Systematical study of the optical potential for systems like A+ 58Ni from sub-barrier data analyses

Systematical study of the optical potential for systems like A¿58Ni from sub-barrier data analyses L. R. Gasques,1L. C. Chamon,1D. Pereira,1M. A. G. Alvarez,1E. S. Rossi, Jr.,1C. P. Silva,1G. P. A. Nobre,1and B. V. Carlson2 1Departamento de Fı ´sica Nuclear, Instituto de Fı ´sica da Universidade de Sa ˜ o Paulo, Caixa Postal 66318, 05315-970 Sa ˜ o Paulo, Sa ˜ o Paulo, Brazil 2Departamento de Fı ´sica, Instituto Tecnolo ´gico de Aerona ´utica, Centro Te ´cnico Aeroespacial, Sa ˜ o Jose ´dos Campos, Sa ˜ o Paulo, Brazil 共Received 19 December 2002; published 20 June 2003兲 Elastic scattering differential cross sections were measured for the 28Si⫹58Ni system at sub-barrier energies. The corresponding nuclear potential was compared with earlier results of systems like A⫹58Ni. The present data also allowed the determination of the 28Si nuclear density through an unfolding method. The experimentally extracted 28Si density values are compared with those previously obtained for the 4,6He, 12C, 16,18O nuclei. We present a critical discussion of the absolute precision obtained for the density parameters extracted from the data analyses. DOI: 10.1103/PhysRevC.67.067603 PACS number共s兲: 24.10.Ht, 21.10.Ft, 21.10.Gv We present elastic scattering angular distributions for the 28Si⫹58Ni system at sub-barrier energies. One of the purposes of the experiment was the determination of the nuclear potential for this system in the surface region. This method was successfully applied to several other systems involving the same target and the 12C, 16,18O nuclei as projectiles 关1–6兴. As discussed in these earlier works, at sub-barrier energies the imaginary part of the optical potential is negligible since the corresponding reaction cross section is very small. Therefore, the elastic scattering data analysis in this energy region unambiguously determines the real part of the interaction. In recent works 关7–10兴, we have developed a nonlocal model for the nuclear interaction that is based on quantum effects related to the exchange of nucleons between the target and projectile. The nonlocal model has provided a good description of the elastic and inelastic scattering, transfer, and fusion processes for several systems in a wide energy range 关3,5,6,8–11兴. The model has also provided good predictions for an extensive systematics of potential strengths extracted from heavy-ion elastic scattering data analyses 关7兴. In the present work, we compare the potential for the 28Si⫹58Ni system with the results of such a systematics and also with other results for systems involving the same target nucleus, i.e., 12C, 16,18O⫹58Ni 共from Refs. 关3,5,6兴兲, and 4,6He⫹58Ni 共the data for these systems can be found in Ref. 关13兴, but the corresponding nuclear potentials were not published so far兲. If the nonlocal model is assumed for the interaction and the density of one nucleus is known, an unfolding method can be used to extract the nuclear density of the other nucleus from the elastic scattering data analyses. The method has already been successfully applied in the experimental determination of densities for the 4,6He, 12C, and 16,18O nuclei 关5,6,12,13兴. In the present work, we use this method in the data analyses of the 28Si⫹58Ni system at sub-barrier energies, with the aim of obtaining the 28Si nuclear density in the surface region. We compare the 28Si extracted density values with those experimentally obtained for the other nuclei. In this work, we also discuss the absolute precision for the experimentally extracted density parameters and the consistency of the results with an earlier extensive systematics of heavy-ion densities 关7兴. The measurements were made using the 28Si beam from the Sa ˜ o Paulo 8UD Pelletron Accelerator, Brazil. The detecting system was already described in Ref. 关1兴. Figure 1 shows the elastic scattering cross sections for the 28Si⫹58Ni system. In the optical model calculations, we have adopted a procedure similar to that described in the analyses of the sub-barrier data for several systems 关1–6兴. As a first step, we 40 60 80 100 120 140 160 180 0.6 0.7 0.8 0.9 1.0 0.9 1.0 0.9 1.0 28Si + 58Ni ELab = 77.0 MeV ELab = 75.5 MeV ELab = 74.0 MeV σel. / σRuth θcm (degree) FIG. 1. Elastic scattering angular distributions for the 28Si ⫹58Ni system at sub-barrier energies. The solid lines represent optical model predictions, in which the nonlocal model is assumed for the real part of the interaction. PHYSICAL REVIEW C 67, 067603 共2003兲 0556-2813/2003/67共6兲/067603共4兲/$20.00 ©2003 The American Physical Society67 067603-1 have assumed a Woods-Saxon shape for the real and imaginary parts of the optical potential. Our experiments indicate that peripheral reaction channels present no significant cross section at the energy region in which the data were obtained. Taking this into account, the parameters of the imaginary part of the potential were taken so as to provide small strengths in the surface region and complete internal absorption from barrier penetration, through a large imaginary potential internal to the barrier and a regular boundary condition at the origin. With these conditions, the elastic scattering cross section predictions are quite insensitive to variations of the imaginary potential parameters. The depth and diffuseness parameters of the real part of the optical potential were searched for the best data fits. For each angular distribution, we found a family of potentials that give equivalent data fits. The point at which these potentials cross defines the sensitivity radius, where the value of the potential is determined without ambiguity 共see, for example, Refs. 关3–6兴兲. At subbarrier energies, the sensitivity radius is energy dependent and the nuclear potential can therefore be obtained over a large range of surface radial distances by varying the incident energy. This dependence determines the diffuseness of the potential, as can be observed in Fig. 2 for the 4,6He, 12C, 16,18O, 28Si⫹58Ni systems. The elastic scattering data analyses for different systems in a wide energy range have resulted in phenomenological optical potentials with significant dependence on the bombarding energies 关14兴. We have developed a model that associates this dependence with nonlocal quantum effects 关7–10兴. Within this model, the bare interaction VNis connected with the folding potential VFthrough VN共R,E兲⬇VF共R兲e⫺8[E⫺VC(R)⫺VN(R,E)]/ ␮ c2.共1兲 In the present work, the nonlocality correction has little effect since we analyze near barrier data. In this energy region the nonlocal potential is almost identical to the folding one 关see Eq. 共1兲兴. The folding potential depends on the densities of the two partners in the collision VF共R兲⫽ 冕 ␳ 1共r1兲 ␳ 2共r2兲u0共R ជ ⫺r ជ 1⫹r ជ 2兲dr ជ 1dr ជ 2.共2兲 In Ref. 关7兴a systematization of nuclear densities has been proposed, based on an extensive study involving experimental and theoretical densities. The two-parameter Fermi 共2pF兲 distribution was adopted to describe the nuclear densities, and a distinction between nucleon and matter distributions was made by taking into account the finite size of the nucleon 关7兴. The radii of the 2pF distributions are well described by R0⫽1.31A1/3⫺0.84 fm, 共3兲 where Ais the number of nucleons in the nucleus. The nucleon and matter densities present average diffuseness values of a ¯ N⫽0.50 fm and a ¯ M⫽0.56 fm, respectively. Owing to specific nuclear structure effects 共single particle and/or collective兲, the parameters R0and ashow small variations around the corresponding average values throughout the periodic table. However, as far as the folding potential is concerned, the effects of the structure of the nuclei are mostly present at the surface 共of the potential兲and are mainly related to the diffuseness parameter 共of the densities兲关7兴. The extensive optical potential systematics of Ref. 关7兴 was performed within this context. The experimental potential strengths were described within 25% precision, by combining Eqs. 共1兲and 共2兲through two different methods: 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 u0(r ជ ). As shown in Ref. 关7兴, both alternatives are equivalent in describing the heavy-ion nuclear potential. In Ref. 关7兴, we also demonstrated that the folding potential arising from 2pF distributions with slightly different diffuseness parameters can be appropriately simulated by considering the folding of two distributions with the same diffuseness value: a⫽(a1⫹a2)/2. This procedure was used in the present work to describe the experimental potential strengths 共see the solid lines in Fig. 2兲. Equation 共3兲gives accurate results for the radii of heavy nuclei, but it fails in the case of light nuclei in which the effects of structure are more significant 关7兴. Therefore, for the 4,6He we have also considered more realistic radii as obtained in Refs. 关13,15兴 共see Table I兲. The values of the matter diffuseness were considered as free parameters and adjusted to reproduce the experimental potential strengths. The resulting values are presented in Table I. The results obtained with the nonlocal model for the elastic scattering cross sections of the 28Si ⫹58Ni system are presented in Fig. 1 共solid lines兲. The theoretical calculations fail to reproduce the oscillations observed in the dataset. We believe that such oscillations are 81012 10-1 100 - V (MeV) R (fm) 4He + 58Ni 6He + 58Ni 12C + 58Ni 16O + 58Ni 18O + 58Ni 28Si + 58Ni FIG. 2. The nuclear potential strength as a function of the sensitivity radius for the 4,6He, 12C, 16,18O, and 28Si⫹58Ni systems. The lines represent the results of the nonlocal model, using the matter distributions of the nuclei within the zero-range approach 共solid lines兲or the nucleon densities with the M3Y interaction 共dashed lines兲. The parameters of the densities involved in the calculations are presented in Table I and Fig. 3. BRIEF REPORTS PHYSICAL REVIEW C 67, 067603 共2003兲 067603-2 due to the coupling to some particular reaction channel. However, we have not investigated this possibility in the present work, because our intent is to show the quality of the predictions of the nonlocal model. An inspection of Fig. 1 shows that such predictions are in agreement with the average behavior of the data. The values for the matter diffuseness presented in Table I should be compared with the average diffuseness obtained in the extensive systematics of Ref. 关7兴:a ¯ M⫽0.56 fm. In the systematics, a dispersion 共standard deviation兲of about 0.025 fm around the average value was found and was associated with effects of the structure of the nuclei throughout the periodic table. The 4He⫹58Ni and 18O⫹58Ni systems present matter diffuseness differing from the average value (a ¯ M ⫽0.56 fm) by about two standard deviations. In the 18O case, this behavior is connected with the two off-shell valence neutrons 关6兴. On the other hand, from electron scattering experiments, the diffuseness of the 4He charge distribution has already been found to be significantly smaller than those for heavy ions 关16兴. We mention that the large uncertainty of the potential strength for the 6He⫹58Ni system 共see Fig. 2兲results in a large uncertainty 共about 0.03 fm兲for the corresponding density diffuseness. If the nonlocal model is assumed for the interaction, and the density of one nucleus is known, an unfolding method can be used to extract the ground-state nuclear density of the other nucleus through the elastic scattering data analyses. Here, the double-folding potential is considered in the usual form: the nucleon densities and the M3Y effective nucleonnucleon interaction are adopted in Eq. 共2兲. The method has been successfully applied in the experimental determination of densities for the 4,6He, 12C, and 16,18O nuclei, and the resulting densities have been shown to be consistent with the experimental electron scattering data 关5,6,12,13兴.Inthe present paper, we describe the method in a quite concise manner, and we invite the reader to obtain further details in a complete discussion presented in Refs. 关5,6,12兴. In the data analyses, we have used a theoretical Dirac-Hartree- Bogoliubov density for the 58Ni nucleus 关17兴, since the resulting predictions for electron scattering cross sections are in very good agreement with the data 关5,12兴. The 28Si density was obtained with a procedure similar to that used in the determination of the potential strengths at the sensitivity radii. We have assumed the 2pF distribution to describe the 28Si density. The diffuseness (aN) and radius (R0) were searched for the best elastic scattering data fits, with the ␳ 0 parameter determined by the normalization condition. For each angular distribution, we have found a family of densities that give equivalent data fits. These densities cross at the sensitivity radius, where the value of the density is determined without ambiguity. Figure 3 共top兲contains the experimental nucleon density values for 4,6He, 12C, 16,18O, and 28Si at the corresponding sensitivity radii, as obtained from elastic scattering data analyses of several angular distributions for A⫹58Ni systems. The statistical error bars for the density values have been determined as described in Ref. 关5兴. We have adjusted 2pF distributions to the experimental density values 共solid lines in Fig. 3, top兲, using the corresponding radii of Eq. 共3兲 共see Table I兲and considering the nucleon diffuseness as a free parameter. The results obtained for the values of the TABLE I. The diffuseness values of the matter distributions and the radii of the projectiles considered in the calculations of the nuclear potentials. System Model R0共fm兲aM共fm兲 4He⫹58Ni Eq. 共3兲1.24 0.50 Ref. 关13兴1.64 0.48 Ref. 关15兴1.26 0.50 6He⫹58Ni Eq. 共3兲1.54 0.54 Ref. 关15兴1.23 0.56 12C⫹58Ni Eq. 共3兲2.16 0.56 16O⫹58Ni Eq. 共3兲2.46 0.57 18O⫹58Ni Eq. 共3兲2.59 0.61 28Si⫹58Ni Eq. 共3兲3.14 0.58 FIG. 3. Top—experimental nucleon density values at the sensitivity radii for the 4,6He, 12C, 16,18O, and 28Si nuclei, extracted from elastic scattering data analyses at sub-barrier energies. The solid lines correspond to two-parameter Fermi distributions, with radii obtained from Eq. 共3兲共see Table I兲and diffuseness values as indicated in the figure. The dashed and dotted lines correspond to realistic SF and 2pF distributions obtained in Refs. 关15,13兴, respectively. Bottom—the density values have been decreased by 25%, and the corresponding diffuseness values have been recalculated. BRIEF REPORTS PHYSICAL REVIEW C 67, 067603 共2003兲 067603-3 diffuseness are indicated in Fig. 3 共top兲. The dashed lines in Fig. 2 represent the nuclear potentials obtained using these 2pF distributions for the nucleon densities of the projectiles and the M3Y effective nucleon-nucleon interaction. As demonstrated earlier 关5,6,12,13兴, information about the density in the surface region is obtained from sub-barrier elastic scattering data analyses, while intermediate energy data analyses probe the density in much inner distances. This fact has already been used 关13兴to determine a 2pF distribution for the 4He nucleus 共dotted line in Fig. 3, top兲. The corresponding results for the radius and diffuseness are indicated in Table I and Fig. 3 共top兲, respectively. In Ref. 关15兴, the nuclear 4,6He densities were obtained from elastic scattering data analyses for the 4,6He⫹psystems at 700 MeV/ nucleon, using the Glauber multiple scattering theory for the interaction. In that work, different parametrizations for the 6He density were tested, including symmetrized Fermi 共SF兲 distributions. For the purpose of comparison, the corresponding SF distributions for the 4,6He nuclei are included as dashed lines in Fig. 3 共top兲, and the corresponding radii and diffuseness are presented in Table I and Fig. 3 共top兲. The values found in the present work for the nucleon diffuseness of the 12C, 16,18O, and 28Si nuclei are somewhat greater than the average value (a ¯ N⫽0.50 fm) obtained in the systematics of Ref. 关7兴. This seems to be an inconsistent result. However, the determination of density values at the sensitivity radii is exposed to systematical errors that arise from several sources: 共i兲the dependence of the results on the shape assumed for the projectile; 共ii兲the theoretical density assumed for the target; 共iii兲the contribution of the polarization potential that has not been included in our analyses, etc. In previous works 关5,6,12,13兴, we have estimated the systematical error of the density values as 20% to 30%. With the aim of illustrating the effect of this error in the determination of the diffuseness of the density distributions, in Fig. 3 共bottom兲we show experimental density values decreased by 25% in comparison with those of Fig. 3 共top兲. The corresponding values of the diffuseness are about 7% smaller than those obtained without the reduction 共see Fig. 3兲, and they approach the average diffuseness obtained in Ref. 关7兴. In this work, we have presented elastic scattering data at sub-barrier energies for the 28Si⫹58Ni system. The corresponding data extracted nuclear potentials at the sensitivity radii have been compared with those for the 4,6He, 12C, and 16,18O⫹58Ni systems. All the results for the potentials are in very good agreement with the nonlocal model in the context of the systematics for the nuclear densities of Ref. 关7兴.We have also applied the nonlocal model to extract density values from data analyses. In this case, larger systematical errors are expected in comparison with those involved in the determination of potential strengths. Within the estimate of the systematical uncertainties, the density diffuseness experimentally extracted from data analyses is also in good agreement with the systematics of Ref. 关7兴. The uncertainty of the absolute values for the diffuseness is about 7%, but the comparison of results for different nuclei is probably less exposed to systematical errors. In fact, the use of the same method 共and same target nucleus兲would provide partial cancellation of the effects of the systematical errors. Taking this into account, our results clearly demonstrate that the 4He and 18O have, respectively, the smallest and the greatest diffuseness among all the nuclei studied in the present work. The 12C, 16O, and 28Si nuclei present similar diffuseness approaching the average value of Ref. 关7兴. Besides the systematical error, the diffuseness of the 6He exotic nucleus has been determined within greater statistical uncertainty (⬇0.03 fm), but our analysis indicates that its diffuseness is comparable with those of heavy nuclei and much greater than the value found for the neighboring stable 4He. This work was partially supported by Fundac¸a ˜ odeAmparo a `Pesquisa do Estado de Sa ˜ o Paulo 共FAPESP兲and Conselho Nacional de Desenvolvimento Cientı ´fico e Tecnolo ´gico 共CNPq兲. 关1兴L.C. Chamon et al., Nucl. Phys. A582, 305 共1995兲. 关2兴L.C. Chamon et al., Nucl. Phys. A597, 253 共1996兲. 关3兴M.A.G. Alvarez et al., Nucl. Phys. A656, 187 共1999兲. 关4兴C.P. Silva et al., Nucl. Phys. A679, 287 共2001兲. 关5兴L.R. Gasques et al., Phys. Rev. C 65, 044314 共2002兲. 关6兴E.S. Rossi, Jr., et al., Nucl. Phys. A707, 325 共2002兲. 关7兴L.C. Chamon et al., Phys. Rev. C 66, 014610 共2002兲. 关8兴M.A. Ca ˆndido Ribeiro et al., Phys. Rev. Lett. 78, 3270 共1997兲. 关9兴L.C. Chamon et al., Phys. Rev. Lett. 79, 5218 共1997兲. 关10兴L.C. Chamon et al., Phys. Rev. C 58, 576 共1998兲. 关11兴L. R. Gasques et al. 共unpublished兲. 关12兴M.A.G. Alvarez et al., Phys. Rev. C 65, 014602 共2002兲. 关13兴L.R. Gasques et al., Phys. Rev. C 67, 024602 共2003兲. 关14兴M.E. Brandan and G.R. Satchler, Phys. Rep. 285, 142 共1992兲, and references therein. 关15兴G.D. Alkhazov et al., Phys. Rev. Lett. 78, 2313 共1997兲. 关16兴H. De Vries, C.W. De Jager, and C. De Vries, At. Data Nucl. Data Tables 36, 495 共1987兲. 关17兴B.V. Carlson and D. Hirata, Phys. Rev. C 62, 054310 共2000兲. BRIEF REPORTS PHYSICAL REVIEW C 67, 067603 共2003兲 067603-4