⇦Chinese Physics Letters, 2016, Vol. 33, No. 7, Article code 072401 A Common Optical Potential for $^{4}$He+$^{12}$C at Intermediate Energies * Li-Yuan Hu(胡力元), Yu-Shou Song(宋玉收)**, Ying-Wei Hou(侯英伟), Hui-Lan Liu(刘辉兰) Affiliations Fundamental Science on Nuclear Safety and Simulation Technology Laboratory, Harbin Engineering University, Harbin 150001 Received 11 April 2016 *Supported by the National Natural Science Foundation of China under Grant No 11205036, and the Fundamental Research Funds for the Central Universities of China under Grant No HEUCF101501.
**Corresponding author. Email: songyushou80@163.com Citation Text: Hu L Y, Song Y S, Hou Y W and Liu H L 2016 Chin. Phys. Lett. 33 072401 Abstract A common optical potential for $^4$He+$^{12}$C at intermediate bombarding energies, which is essential in analyzing exotic nuclei with $^4$He clusters, is obtained based on the São Paulo potential. Among systematic optical potentials for $^4$He+$^{12}$C, this potential has the merit of using a fixed imaginary part of the Woods–Saxon form. By optical-model calculations, this potential reproduces the experimental elastic scattering angular distributions of $^4$He+$^{12}$C well within the energy range of 26$A$–60$A$ MeV. It is also applied successfully in calculations of the breakup reactions of $^6$Li+$^{12}$C and $^6$He+$^{12}$C with a three-body continuum discretized coupled-channel method. DOI:10.1088/0256-307X/33/7/072401 PACS:24.10.Ht, 25.55.Ci, 24.10.Eq © 2016 Chinese Physics Society Article Text As one of the simplest nuclei, $^4$He is also an important cluster in exotic nuclei, such as $^{6,8}$He[1,2] and $^{12,14}$Be.[3,4] The breakup reaction of these nuclei on $^{12}$C targets is an effective way to extract their cluster information.[5] The continuum discretized coupled-channel (CDCC) method[6] is traditionally employed to handle this reaction calculation, where the optical potential (OP) between $^4$He and $^{12}$C targets is necessary. Unfortunately, the experimental data of elastic scattering angular distribution (ESAD) for some energy used to extract OPs does not exist. Alternatively, we can choose the OP of an adjacent energy, which is in rough approximation. The OP between $^4$He and $^{12}$C is supposed to be studied systematically. A few global OPs[7,8] have been presented based on existent experimental data. However, satisfactory ESADs have not been given for a particular projectile-target system at some energies. There are also some microscopic OPs for $^4$He[9-11] obtained by double- or single-folding approaches. The real and imaginary parts are both determined, while their strengths need to be modified for different energies. Guo et al. proposed a microscopic OP by employing the Green function method. A substantial difference was observed between the calculated ESADs and the experimental results at larger angles.[12] The São Paulo potential,[13] a nonlocal double-folding model for the heavy-ion nuclear interaction, is a good choice for the systematic study of the OP. In Ref. [14], it was used to develop a parameter-free OP with the nuclear component in the form of $V_{\rm SPP}+i0.78V_{\rm SPP}$. This OP reproduces the ESADs of seven heavy-ion systems (for example, $^{12}$C+$^{12}$C and $^{16}$O+$^{208}$Pb) at different energies fairly well without any adjustments. However, it is not suitable for $^4$He+$^{12}$C due to the fact that the experimental data of refractive $^4$He-nucleus systems require a different shape for the absorptive potential.[15] Based on the SPP, we analyze the $^4$He+$^{12}$C elastic scattering at intermediate energies with the optical model (OM). The SPP is selected as the real part of the nuclear component of the OP, and an adjustable Woods–Saxon (WS) form is assumed for the imaginary part. The existing experimental ESADs obtained from EXFOR[16] are fitted in the OM calculations by varying the parameters of the imaginary potential. The best-fit parameters are determined and the ESADs calculated with these parameters are in good agreement with the experimental data. Furthermore, a fixed imaginary part is found to be possible to construct a common OP (COP) for $^4$He+$^{12}$C at energies ranging from 26$A$ MeV to 60$A$ MeV. This COP is tested in the CDCC calculations for the reactions of $^6$Li+$^{12}$C and $^6$He+$^{12}$C. The ESADs of the former are reproduced well. The ESAD of the latter system is underestimated by the calculation at larger angles, which is consistent with the work of Keeley et al.[17] The calculations based on OM and CDCC are performed by the code FRESCO.[18] As usual, the OP between $^4$He and $^{12}$C is composed of a Coulomb part and a complex nuclear part. The nuclear potential $U_{\rm N}$ is expressed as $$\begin{align} U_{\rm N}(R)=V(R)+iW(R).~~ \tag {1} \end{align} $$ The SPP adopting a zero-range approach[13] is chosen as the real part ($V$), $$\begin{align} V_{\rm SPP}(R)=\,&V_{0}\int\rho_{\rm m1}({\boldsymbol r}_{1})\rho_{\rm m2}({\boldsymbol r}_{2})\delta({\boldsymbol R}-{\boldsymbol r}_{1}+{\boldsymbol r}_{2})\\ &\cdot e^{-4v^2/c^2}d{\boldsymbol r}_{1}d{\boldsymbol r}_{2},~~ \tag {2} \end{align} $$ where the effective interaction is in the form of a delta function, and $V_{0} =-456$ MeV$\cdot$fm$^3$. The matter density $\rho_{\rm m}$ is described by the two-parameter Fermi (2 pF) distribution[13] $$ \rho_{\rm m}(r)=\frac{\rho_{\rm m0}}{1+\exp\Big({\frac{r-R_{\rm m0}}{a_{\rm m}}}\Big)},~~ \tag {3} $$ where $\rho_{\rm m}$ of $^{12}$C is determined by the SPP systematics for the densities of heavy ions ($R_{\rm m0}=2.159$ fm, $a_{\rm m}=0.56$ fm).[13] $^4$He is too light to be covered by this systematics, thus specific values of $R_{\rm m0}=1.162$ fm and $a_{\rm m}=0.42$ fm are used.[19] The exponential term includes the Pauli nonlocality between the projectile and the target nuclei. The energy dependence of the potential is implied in the relation between the local relative velocity $v$ and the kinetic energy.[19] The imaginary part ($W$) uses a WS form factor with the radius $R_{\rm I0}=r_{\rm I0}A_{\rm t}^{1/3}$. For the Coulomb interaction $U_{\rm C}$, we assume that the Coulomb radius parameter $R_{\rm C}=1.3A_{\rm t}^{1/3}$ fm. With this OP, experimental ESADs of $^4$He on a $^{12}$C target at different energies[20-24] (Table 1) are analyzed in the OM framework. The adjustable parameters in the imaginary part, i.e., $W_{0}$, $r_{\rm I0}$ and $a_{\rm I}$, are determined by fitting the experimental data. Good agreement between the theoretical calculations and the experimental data is achieved (Fig. 1). The best-fit $W_{0}$, $r_{\rm I0}$ and $a_{\rm I}$ (set 1) fall into the ranges of 14.2–16.6 MeV, 1.70–1.80 fm and 0.50–0.60 fm, respectively, except the data at 30$A$ MeV and 34.75$A$ MeV. Certain elastic scattering data may lead to potentials with different parameters within the error limit. To keep consistency with the parameters of the other data sets, the diffuseness $a_{\rm I}$ of the imaginary potential derived from the data at 30$A$ MeV is reduced to 0.55 fm, the middle value of the diffuseness fitted by the other data sets. The fitting procedure is redone for this data. The obtained values of $r_{\rm I0}$ and $W_{0}$ enter the ranges of the other data (see Table 1) as well. This can be understood as the correlation among $r_{\rm I0}$, $a_{\rm I}$ and $W_{0}$ within certain limits.[25] We carry out the same thing for the data at 34.75$A$ MeV except that $W_{0}$ is tuned manually to make the fit better at larger angles. The two new sets of parameters are tagged with set 2 in Table 1. As listed in Table 1, there is no explicit dependence of the imaginary-part parameters on energy, and their values of each data set are close to each other. This makes it possible to use a fixed-WS imaginary part for these data sets. To obtain its appropriate parameters, we perform a WS-function fit based on the points given by the discretization of these WS functions plotted in Fig. 2 (set 2 of data sets at 30$A$ MeV and 34.75$A$ MeV, and set 1 of the other data sets). The outcomes $W_0=15.48$ MeV, $r_{\rm I0}=1.760$ fm and $a_{\rm I}=0.552$ fm. With this fixed imaginary part and the SPP used as the real part, the COP was constructed and applied to the OM calculations of $^4$He+$^{12}$C. The calculated ESADs agreed with the experimental data well without adjustments (Fig. 1) and the $\chi^2/N$ values only became slightly larger (Table 1).
cpl-33-7-072401-fig1.png
Fig. 1. Comparisons between OM calculations with WS imaginary part and experimental ESAD data of $^4$He+$^{12}$C at different energies.[20-24] The data set at 36.25$A$ MeV is on the real scale, and the others are offset by factors of 100 for good views.
Table 1. The best-fit parameters of the imaginary parts at different energies. Here $\chi_{\rm best}^2/N$ and $\chi_{\rm COP}^2/N$ correspond to the best-fit OPs and the COP, respectively. The last column gives the references of the experimental data.
$E$ ($A$ MeV) Set $W_{0}$ (MeV) $r_{{\rm I}0}$ (fm) $a_{\rm I}$ (fm) $\chi_{\rm best}^2/N$ $\chi_{\rm COP}^2/N$ References
26 1 16.63 1.707 0.550 31.161 32.652 Ref. [20]
30 1 19.44 1.595 0.673 20.978 – Ref. [21]
2 14.77 1.799 0.550 23.959 24.421
34.75 1 20.54 1.596 0.644 362.158 – Ref. [22]
2 16.00 1.748 0.550 494.424 521.180
36.25 1 15.33 1.788 0.539 11.106 12.945 Ref. [21]
41.5 1 14.23 1.795 0.504 13.149 18.881 Ref. [23]
43.125 1 15.44 1.778 0.544 12.072 13.260 Ref. [21]
60 1 15.99 1.708 0.608 2.760 6.127 Ref. [24]
The COP was utilized in the OM calculations of $^4$He+$^{12}$C at other energies as well to analyze the applicable energy range. Using the COP in the reaction at 27.5$A$ MeV,[26] the experimental ESAD was reproduced successfully as those data sets in Fig. 1. At 12.625$A$ MeV,[27] 13.525$A$ MeV[28] and 16.25$A$ MeV,[29] the calculated ESADs are acceptable before 30$^{\circ}$ (c.m. system), however, it underestimates the experimental data at larger angles. At 96.5$A$ MeV,[30] there is also an evident discrepancy between the computed ESAD and the experimental one. As a result we infer that this COP is appropriate within the energy range roughly from 26$A$ MeV to 60$A$ MeV.
cpl-33-7-072401-fig2.png
Fig. 2. The imaginary parts of the WS form of OPs obtained by fitting the experimental ESADs of $^4$He+$^{12}$C at different energies (set 2 of data sets at 30$A$ MeV and 34.75$A$ MeV and set 1 of the other data sets).
Further, the COP was also used in the three-body CDCC calculations of $^6$Li+$^{12}$C at 28.1$A$ MeV,[31] 35$A$ MeV[32] and 53$A$ MeV,[33] and $^6$He+$^{12}$C at 38.3$A$ MeV.[34] The breakup couplings to the elastic scattering are considered. Two-body cluster structures (core+valence) are assumed for projectiles $^6$Li=$^4$He+$d$[35] and $^6$He=$^4$He+2$n$.[1] In these calculations, the OPs for core+target and valence+target are needed. The COP is used as the interaction between $^4$He and $^{12}$C. The global OP proposed in Ref. [36] is used for $d+^{12}$C and 2$n+^{12}$C. The spin of the valence $d$ of $^6$Li is ignored, which leads to the degeneration of three original $l=2$ resonances at the relative energy $E_{\rm rel}\approx 2$ MeV.[35] The binding potential (BP) of $^4$He+$d$ is of the WS form with a radius $R_{\rm 0}=1.9$ fm and a diffuseness $a_{\rm 0}=0.65$ fm.[17,37,38] For the ground state, a depth of 77.46 MeV is used in the BP to give the binding energy of 1.471 MeV.[37] The depths for the non-resonant continuum and the resonance are 77.5 MeV and 79.44 MeV, respectively. In the construction of the model space, relative $s$-, $p$- and $d$-waves between $^4$He and $d$ are included. The $^4$He+$d$ continuum is discretized in the momentum space with a step $\Delta k=0.15$ fm$^{-1}$ up to 1.5 fm$^{-1}$. The resonance of $l=2$ is specifically set as a single bin with a width of 3 MeV. These bin states are generated by using the mid-point method. To reproduce the experimental ESADs better, renormalization factors $N_{\rm R}$ for the real parts of nuclear potentials are introduced[38] (Table 2), and good results are obtained (Fig. 3).
Table 2. Renormalization factors $N_{\rm R}$ for the real parts of the OPs used in CDCC calculations of $^6$Li+$^{12}$C.
$E$ ($A$ MeV) $N_{\rm R}$
28.1 0.85
35, 53 0.9
The improved di-neutron cluster model of $^6$He given by Moro et al.[1] is used for $^6$He+$^{12}$C. The BPs for the ground state, the non-resonant continuum and the $l=2$ resonance are all taken from Ref. [1]. Considering the similarities between the structures of $^6$He and $^6$Li, the calculation settings similar to those of $^6$Li+$^{12}$C are applied to the calculation of $^6$He+$^{12}$C. Particularly, $N_{\rm R}=0.9$ is assumed, which is analogous to $^6$Li+$^{12}$C at 35$A$ MeV. Compared with the experimental ESAD, the calculation strongly underestimates the elastic differential cross section at angles larger than 10$^{\circ}$ (Fig. 3). This means that the calculation considering the breakup couplings overestimates the absorption of $^4$He+$^{12}$C heavily. This phenomenon has been reported in Ref. [17], where the OP used for $^4$He+$^{12}$C is derived from the experimental ESAD at an adjacent energy 34.75$A$ MeV.[22] With the CDCC calculation using the COP, we have confirmed this phenomenon. Moreover, as an unstable halo nucleus, $^6$He was expected to cause a stronger absorption than $^6$Li when scattering on the carbon target at similar energies. However, $^6$He+$^{12}$C at 38.3$A$ MeV has a larger elastic differential cross section (i.e., a weaker absorption) than $^6$Li+$^{12}$C at 35$A$ MeV (Fig. 3). Both the calculated results and the experimental data have this behavior, which is also consistent with Ref. [17].
cpl-33-7-072401-fig3.png
Fig. 3. Comparisons between the CDCC calculations and the experimental ESAD data of $^6$Li+$^{12}$C[31-33] and $^6$He+$^{12}$C[34] at different energies. The data set at 53$A$ MeV is on the real scale, and the others are offset by factors of 10000.
In summary, the elastic scattering of $^4$He on $^{12}$C has been studied comprehensively with an OM. It is found that the ESAD is more sensitive to the real part than to the imaginary part of the optical potential. Meanwhile, the SPP considering the Pauli nonlocality includes the energy dependence naturally. Using a nuclear potential with the SPP as its real part and a fixed WS function as its imaginary part, the optical potential reproduces the ESADs of existing data within the energy range of 26–60$A$ MeV. It interprets an ultra weak energy dependence of the imaginary part in this energy range. The CDCC calculations by using the COP indicate the reliability of the COP in the $^4$He-cluster-involved breakup reactions. On the other hand, the puzzle of the weaker absorption of $^6$He+$^{12}$C than of $^6$Li+$^{12}$C is confirmed. We thank Thompson I J, Moro A M and Pang D Y for their help in the usage of FRESCO. Thanks are also given to Chamon L C for his help in the consideration of the SPP. References Improved di-neutron cluster model for He 6 scatteringTwo-neutron correlations in microscopic wave functions of 6 He, 8 He, and 12 CCluster Structure in Be Isotopes within Point-Coupling Covariant Density FunctionalObservation of Enhanced Monopole Strength and Clustering in Be 12 New Measurements for 8 He Excited StatesChapter III. Effects of Deuteron Virtual Breakup on Deuteron Elastic and Inelastic ScatteringGlobal alpha-nucleus optical potentialApplication of the Jeukenne-Lejeune-Mahaux folding model to α -nucleus elastic scatteringMicroscopic optical potential for α -nucleus elastic scattering in a Dirac-Brueckner-Hartree-Fock approachEnergy-dependent optical model potentials for α and deuteron with 12 C He 4 microscopic optical model potentialToward a global description of the nucleus-nucleus interactionA parameter-free optical potential for the heavy-ion elastic scattering processMissing monopole strength in 58 Ni and uncertainties in the analysis of α -particle scatteringTowards a More Complete and Accurate Experimental Nuclear Reaction Data Library (EXFOR): International Collaboration Between Nuclear Reaction Data Centres (NRDC) Li 6 and He 6 elastic scattering from C 12 and the effect of direct reaction couplingsCoupled reaction channels calculations in nuclear physics α + α scattering reexamined in the context of the São Paulo potentialElastic scattering of 104 MeV alpha particlesThe (α, α), (α, α′) and (α, 3He) reactions on 12C at 139 MeV166 MeV elastic and inelastic alpha-particle scattering; macroscopic and microscopic analysisIsoscalar electric multipole strength in 12 C The form factor of the real part of the α-nucleus potential studied over a wide energy rangeα + 12C scattering at 110 MeV: Has exotic 7.65 MeV Hoyle’s state signatures “alpha-condensate” structure?Measurement and folding-potential analysis of the elastic ?-scattering on light nucleiDeformation parameter of 12C via 12C(α, α') and 12C(α, α'α) reactionsSystematics of the bimodal isoscalar giant dipole resonanceBreakup effects of 6,7li on elastic and inelastic scattering from 12C at 18–28 MeV/nucleonElastic scattering of 210 MeV Li 6 ions from C 12 and Ni 58 and unique nucleus 6 optical potentialsElastic scattering of 318 MeV Li 6 from C 12 and Si 28 : Unique phenomenological and folding-model potentialsCoupling effects in the elastic scattering of 6 He on 12 C Energy and target dependence of projectile breakup effect in elastic scattering of Li 6 Global deuteron optical model potential for the energy range up to 183 MeVThe 6Li exclusive breakup on 28Si at 13 MeVDynamic polarization potential due to 6 Li breakup on 12 C
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