Global and consistent analysis of the heavy-ion elastic scattering and fusion processes

Global and consistent analysis of the heavy-ion elastic scattering and fusion processes L. R. Gasques,1L. C. Chamon,1D. Pereira,1M. A. G. Alvarez,1E. S. Rossi, Jr.,1C. P. Silva,1and B. V. Carlson2 1Laboratório Pelletron, Instituto de Física da Universidade de São Paulo, 05315-970 São Paulo, São Paulo, Brazil 2Departamento de Física, Instituto Tecnológico de Aeronáutica, Centro Técnico Aeroespacial, São José dos Campos, São Paulo, Brazil (Received 17 October 2003; published 10 March 2004) We have developed a model for the nuclear interaction which is based on the effects of the Pauli nonlocality. In earlier works, we have successfully used this interaction to describe the elastic scattering for several systems in a very wide energy range. In the present work, we have checked the validity of the same interaction in the description of about 2500 fusion cross section data for 165 different systems. By introducing only one energyand system-independent effective parameter, the nonlocal model describes the global behavior of the fusion process with good precision. DOI: 10.1103/PhysRevC.69.034603 PACS number(s): 24.10.Ht, 25.60.Pj, 25.70.Jj The heavy-ion fusion process has been extensively studied over the last decades [1]. It is well known that fusion cross sections for heavy-ion systems have shown large enhancements at sub-barrier energies in comparison with theoretical predictions from the barrier penetration model [2]. These enhancements can be described by introducing effective barrier parameters, which have been studied in a global way for a large number of systems [2–4]. On the other hand, the enhancements have been explained for several particular systems by considering the internal structure of the participating nuclei through couped-channel calculations (e.g., Refs. [5,6]). A few models have also been presented to describe the elastic scattering process and the energy and system dependences of the corresponding optical potential [7]. However, the consistency between the models for the fusion and elastic scattering processes has only been verified in certain particular cases (e.g., Ref. [8]). We have developed a model for the nuclear interaction which has been successful in describing the elastic scattering for several systems in a wide energy range [9–15]. The model is based on the effects of the Pauli nonlocality and is totally parameter free. In this work, we use this interaction in the description of about 2500 fusion cross section data [16–53]for 165 different systems from sub-barrier to high energies. Within the nonlocal model, the bare interaction VNis connected with the folding potential VFthrough [54] VN共R,E兲=VF共R兲e−4v2/c2,共1兲 where cis the speed of light and vis the local relative velocity between the two nuclei, v2共R,E兲=2 ␮ 关E−VC共R兲−VN共R,E兲兴.共2兲 The folding potential 关Eq. 共3兲兴 can be obtained in two different ways 关54兴:共i兲using the nucleon distributions of the nuclei and an appropriate form for the nucleon-nucleon interaction, and 共ii兲using the matter distributions of the nuclei with a zero-range approach for v共r ជ 兲. We distinguish the matter density from the nucleon one by taking into account the finite size of the nucleon. Both alternatives are equivalent in describing the heavy-ion nuclear potential 关54兴, and we have adopted the zero-range approach to describe the fusion process: VF共R兲= 冕 ␳ 1共r1兲 ␳ 2共r2兲v共R ជ −r ជ 1+r ជ 2兲dr ជ 1dr ជ 2.共3兲 With the aim of providing a global description of the nuclear interaction, we proposed 关54兴an extensive systematization of nuclear densities, based on experimental charge distributions and theoretical densities calculated through the DiracHartree-Bogoliubov model. In that work, we adopted the two-parameter Fermi 共2pF兲distribution to describe the nuclear densities. The radii of the 2pF distributions are well described by R0= 1.31A1/3 − 0.84 fm, 共4兲 where Ais the number of nucleons of the nucleus. The matter densities present an average diffuseness value a =0.56 fm. Owing to specific nuclear structure effects 共single particle and/or collective兲, the parameters R0and a show small variations around the corresponding average values throughout the periodic table. This systematization of the nuclear distributions is essential to obtain a parameter-free interaction, since the folding potential depends on the densities of the partners in the collision. In the present work, we use the nonlocal model in the context of this systematics, i.e., assuming the average diffuseness value and Eq. 共4兲for the radii of the distributions. Therefore, the interaction does not contain any free parameter and is quite appropriate to connect experimental results for different systems in a realistic manner. In the present work, we have also calculated the Coulomb potential through a folding procedure using the realistic charge densities of Ref. 关54兴. In the context of the barrier penetration model (BPM), the effective potential is a sum of the Coulomb, nuclear, and centrifugal parts: Veff共R,E兲=VC共R兲+VN共R,E兲+ᐉ共ᐉ+1兲ប2 2 ␮ R2.共5兲 The fusion cross section is associated with the transmitted flux through PHYSICAL REVIEW C 69, 034603 (2004) 0556-2813/2004/69(3)/034603(5)/$22.50 ©2004 The American Physical Society69 034603-1 ␴ BPM共E兲= ␲ k2兺共2ᐉ+1兲Tᐉ.共6兲 In our calculations, the sum in Eq. 共6兲is performed up to a maximum ᐉwave, which is the greatest ᐉvalue that results a pocket 共and a barrier兲in the corresponding effective potential. For ᐉwaves with effective barrier heights below the center of mass energy, we have approximated the effective potential by a parabola with curvature ប ␻ ᐉ. In such cases, the transmission coefficients can be obtained through the HillWheeler formula 关55兴: Tᐉ= 再 1 + exp 冋 2 ␲ 共VBᐉ−E兲 ប ␻ ᐉ 册 冎 −1,共7兲 ប ␻ ᐉ=冏ប2 ␮ d2Veff dR2冏RBᐉ 1/2 ,共8兲 where VBᐉand RBᐉare the barrier height and the corresponding radius, respectively. On the other hand, for ᐉwaves with effective barriers above the center of mass energy, instead of the Hill-Wheeler formula we have used the more appropriate WKB method: Tᐉ=关1 + exp共Sᐉ兲兴−1,共9兲 Sᐉ= 冕 R1 R2冑8 ␮ ប2关Veff共R,E兲−E兴dR,共10兲 where R1and R2are the classical turning points. At low energies, the WKB method results in values for the transmission coefficients quite different from those of the HillWheeler formula. In this case, we have defined the barrier curvature connecting expressions 共7兲and 共9兲as ប ␻ ᐉ=2 ␲ 共VBᐉ−E兲 Sᐉ .共11兲 Within the context of parabolic transmission coefficients, and considering RBᐉ⬇RB0and ប ␻ ᐉ⬇ប ␻ 0, Wong has demonstrated 关56兴that ␴ Wong共E兲=RB0 2ប ␻ 0 2Eln 再 1 + exp 冋 2 ␲ 共E−VBᐉ兲 ប ␻ 0 册 冎 .共12兲 With the aim of obtaining system-independent quantities, we have defined the following reduced cross section and energy: ␴ red =2E RB0 2ប ␻ 0 ␴ fus,共13兲 Ered =E−VB0 ប ␻ 0.共14兲 By using these adimensional quantities, Eq. 共12兲can be recast in the following system-independent form: ␴ red Wong =ln关1 + exp共2 ␲ Ered兲兴.共15兲 However, we have verified that, in some cases, the results obtained from the Wong expression 关Eq. 共12兲兴 present significant differences in comparison with those from the full BPM calculations 关Eq. 共6兲兴. Thus, in order to compare experimental results for very different systems, we have defined the experimental reduced fusion cross section through the following trivial equation: ␴ red exp = ␴ fus ␴ BPM ␴ red Wong.共16兲 The reduced fusion cross section data, according to Eq. (16), are presented in Fig. 1, as a function of the reduced energy [Eq. (14)]. The data set includes 165 quite different systems ranging in reduced mass from about 4 to 40 amu. A very good agreement between data and theoretical predictions is obtained for energies above the s-wave barrier 共Ered艌0兲for the whole set of systems, while sub-barrier data present large enhancements in comparison with the BPM calculations. The sub-barrier deviation is negligible for ␮ 艋8 and increases as a function of the reduced mass of the system in a relatively smooth manner. A similar behavior of the subbarrier fusion data has already been very well established [2]. An inspection of the slopes of the data in Fig. 1 indicates that the enhancements are more closely connected with the barrier curvature than with the barrier height itself. Taking into account all these considerations, we propose a simple model to describe the enhancements by introducing effective barrier curvatures. This effective parameter is assumed to be a simple linear function of the reduced mass of the system, as given in Eq. (17). The fit to the data results in ␭=0.1/amu: ប ␻ ᐉeff = 再 ប ␻ ᐉfor ␮ 艋8 amu ប ␻ ᐉ关1+␭共 ␮ −8兲兴 for ␮ 艌8 amu. 共17兲 Using Eq. 共17兲with ␭=0.1, the barrier penetration model describes with good precision the global behavior of the 2500 experimental fusion cross section data in the whole energy range 共see Fig. 2兲. In fact, it is possible to describe FIG. 1. The reduced fusion cross section as a function of the reduced energy, using barrier parameters obtained with the nonlocal model. The figure shows the results for different ranges of the reduced mass of the system. The solid lines represent the reduced Wong equation. L. R. GASQUES et al. PHYSICAL REVIEW C 69, 034603 (2004) 034603-2 the above-barrier data within 20% precision 共standard deviation兲and the sub-barrier data within an average factor of about 3. We consider this dispersion rather small because our analysis has been performed for a large number of different systems and over a very wide energy range 共from about 18 MeV below the barrier to 120 MeV above it兲with only one systemand energy-independent free parameter ␭. With the aim of including the whole data set, Figs. 1 and 2 have been made with reduced scales. Figures 3–5 present the results in usual scales for a few particular systems, in order to provide further examples of the quality of our predictions. We believe that the effective curvature simulates, in some way, the average effect of a large number of coupled channels that contribute to the fusion process. The effect of the couplings depends on the reduced mass of the system because heavier systems present a greater number of reaction channels and also larger coupling amplitudes (which are connected with the size of the nuclei). The dispersion observed in Fig. 2 is probably due [57]to particular strong couplings that were not included in our model, and is also connected with variations of the densities, arising from nuclear structure effects, which have an influence on the nuclear potential [54]. For example, the great isotopic dependence (see Fig. 6—top)of the sub-barrier fusion cross section for the 16O+144,148,150,152,154Sm systems is decreased by using the effective curvature (Fig. 6—bottom). Even so, the difference is not totally eliminated probably due to structure effects not included in our model. The effect of the nonlocality is not very significant at near-barrier energies (low velocities), where Eq. (1)indicates that VN共R,E兲⬇VF共R兲. The effect becomes greater as the energy increases, such that at energies of about 200 MeV/nucleon VN共R,E兲is about one order of magnitude less intense than the corresponding folding potential [9,10]. FIG. 2. The same as Fig. 1, considering the correction [Eq. (17) with ␭=0.1/amu]for the barrier curvatures. FIG. 3. The fusion cross section for the 12C+12C, 12C+16O, 16O+208Pb, and 58Ni+64Ni systems. The lines represent full barrier penetration model calculations with (solid lines)or without (dashed lines)including the effect of the effective curvature. FIG. 4. The same of Fig. 3 for the 16O+144,148,150,154Sm systems. FIG. 5. The fusion cross section for the 12C+27Al, 32S systems. The lines represent full barrier penetration model calculations with (dotted lines)or without (solid lines)the effect of the Pauli nonlocality. GLOBAL AND CONSISTENT ANALYSIS OF THE HEAVY-ION…PHYSICAL REVIEW C 69, 034603 (2004) 034603-3 However, there are few available fusion data at high energies. Thus, for most of the cases considered in this work, the theoretical results for the BPM cross sections obtained using Eq. (1)differ by less than 5% in comparison from those obtained considering VN共R,E兲=VF共R兲. Even so, we have found two cases in the present data set in which the energy is high enough to emphasize such a difference (see Fig. 5). Here the decrease of the cross section for higher energies is connected with the competition between centrifugal repulsion and nuclear attraction. The energy dependence of the nuclear interaction that arises from nonlocality plays an important role by providing further reduction of the cross section. On the other hand, the nonlocality has been fundamental in our description of the heavy-ion elastic scattering process, in which the energy dependence of the potential has been successfully taken into account by Eq. (1). Therefore, we consider the major reason for using the nonlocal interaction to be the goal of obtaining a unified description of both the elastic scattering and fusion processes, within a consistent model from the sub-barrier region to intermediate energies. In summary, our parameter-free nonlocal model for the nuclear interaction has been successful in describing the heavy-ion elastic scattering process from sub-barrier energies up to 200 MeV/nucleon. This interaction has also been successfully tested in some cases for inelastic scattering and transfer at sub-Coulomb and intermediate energies [11,12,15]. In the present work, we have also obtained good predictions for fusion cross sections, using the nonlocal interaction in the context of the barrier penetration model, for a very large number of different systems and over a wide energy range (from the sub-barrier region to energies about 15 MeV/nucleon, where there are still a few existing experimental data). We emphasize that the theoretical calculations have no free parameters, except for the systemand energyindependent one connected with the effective barrier curvature. We also emphasize that our model provides remarkable predictions for the light heavy-ion systems over nine orders of magnitude (see Fig. 1 for ␮ 艋8). In this range of reduced mass there is no need to include any correction in the barrier curvatures. Therefore, the parameter-free nonlocal model seems to be a good basis for studying the fusion process in important cases for astrophysics, which involve mainly light systems. 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