⇦Chinese Physics Letters, 2017, Vol. 34, No. 4, Article code 042501 Positive $Q$-Value Neutron Transfer Mediated Sub-Barrier Fusion Reactions * Pei-Wei Wen(温培威)1,2, Zhao-Qing Feng(冯兆庆)3, Fan Zhang(张凡)1,2, Cheng Li(李成)1,2, Cheng-Jian Lin(林承键)4, Feng-Shou Zhang(张丰收)1,2,5 Affiliations 1Key Laboratory of Beam Technology and Material Modification of Ministry of Education, College of Nuclear Science and Technology, Beijing Normal University, Beijing 100875 2Beijing Radiation Center, Beijing 100875 3Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000 4China Institute of Atomic Energy, Beijing 102413 5Center of Theoretical Nuclear Physics, National Laboratory of Heavy Ion Accelerator of Lanzhou, Lanzhou 730000 Received 24 November 2016 *Supported by the National Natural Science Foundation of China under Grant Nos 11635003, 11025524, 11161130520, 11175218 and U1332207, the National Basic Research Program of China under Grant No 2010CB832903, and the European Commission's 7th Framework Programme (Fp7-PEOPLE-2010-IRSES) under Grant No 269131.
**Corresponding author. Email: fszhang@bnu.edu.cn Citation Text: Wen P W, Feng Z Q, Zhang F, Li C and Lin C J et al 2017 Chin. Phys. Lett. 34 042501 Abstract Positive $Q$-value neutron transfer mediated sub-barrier fusion reactions are studied with an empirical coupled channels model, which takes into account neutron rearrangement related only to the dynamical matching condition with no free parameters. Fusion cross sections of collision systems $^{32}$S+$^{90,94,96}$Zr are calculated and analyzed. Logarithmic residual enhancement (LRE) is proposed to evaluate the discrepancy between calculated results and experimental data. The experimental data can be described well with this model for the first time as a whole, while the LRE analysis shows that there are still theoretical systematic deviations. DOI:10.1088/0256-307X/34/4/042501 PACS:25.70.Hi, 24.10.Eq, 25.70.Jj © 2017 Chinese Physics Society Article Text Understanding the fusion reaction dynamics is indispensable to understand the nucleosynthesis in stellar evolutions and the extending of the periodic table.[1-3] Studying fusion reactions will also bridge the gap between nuclear structures and nuclear reactions due to strong couplings between the degrees of freedom of internal movements and those of relative movements. Unifying the microscopic nuclear structure calculations and reaction models is very important in understanding many experimental data.[4,5] Moreover, nuclear fusion is also closely related to the quantum decoherence phenomenon which is beyond the usual description of coherent coupled channels.[6,7] Recently, neutron transfer mediated sub-barrier fusion reactions have drawn extensive attention. Many theoretical works[1,8-11] and experiments[12-17] have been devoted to the study of this phenomenon. However, there is no unified theoretical interpretation on the dynamics of neutron transfer mediated sub-barrier fusion reactions yet. Even the theory for pure transfer process should be developed with more complex two-particle correlations.[18] The mechanism of how sub-barrier fusion reaction with positive $Q$-value is enhanced and why the enhancement is not proportional to the magnitudes of the $Q$-values is unclear yet.[1,19] Enhancement of certain sub-barrier fusion reactions can be interpreted by different theories based on completely different grounds. For example, sub-barrier fusion enhancement of $^{40}$Ca+$^{96}$Zr can be interpreted by not only coupling of neutron transfer,[10,11] but also by coupling of the strong octuple vibrational state of $^{96}$Zr.[20] The role of neutron transfer in the capture process at sub-barrier energies has also been studied and explained within the quantum diffusion framework, which shows that transfer of two neutrons influences the sub-barrier capture process through the change of the deformations of the colliding nuclei.[19] Recently, an energy scaling approach was proposed to account for sub-barrier enhancement problems via considering the compound nucleus aspect, where no microscopic tunneling theory has incorporated[21,22] and this theory is still under debate.[23] The empirical coupled channels model (ECC) with widths of the barrier distribution taking into account positive $Q$-value neutron pair transfer channel achieved a global fit of fusion data recently.[24,25] Moreover, other theoretical models can explain certain sub-barrier enhancement phenomena, such as quantum molecular dynamics (QMD)[26-28] and the time-dependent Hartree–Fock model (TDHF).[29] In this study, the ECC model is adopted for the study of fusion reactions, which is designed to predict the penetration probability in the first step of superheavy nuclei synthesis.[10,26,30-32] The $Q$-value dependent dynamical matching factor is expressed as $$\begin{align} \alpha_x(E,l,Q)=N_x^{-1}\exp(-C_x Q^2),~~ \tag {1} \end{align} $$ where $x$ denotes the $x$th neutron transfer channel, and $C_x$ determines the variance of the distribution of neutron transfer probability with respect to $Q$, which is given by the semiclassical first-order perturbation theory for transfer and inelastic reactions between heavy ions.[33] Here $N_x$ is a normalization factor, namely $$\begin{align} N_x=\int^{Q_{x}}_{-E}\exp(-C_x Q^2)dQ,~~ \tag {2} \end{align} $$ where $Q_{x}$ is the $Q$-value of the ground-to-ground $x$th neutron transfer channel with $n_x$ transferred neutrons. By incorporating the neutron rearrangement factor as a kind of barrier distribution, the total penetration probability with averaging over different neutron transfer channels is written as $$\begin{align} T_l(E)=\,&N_{\rm tr}^{-1}\int_{-E}^{Q_{x}}[\delta(Q)+\alpha_{\rm tr} (E,l,Q)] \\ &\times T_l^{\rm CC}(E+Q) dQ,~~ \tag {3} \end{align} $$ where the first term in square brackets corresponds to the no-transfer channel. The second term is defined as $$\begin{align} \alpha_{\rm tr}(E,l,Q)=\,&\sum_{x=1}^{x_{\max}}\alpha'_x(E,l,Q) \\ =\,&\sum_{x=1}^{x_{\max}}N_x^{-1} \exp(-C_x Q^2),~~ \tag {4} \end{align} $$ where $x_{\max}$ is the maximal number of the included neutron transfer channels, and $N_{\rm tr}^{-1}$ is the normalization constant. The partial penetration probability $T_l^{\rm CC}$ in Eq. (3) is predicted by the ECC model, which is obtained by taking into account a multidimensional character of the realistic barrier, $$ T_l^{\rm CC}(E)=\int f(B){T}_l^{\rm HW}(B,E)dB,~~ \tag {5} $$ where $f(B)$ is the empirical asymmetric Gaussian barrier distribution function the same as in Ref. [34], and ${T}_l^{\rm HW}$ is the usual Hill–Wheeler formula regarding the radial dependence of the barrier as a parabola one. The nuclear potential used in this model is the Woods–Saxon potential or proximity potential with no free parameters, except that the diffuseness parameter is equal to 1 fm for heavy nuclei and 1.1 fm for light projectiles. In this study, the double folding potential based on the Skyrme interaction force without considering the momentum and the spin dependence is used to testify the reliability of this model,[35,36] which is $$\begin{align} V_{N}=\,&C_{0}\{\frac{F_{\rm in}-F_{\rm ex}}{\rho_{0}}[\int\rho_{1}^{2}({\boldsymbol r})\rho_{2}({\boldsymbol r} -{\boldsymbol R})d{\boldsymbol r} \\ &+\int\rho_{1}({\boldsymbol r})\rho_{2}^{2}({\boldsymbol r}-{\boldsymbol R})d{\boldsymbol r}] \\ &+F_{\rm ex}\int\rho_{1}({\boldsymbol r})\rho_{2}({\boldsymbol r}-{\boldsymbol R})d{\boldsymbol r}\},~~ \tag {6} \end{align} $$ with $$ F_{\rm in,ex}=f_{\rm in,ex}+f'_{\rm in,ex} \frac{N_{1}-Z_{1}}{A_{1}}\frac{N_{2}-Z_{2}}{A_{2}},~~ \tag {7} $$ which is dependent on the nuclear densities and on the orientations of deformed nuclei in the collision. The parameters $C_{0}=300$ MeV$\cdot$fm$^{3}$, $f_{\rm in}=0.09$, $f_{\rm ex}=-2.59$, $f'_{\rm in}=0.42$, $f'_{\rm ex}=0.54$, and $\rho_{0}=0.16$ fm$^{-3}$ are used in the calculation. The Woods–Saxon density distributions are expressed for two nuclei as $$\begin{align} \rho_{1}({\boldsymbol r})=\,&\frac{\rho_{0}}{1+\exp[(r-\Re_{1}(\theta_{1}))/a_{1}]},~~ \tag {8} \end{align} $$ $$\begin{align} \rho_{2}({\boldsymbol r}-{\boldsymbol R})=\,&\frac{\rho_{0}}{1+\exp[(|{\boldsymbol r}-{\boldsymbol R}|-\Re_{2}(\theta_{2}))/a_{2}]},~~ \tag {9} \end{align} $$ where $\Re_{i}(\theta_{i})$ $(i=1,2)$ is the surface radii of the nuclei with $\Re_{i}(\theta_{i})=R_{i}(1+\beta_{i}Y_{20}(\theta_{i}))$, and the spherical radii $R_{i}$. The parameter $a_i$ for target or projectile nuclei represent the surface diffusion coefficients, which is taken as 0.50 fm in the calculation.
cpl-34-4-042501-fig1.png
Fig. 1. (Color online) Fusion cross sections for $^{32}$S+$^{90,94,96}$Zr with respect to the incident energy in the center-of-mass frame. The results are calculated by the ECC model with double folding potential. The no-coupling limits are represented by blue dotted lines (w/o). The red solid lines (+DN) correspond to the calculations with both dynamical deformation and neutron transfer, while the blue dashed lines (+D) take account of only dynamical deformation. The experimental fusion cross sections denoted by open circles are taken from Ref. [12] for $^{32}$S+$^{90,96}$Zr and Ref. [14] for $^{32}$S+$^{94}$Zr. The Coulomb barrier is represented by the black arrow.
Fusion cross sections for $^{32}$S+$^{90,94,96}$Zr calculated by the ECC model with double folding potential are shown in Fig. 1. The parameters used for calculation of these three reactions are kept the same to retain consistency. It can be seen from the figure that the calculated results describe well the general trends of all the experimental values of these three reactions when dynamical deformation and neutron arrangement are considered (blue dashed lines in Fig. 1 (a) and red solid lines in Figs. 1(b) and 1(c)). The overall trends of the red lines for $^{32}$S+$^{94}$Zr and $^{32}$S+$^{96}$Zr are almost the same. In the sub-barrier energy region, theoretical predictions are all slightly lower than experimental data for $^{32}$S+$^{94}$Zr and $^{32}$S+$^{94}$Zr. In previous studies,[14,37] residual enhancement (RE) with no free parameters is proposed and defined by excluding the inelastic coupling effects calculated by the coupled channel (CC) theory, as the ratio of the experimental fusion data to the CC calculation, that is, ${\rm RE}=\sigma_{\rm exp}/\sigma_{\rm cc}$. In this study, the RE is generalized to logarithmic residual enhancement (LRE), which is defined as the logarithmic ratio of the experimental fusion data to any theoretical calculation, that is, ${\rm LRE}=\log(\sigma_{\rm exp}/\sigma_{\rm cal})$.
cpl-34-4-042501-fig2.png
Fig. 2. Modified residue enhancement (MRE) (a) and logarithmic residue enhancement (LRE) (b) for $^{64}$Ni+$^{64}$Ni, which are denoted by open squares. The experimental fusion cross sections are taken from Ref. [38]. The corresponding coupled channel calculations are obtained from the solid line of Fig. 1 in the same reference. Here MRE equals to $-1$ and LRE equals to 0 are denoted by the dashed line, which means that the experimental data agree with the theoretical results.
cpl-34-4-042501-fig3.png
Fig. 3. (Color online) LRE for $^{32}$S+$^{94,96}$Zr by the ECC model with respect to ratio of incident energy over Coulomb barrier. Neutron rearrangement is considered for all the cases. LRE of $^{32}$S+$^{96}$Zr is denoted by red open rectangles, while the blue open circles represent LRE of $^{32}$S+$^{94}$Zr. Experimental data and theoretical results used to extract LRE are the same as those in Fig. 1. The dashed line in the figure denotes that LRE is equal to zero, where the experimental data agree with the theoretical results.
There are several merits to using LRE instead of RE to evaluate the deviation of calculated results from experimental data. First, if the theoretical calculation is much larger than the experimental data, RE will be close to zero while LRE can be naturally developed to a negative value, which is a hindrance actually. This advantage is important for evaluation of different kinds of calculation, especially for describing the hindrance of fusion far below the Coulomb barrier. Secondly, LRE is able to display not only enhancement in the lower energy region but also variation in higher energy region. RE can also be modified as ${\rm MRE}=-\sigma_{\rm cc}/\sigma_{\rm exp}$ to evaluate hindrance. Comparison of MRE and LRE is displayed in Fig. 2 for fusion reaction $^{64}$Ni+$^{64}$Ni.[38] MRE is almost a straight line and no error bars can be seen in the higher energy region, while LRE is able to display that there are fluctuations and clear error bars in the higher energy region. In the lower energy region, LRE and its error bars change more smoothly than MRE. Therefore, LRE is engaged to analyze the discrepancy between theoretical results and experimental data in the following. The deviation of calculated data implemented with neutron rearrangement (red solid lines in Fig. 1) with respect to experimental data can be seen more specifically by plotting the LRE, which is shown in Fig. 3. As for $^{32}$S+$^{94,96}$Zr, changes of LRE with respect to ratio of energy over the Coulomb barrier are not synchronous in the sub-barrier region. It can be seen that theoretical predictions agree well with the experimental data in higher energy region for both models, though with some fluctuations. As the energy decreases, the variation from the zero line is larger. The LRE scatters tend to distribute on the opposite side with respect to the dashed line for these two reactions in the lower energy tail region. These results suggest that more work should be carried out for completely consistent explanation of $^{32}$S+$^{94,96}$Zr in the empirical framework of coupled channels with the same parameter set, and the question addressed about the anonymous enhancement of $^{32}$S+$^{94}$Zr in Ref. [14] is not solved thoroughly yet. In conclusion, positive $Q$-value neutron transfer mediated sub-barrier fusion reaction has been studied with the ECC model. Neutron rearrangement related only to the dynamical matching condition with no free parameters is implemented in the ECC model. Experimental data of fusion cross sections $^{32}$S+$^{90,94,96}$Zr have been described well with this model for the first time as a whole. Moreover, to evaluate the discrepancy between calculated results and experimental data, LRE is put forward. It is quite convenient to analyze experimental data for both sub-barrier and deep-barrier fusion reactions. By analyzing the LRE of these reaction systems, it is found that there are still theoretical systematic deviations. Further improvement on the ECC model and exploring the implications of LRE on deep barrier fusion reactions are needed in future. We thank Hui-Ming Jia and Bing Wang for helpful discussions. 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