in Fig. 2, we display the percentage difference of the fusion barrier heights $ \Delta V_B $(%) as a function of the charge product $ Z_1Z_2 $ for all systems with both negative (upper panel) and positive (lower panel) $ Q $-values. It can be defined as follows,

      Figure 2.  (color online) The percentage difference $ \Delta V_B $(%) between the theoretical and experimental barrier heights as a function of the charges product of reacting nuclei based on the original and modified forms of the Porx. 77 model for the fusion reactions under categories I (upper panel) and II (lower panel)

      $ \Delta V_B(\%) = \Biggl(\frac{V_B^{\rm{Theor}}-V_B^{\rm{Expt}}}{V_B^{\rm{Expt}}}\Biggl)\times 100. $

      (7)

      From this figure, we can see that the original version of the proximity potential formalism fails to reproduce the experimental/empirical barrier heights for the studied reactions ($ \chi^2_{V_B} = 3.254 $ and 3.509 for fusion systems under categories I and II, respectively), whereas the Prox. 77 (Mod-1) model gives an acceptable description of the data ($ \chi^2_{V_B} = 1.641 $ and 1.442 for fusion systems under categories I and II, respectively).

      We have also repeated the calculations of the fusion cross sections with the Prox. 77 (Mod-1) potential. The results are shown in Fig. 3 by the short-dashed (black) curves. One can see that the CC calculations performed with this modified potential are in good agreement with the experimental data around and sub-barrier energies but unable to recover the lowest measured energy points at deep sub-barrier energies which give strong evidence of the phenomenon of heavy-ion fusion hindrance. The energy threshold $ E_s $ for the onset of the deep sub-barrier fusion hindrance for the present cases can be estimated from Fig. 3. The extracted values of the energy $ E_s $ are compared in Fig. 4 with the empirical analysis of Ref. [4], as shown with gray solid curve, as a function of the system parameter $ \zeta = Z_1Z_2\mu^{1/2} $. In the two previous studies [26,27], we demonstrated that the fusion hindrance appears at the energy threshold $ E_s$ = 87.29, 88.9, 67.5, 73.42 and 72.95 MeV for 58Ni+56Fe, 64Ni+64Ni, 28Si+100Mo, 32S+89Y and 34S+89Y systems, respectively. These values are presented in Fig. 4, for comparison. We observe that the location of the hindrance threshold points for the present systems are in reasonable agreement with the empirical prediction. As can be seen, the effect of surface energy coefficient $ \gamma $ and thermal properties of liquid and hot nuclei have so far been successful to cure the deficiencies of the nuclear proximity potential in the energy region $ E_{\rm{c.m.}}> E_s $.

      Figure 3.  (color online) Measured fusion cross sections for different heavy-ion fusion reactions under categories I (upper panels) and II (lower panels) are compared to the CC calculations based on the original and modified versions of the proximity potential

      Figure 4.  (color online) Comparison of the extracted values of the energy $ E_s $ where the S factor has a maximum with the empirical expression given by Ref. [4] as a function of the system parameter $ \xi = Z_1Z_2 \sqrt{\mu} $ for the present reactions and those studied in Refs. [26,27]

      We would like to mention that the characteristics of the nuclear ion-ion potential at extremely close distances affect the measurements of fusion cross sections in the range of microbarn and nanobarn levels [7-9,56]. On the other hand, it can be shown that the adjustable parameter $ \acute{\gamma_0} $ in Eq. (5) directly affects the shape of the inner side of the potential barrier. In order to give the best fit to the experimentally observed fusion data at extreme low energies $ (E_{\rm{c.m.}}\leqslant E_s) $, therefore, we attempt to investigate the fusion mechanisms of the selected reactions by reducing the strength of surface energy coefficient $ \acute{\gamma_0} $ at each bombarding energy. Under these conditions, we encounter an energy-dependent behavior of this constant. It must be pointed out that by imposing this physical effect, we obtain a modified form of the Prox. 77 (Mod-1) potential which is marked as "Prox. 77 (Mod-2)". It is worth noting that the calculated ion-ion potentials and thus fusion cross sections based on the Prox. 77 (Mod-2) are quite similar to those obtained by the Prox. 77 (Mod-1) at the energy range $ E_{\rm{c.m.}} > E_s $. One can conclude that the energy-dependent parameter $ \acute{\gamma_0}(E) $ leads to a spectrum of fusion barriers of variable weights and heights in the energy region $ E_{\rm{c.m.}}\leqslant E_s $ as shown in Fig. 5 for 28Si+64Ni reaction as an example. In this spectrum, the heights of the energy-dependent fusion barriers increase by decreasing the incident energy of relative motion in the center of mass frame. Consequently, it can be seen that the Prox. 77 (Mod-2) model calculations (green solid curve) predict lower values for sub-barrier fusion cross sections in comparison with the outcomes of the Prox. 77 (Mod-1) model (red dotted curve), so that this model is able to achieve a close agreement with experimental cross sections at deep sub-barrier energies. Note that the adjusted values of the $ \acute{\gamma_0} $ parameter to gain better agreement with the fusion data of the 28Si+64Ni reaction are presented in Fig. 5. To further explore the influence of the energy-dependent interaction potential on the fusion cross sections at deep sub-barrier energy region, in Fig. 3, the available experimental fusion excitation functions are compared with CC calculations performed with this modified form of the Prox. 77 model for all fusion systems under study with both negative and positive $ Q $-values. It is seen that the best fits to the data at low energies are achieved by using the calculations based on the Prox. 77 (Mod-2) model.

      Figure 5.  (color online) Plot of the fusion excitation function for the system 28Si+64Ni compared with several calculations described in the text. The energy-dependent behavior of the ion-ion potential based on the Prox. 77 (Mod-2) model at $ E_{\rm{c.m.}}\leqslant E_s $ region is also displayed in the insert of the figure

      In Fig. 6, we display variation of energy-dependent ratio $ \acute{\gamma_0}(E)/\acute{\gamma_0} $ with the energy ratio $ E/E_s $ for all of the colliding systems. The results are compared with the available theoretical data taken from Refs. [26,27]. One can realize that the extracted values of surface energy constant at bombarding energies $ E\leqslant E_s $ follow a linear increasing trend with the increase of energy in the center of mass frame. In the present study, we formulate this observed trend using the following relation,

      Figure 6.  (color online) Variation of the $ \acute{\gamma_0}(E)/\acute{\gamma_0} $ ratio as a function of the $ E/E_s $ within the framework of the Prox. 77 (Mod-2) proximity potential for different considered fusion systems under categories I (upper panel) and II (lower panel). Note that the calculations are restricted to the energy range $ E\leqslant E_s $. The data extracted from our previous works [26,27] have also been presented, for comparison

      $ \acute{\gamma_0}(E) = \acute{\gamma_0}\Biggl[a\bigg(\frac{E}{E_s}\bigg)-b\Biggl]\; \; {{\rm{MeV. fm}}}^{-2}, $

      (8)

      where the extracted values of the $ (a,b) $ constants are equal to (3.2416,2.4601) and (2.7474,1.9062) for the fusion systems with negative and positive $ Q $-values, respectively. The results shown in Fig. 6 reflect a slight difference in the slope and intercept of the fitted lines due to the sign of the $ Q $-values. By considering the above energy-dependent formula in the proximity potential formalism, we suggest a modified form of the surface energy coefficient as follows

      $ \begin{array}{l} \gamma^{\rm{mod}}(T,E) = \left\{\begin{array}{rl}\gamma(T)\biggl[a\bigg(\dfrac{E}{E_s}\bigg)-b\biggl],\; \; \; \; \; {\rm{for}} \; \; {\rm{E}}_{\rm{c.m.}}\leqslant {\rm{E}}_{\rm{s}},\\ \gamma(T),\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; {\rm{for}} \; \; {\rm{E}}_{\rm{c.m.}} > {\rm{E}}_{\rm{s}},\\ \end{array} \right. \end{array} $

      (9)

      where $ \gamma(T) $ can be calculated from Eq. (5). The presently suggested relation, in fact, is an extension of calculations caried out in our previous work [26].

      In Figs. 7 and 8, the experimental data of the representations of astrophysical $ S(E) $ factor and logarithmic derivative $ L(E) $ for the heavy-ion fusion reactions studied here are respectively compared with those calculated with different proximity potentials. In the calculations of the astrophysical $ S $ factor representations of the fusion cross sections, we take $ \eta_0 = $ 30.5, 34.0, 47.0, 39.5, 51.0, 68.0, 92.5, 35.5, 39.0, 43.5, 46.0, 59.0, 75.4 and 68.0 for the 11B+197Au, 12C+198Pt, 16O+208Pb, 28Si+64Ni, 28Si+94Mo, 58Ni+58Ni, 60Ni+89Y, 12C+204Pb, 40Ca+40Ca, 48Ca+48Ca, 36S+64Ni, 36S+90Zr, 48Ca+96Zr and 40Ca+90Zr fusion systems, respectively. We take $ \eta_0$ = 11.50, 20.0, 22.7, 36.2 and 42.2 for fusion systems with positive $ Q $-values 12C+30Si, 24Mg+30Si, 28Si+30Si, 36S+48Ca, and 40Ca+48Ca, respectively. It is observed in Fig. 7 that the $ S $ factors calculated by the original proximity potential 1977 are consistent with the corresponding experimental data at energies near and above the Coulomb barrier. This model underestimates the lowest-energy points for the considered reactions. In addition, as depicted in Fig. 7, drastic improvements have been made by taking into account the effect of surface energy coefficient as well as thermal effects of hot nuclei in the theoretical results caused by the Prox. 77 (Mod-1). It is seen that the clear maximums in the measured $ S $ factors can be reproduced only by the Prox. 77 (Mod-2) calculations. For the logarithmic derivative of the heavy-ion systems of interest, we conclude from an inspection of Fig. 8 that the results provided by the Prox. 77 and Prox. 77 (Mod-1) models are saturated at extremely low incident energies. Conversely, the results calculated with the Prox. 77 (Mod-2) model increase with decreasing incident energy and a nice agreement with the experimental data of $ L(E) $ is obtained in the CC calculations that are based on this modified form of the proximity potential.

      Figure 7.  (color online) Same as Fig. 3 but for the astrophysical $ S(E) $ factor

      Figure 8.  (color online) Same as Fig. 3 but for the logarithmic derivative $ L(E) $. Prediction for a constant $ S $ factor $ L_{CS} $, is presented by short-dotted (purple) line

    • B.   The fusion barrier distribution

    • It is well known that in the presence of the coupling between the relative motion of the colliding nuclei and their internal structure the single fusion barrier can be thought of as splitting into a distribution of barrier heights. The concept of the so-called barrier distribution (BD) can be specified as the second derivative of the energy-weighted cross sections [57],

      $ B(E) = \frac{d^2(E \sigma_{\rm{fus}})}{dE^2}\; \; {\rm{mb.MeV}}^{-1}. $

      (10)

      On the basis of the first derivative using point difference formula, the second derivative of $ E \sigma_{\rm{fus}} $ can be numerically computed at energy $ (E_1+ 2E_2 + E_3)/4 $ as follows,

      $\begin{aligned}[b] \frac{d^2(E \sigma_{\rm{fus}})}{dE^2} =& 2\Biggl[ \frac{(E \sigma_{\rm{fus}})_3-(E \sigma_{\rm{fus}})_2}{E_3-E_2}-\frac{(E \sigma_{\rm{fus}})_2-(E \sigma_{\rm{fus}})_1}{E_2-E_1}\Biggl]\\&\times\frac{1}{(E_3-E_1)}, \end{aligned}$

      (11)

      where $ (E \sigma_{\rm{fus}})_i $ are appraised at energies $ E_i $. In the case of equal energy steps $ \Delta E = (E_3-E_2) = (E_2-E_1) $, we notice that the relation reduces to the following form,

      $ \frac{d^2(E \sigma_{\rm{fus}})}{dE^2} = \frac{(E \sigma_{\rm{fus}})_3-2(E \sigma_{\rm{fus}})_2+(E \sigma_{\rm{fus}})_1}{(\Delta E)^2}. $

      (12)

      Then the statistical error $ \delta_c $ related to the second derivative at energy $ E $ is approximately given by,

      $ \delta_c = \Biggl(\frac{E}{{\Delta E} ^2}\Biggl)\Biggl[ (\delta \sigma_{\rm{fus}})^2_1 - 4(\delta \sigma_{\rm{fus}})^2_2 + (\delta \sigma_{\rm{fus}})^2_3 \Biggl]^{\frac{1}{2}}, $

      (13)

      where $ (\delta \sigma_{\rm{fus}})_i $ are the errors in the fusion cross sections [58]. Comparisons of experimental fusion BDs with the calculations based on the various versions of proximity potential formalism are displayed in Fig. 9 for the heavy-ion fusion reactions under categories I and II. While determining the barrier distributions from the theoretical cross sections, we have used the same energy step of around $ \Delta E \approx 1 $ MeV. One can find that the BDs calculated with the original proximity potential 1977 are not in good agreement with the data for the studied reactions. It is obvious from Fig. 9, however, that the best fit to experimentally observed BD has been obtained with the CC calculations for the Prox. 77 (Mod-2). This means that the effect of the nuclear surface tension coefficient and also temperature dependence play an important role in the shape of the fusion barrier distribution.

      Figure 9.  (color online) Comparison between the experimental BDs and those obtained using the CC calculations for the various versions of the proximity potential in heavy-ion fusion reactions under categories I (upper panels) and II (lower panels)

    IV.   SUMMARY AND CONCLUSIONS
    • By using the CC approach and the proximity formalism, we have systematically analyzed the steep falloff phenomenon of the measured cross sections for fourteen fusion systems with $ 392\leqslant Z_1Z_2\leqslant 1029 $ under category I (with $ Q<0 $) and five fusion systems with $ 84\leqslant Z_1Z_2\leqslant 400 $ under category II (with $ Q>0 $). The standard coupled-channels calculations include couplings among low-lying surface vibration modes $ 2^+ $ and $ 3^- $ of projectile and target as well as their mutual and multiphonon excitations. In the first step, we calculated the nuclear potentials and fusion cross sections within the framework of the original proximity potential 1977. Our results show that the deviations of the experimental excitation functions of different colliding systems from the CC calculations using the Prox. 77 potential are large, especially at far sub-barrier energies. For a better description of the fusion cross sections at this energy range, we have supplemented the proximity formalism with the effect of surface energy coefficient $ \gamma $ and temperature dependence. It is shown that the agreement of our results with the available fusion data enhances when we use the Prox. 77 model supplemented with these physical effects. As a consequence, the extracted values of the energy at the maximum $ E_s $ are in good agreement with the phenomenological curve for colliding systems with both negative and positive $ Q $-values. At the energy range which corresponds to the steep fall-off phenomenon, we have attempted to reproduce the experimental data of the fusion cross sections by considering the energy-dependent potentials via adjusting the surface energy constant $ \gamma_0 $. We can conclude that there are excellent agreements between our calculations based on the Prox. 77 (Mod-2) model for the fusion cross sections $ \sigma_{\rm{fus}}(E) $, astrophysical $ S(E) $ factors and logarithmic derivatives $ L(E) $ and their corresponding experimental data. In the present study, a new energy-dependent form of the nuclear surface tension coefficient $ \gamma^{\rm{mod}}(T,E) $ of the proximity formalism has been derived from fitting the measurements of the fusion-evaporation excitation functions at low energies. The obtained results reveal that the effect of the sign of the $ Q $-values on the energy-dependent behavior of the coefficient $ \gamma $ is insignificant. Eventually, it is remarkable that there are some other fusion reactions with either $ Q<0 $ or $ Q>0 $ for consideration. However, it can be shown that the required corrections for interaction potential and thus fusion cross sections for these colliding systems are quite different from those seen by the present procedure. For example, our primary analysis based on the original version of the proximity formalism shows that the calculated fusion cross sections for 16O+16O (with $ Q = 16.5415 $ MeV) are overpredicted with respect to the corresponding experimental data in the entire energy range. While, as shown in Fig. 1, we encounter different results in the fusion systems considered presently. Hence, one can find that the proximity approach along with the present corrective effects is not suitable to deal with the fusion hindrance phenomenon in these reactions.

Reference (70)

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