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In Fig. 1, we give the fusion cross sections for negative-Q-value system
$ ^{64} {\rm{Ni}}$ +$ ^{64} {\rm{Ni}}$ and positive-Q-value system$ ^{24} {\rm{Mg}}$ +$ ^{30} {\rm{Si}}$ . All input parameters for the CC calculations are the same as those in Refs. [17, 43]. The experimental data denoted by solid circles are taken from Refs. [44-46]. The dotted and dashed lines indicate the results calculated without and with the CC effects, respectively. The solid line indicates the results calculated by CC model with the diffusion factor. The diffusing radius parameters$ r_{\rm{d}} $ are 1.23 and 1.40 fm for fusion systems$ ^{64} {\rm{Ni}}$ +$ ^{64} {\rm{Ni}}$ and$ ^{24} {\rm{Mg}}$ +$ ^{30} {\rm{Si}}$ , respectively. Note that the experimental data at the deep sub-barrier energies are overestimated by the calculated cross sections only resulting from CC model. By considering the diffusion factor after the overlap of projectile and target, the calculated fusion cross sections have a rapid decrease at deep sub-barrier energies and describe the fusion hindrance well.Figure 1. (color online) The fusion cross sections for the (a) 64Ni+64Ni and (b) 24Mg+30Si systems. The solid circles denote the experimental data[44-46]. The dotted and dashed lines indicate the results calculated without and with the CC effects, respectively. The solid line indicates the results calculated by CC model with diffusion factor. The inset gives the corresponding diffusion factor and the shallow region denotes the experimental threshold energy.
The insert in Fig. 1 shows the diffusion factors versus the incident energies. At high incident energies (
$ E>88.93 $ MeV for$ ^{64} {\rm{Ni}}$ +$ ^{64} {\rm{Ni}}$ and$ E>23.51 $ MeV for$ ^{24} {\rm{Mg}}$ +$ ^{30} {\rm{Si}}$ ), the compound nucleus is formed automatically once the Coulomb barrier is penetrated owing to the strong attractive nuclear force in the classical allowed region. Thus, the influence of diffusion process is week and the diffusion factor is assumed to be 1. With the decrease of incident energies, two colliding nuclei touch each other during the tunneling process and the diffusion process has a significant hindrance effect as shown in insert. In addition, the experimental threshold energy, the onset of fusion hindrance, is denoted by the shallow region. Notice that the diffusion factor plays a main role near the threshold energies.Fig. 2 shows the astrophysical S factor and logarithmic derivative representations of the fusion cross sections for
$ ^{64} {\rm{Ni}}$ +$ ^{64} {\rm{Ni}}$ [44] and$ ^{24} {\rm{Mg}}$ +$ ^{30} {\rm{Si}}$ [45-46] systems. The dashed and solid lines denote the results calculated by CC model without and with diffusion factor, respectively. In both systems, the results calculated with the diffusion factor have a significant improvement and are in good agreement with the experimental data. For comparison, in Fig. 2(a) and (b), the results calculated by using the sudden model from Ref. [11] and the adiabatic model from Ref. [15] are plotted by the dotted and dash-dotted lines, respectively. Because of totally different assumptions, two models have the diverse results at much deeper incident energies. For example, in Fig. 2(b) the logarithmic derivative resulting from adiabatic model is saturated below a certain incident energy and, conversely, the result of sudden model increases rapidly with decreasing energies. Interestingly, a hybrid result, which indicates the density variation may be simultaneous with the fusion reactions, is presented by combining the CC approach and the diffusion process as shown in Fig. 2(a) and (b).Figure 2. (color online) Astrophysical S factor and logarithmic derivative representations of the fusion cross sections versus colliding energies for the 64Ni+64Ni and 24Mg+30Si fusion systems. The solid circles denote the experimental data[44-46]. The dashed and solid lines denote the results calculated by CC model without and with diffusion factor, respectively. In (a) and (b), the dotted and dot-dashed lines are the results calculated by using the sudden model[11] and the adiabatic model[20].
Next we extend the calculations to other 19 fusion systems in which the fusion hindrance phenomena are also observed. Table 1 specifies these fusion systems and summarizes all results of diffusing radius and initial energy. The third column,
$ E_{\rm{s}} $ , is the experimental threshold energy where the curve of S factor shows a maximum. The Q values of the entrance channel are listed in the fourth column. The symbols$ r_{\rm{d}} $ ,$ R_{\rm{d}} $ and$ E_{\rm{d}} $ are the diffusing radius parameters and the initial energies used in Eq. (9). Note that the energies$ E_{\rm{d}} $ indicating the onset of the diffusion process for total 21 systems studied are almost consistent with the threshold energies of fusion hindrance$ E_{\rm{s}} $ and this suggests that the fusion hindrance is strongly associated with the dynamic diffusion after the projectile and target overlap. The$ \chi^2 $ values calculated by CC model without and with diffusing factors are given in the eighth and ninth columns, respectively. It is seen that a better fit is achieved by considering the diffusing factors. The last column is the references where the experimental fusion cross sections used in calculations and the corresponding threshold energies are taken.System $A_{\rm{CN}}$ $E_{\rm{s}}$/MeV $Q$/MeV $r_{\rm{d}}$/fm $R_{\rm{d}}$/fm $E_{\rm{d}}$/MeV $\chi^2_1/N$ $\chi^2_2/N$ Refs. 90Zr+92Zr 182 170.7±1.7 –153.7 1.14 10.26 171.16 214.5 11.92 [48] 90Zr+90Zr 180 175.2±1.8 –157.3 1.183 10.60 175.46 450.1 25.01 [48] 90Zr+89Y 179 170.8±1.7 –151.5 1.18 10.56 170.41 152.7 10.18 [48] 60Ni+89Y 149 122.9±1.2 –90.5 1.212 10.16 123.48 105.0 9.55 [9] 48Ca+96Zr 144 88.1±1.3 –45.9 1.25 10.27 90.63 60.96 59.17 [49] 64Ni+64Ni 128 87.5±0.9 –48.8 1.23 9.84 88.93 49.37 3.09 [44] 58Ni+58Ni 116 94.0±0.9 –65.8 1.238 9.58 93.69 4.13 1.77 [50] 40Ca+90Zr 130 91.9±0.9 –57.0 1.225 9.68 92.48 179.0 178.8 [51] 54Fe+58Ni 112 86.7±0.9 –56.5 1.24 9.48 87.04 14.22 2.07 [52] 34S+89Y 123 72.6±0.7 –36.6 1.265 9.75 73.89 1374 42.97 [53] 32S+89Y 121 73.1±0.7 –36.6 1.265 9.66 74.56 759.3 33.01 [53] 16O+208Pb 224 69.0±2.0 –46.5 1.25 10.32 67.87 1155 18.42 [18] 48Ca+48Ca 96 48.2±1.0 –2.98 1.31 9.52 48.60 9.59 5.41 [54] 40Ca+48Ca 88 47.0±0.5 4.56 1.285 9.06 48.31 90.75 90.65 [55] 40Ca+40Ca 80 48.0±1.0 –14.2 1.275 8.72 49.97 1.62 1.61 [56] 28Si+64Ni 92 45.6±0.5 –1.78 1.281 9.01 47.33 273.4 19.89 [57] 12C+198Pt 210 48.2±1.0 –14.0 1.218 10.06 48.27 14.20 10.39 [58] 11B+197Au 208 40.0±1.5 –5.0 1.252 10.07 40.73 9.95 9.03 [59] 28Si+30Si 58 24.2±3.6 14.3 1.37 8.42 26.62 5.25 3.85 [60] 24Mg+30Si 54 20.8±0.7 17.9 1.40 8.39 23.51 11.00 4.65 [45] 12C+30Si 42 10.5±0.75 14.1 1.38 7.45 10.38 6.98 5.31 [61] Table 1. The radius and energy used in diffusing factor for the systems discussed in Refs. [43, 47]. The symbol
$A_{\rm{CN}}$ denotes the mass number of the compound nuclei. The symbol$E_{\rm{s}}$ in the third column is the experimental threshold energies of fusion hindrance. The fourth column lists the Q value of the entrance channel. The symbols$r_{\rm{d}}$ ,$R_{\rm{d}}$ and$E_{\rm{d}}$ are the diffusing radius and energy parameters used in Eq. (9). The eighth and ninth columns denote the$\chi^2$ values of fusion cross sections calculated by CC model without and with diffusing factor, respectively. The last column is the references where the experimental fusion cross sections and threshold energies are taken.In Fig. 3, the radius parameters in the diffusion factor versus the mass numbers of the compound nuclei are displayed. The circles denote the values of diffusing radius extracted from 21 fusion systems with Q value ranging from
$ -157.3 $ to 17.9 MeV. The diffusing radius parameters$ r_{\rm{d}} $ and$ R_{\rm{d}} $ versus the mass number of compound nuclei are shown in Fig. 3(a) and (b), respectively. A good linear relationship between the diffusing radius and the mass numbers of compound nuclei is shown and the dashed line denotes the fitting result of all diffusing radius values obtained here. In addition, the stars are the distances of the closest approach evaluated with the time-dependent Hartree-Fock (TDHF) method and then the dynamic diffusion process plays an important role in the synthesis of superheavy nuclei[62]. Note that the calculated results of our method agree with the prediction of TDHF method for superheavy nuclei.
Investigation of Diffusion Process in Deep Sub-barrier Fusion Reactions
doi: 10.11804/NuclPhysRev.37.2020039
- Received Date: 2020-06-23
- Rev Recd Date: 2020-07-23
- Available Online: 2021-11-22
- Publish Date: 2020-12-20
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Key words:
- fusion hindrance /
- coupled-channels model /
- diffusion process
Abstract: We introduce a new diffusion factor to describe the hindrance phenomenon in fusion reactions at deep sub-barrier energies based on the Pauli blocking effect after the projectile and target overlap. In this approach, the fusion cross sections are assumed to be the product of two parts: the tunneling factor to overcome the Coulomb barrier and the diffusion factor after two colliding nuclei contact. The former is described by coupled-channels approach and the latter depends on the colliding energies as well as the temperature of system. In total, 21 fusion systems with hindrance phenomenon are analyzed in details and it is found that the diffusion factor plays an important role near the experimental threshold energies of fusion hindrance. In addition, taking negative-Q-value system 64Ni+64Ni and positive-Q-value system 24Mg+30Si as examples, not only fusion cross sections but also two important representations, namely, astrophysical S factor and logarithmic derivative, are found to be in good agreement with experimental data.
Citation: | Kaixuan CHENG, Chang XU. Investigation of Diffusion Process in Deep Sub-barrier Fusion Reactions[J]. Nuclear Physics Review, 2020, 37(4): 809-815. doi: 10.11804/NuclPhysRev.37.2020039 |