Determination of the 12C nuclear density through heavy-ion elastic scattering experiments

Determination of the 12C nuclear density through heavy-ion elastic scattering experiments L. R. Gasques,1L. C. Chamon,1C. P. Silva,1D. Pereira,1M. A. G. Alvarez,1E. S. Rossi, Jr.,1V. P. Likhachev,2B. V. Carlson,3 and C. De Conti3 1Departamento de Fı ´sica Nuclear, Instituto de Fı ´sica da Universidade de Sa ˜ o Paulo, Caixa Postal 66318, 05315-970 Sa ˜ o Paulo, SP, Brazil 2Departamento de Fı ´sica Experimental, Instituto de Fı ´sica da Universidade de Sa ˜ o Paulo, Caixa Postal 66318, 05315-970 Sa ˜ o Paulo, SP, Brazil 3Departamento de Fı ´sica, Instituto Tecnolo ´gico de Aerona ´utica, Centro Te ´cnico Aeroespacial, Sa ˜ o Jose ´dos Campos, SP, Brazil 共Received 14 December 2001; published 25 March 2002兲 Precise elastic scattering differential cross sections have been measured for the 12C⫹58Ni,208Pb systems at sub-barrier energies. The corresponding bare potentials have been determined at interaction distances larger than the respective barrier radii, and the results have been compared with those from an early extensive systematics for the nuclear potential. The present data have been combined with others for the 12C ⫹12C,208Pb systems at intermediate energies, in order to extract the 12C ground-state nuclear density through an unfolding method. DOI: 10.1103/PhysRevC.65.044314 PACS number共s兲: 24.10.Ht, 21.10.Ft, 21.10.Gv I. INTRODUCTION In this work, we present elastic scattering differential cross sections for the 12C⫹58Ni,208Pb systems at sub-barrier energies. One of the purposes of the experiments was the determination of the corresponding nuclear potentials in a surface region near the respective barrier radii. The method was earlier applied to several systems involving the 16O nucleus as projectile 关1–4兴. As discussed in these previous works, the imaginary part of the optical potential is negligible at sub-barrier energies due to the corresponding very small reaction cross sections. Thus, provided that a realistic shape in the surface region is assumed for the potential, the elastic scattering data analysis in this sub-barrier energy region unambiguously determines the real part of the interaction. The optical potential is composed of the bare and polarization potentials, the latter containing the contribution arising from nonelastic couplings. The real part of the polarization has been estimated earlier 关1–3兴through extensive coupled-channel calculations for the 16O sub-barrier data set, and represents about 10% in comparison with the bare interaction. A quite complete coupled-channel calculation has also been performed for the 12C⫹208Pb system 关5兴, and a good description of elastic, inelastic, transfer, and fusion cross section data has been obtained for energies above the barrier. An extrapolation of the calculation 共see Fig. 8 of Ref. 关5兴兲 to 54.5⭐Elab⭐57 MeV, an energy region which corresponds to the data of the present work, indicates that the polarization represents about 13% of the real part of the optical potential. Therefore, it is reasonable to assume that the experimentally extracted potential strengths for the 12C ⫹58Ni,208Pb systems at sub-barrier energies can be associated to the bare potential within about 10% precision. In a recent work 关6兴, an extensive systematics of optical potential strengths extracted from heavy-ion elastic scattering data analyses at low and intermediate energies was presented. The energy dependence of the nuclear potential has been accounted for within a model based on the nonlocal nature of the interaction 关6–9兴. The systematics indicates that the heavy-ion potential can be described in a global way, through a double-folding shape which basically presents a simple dependence only on the number of nucleons of the colliding nuclei. The results for the nuclear potential of the 12C⫹58Ni,208Pb systems obtained in the present work are in good agreement with such systematics for the bare interaction. If the nonlocal model is assumed for the interaction, an unfolding method can be used to extract ground-state nuclear densities from heavy-ion elastic scattering data analyses. The method has been successfully applied in the experimental determination of densities for the 16,18O nuclei 关10兴.Inthe present work, we apply the same procedure in the data analyses for the 12C⫹58Ni,208Pb systems at sub-barrier energies, with the aim of obtaining the 12C nuclear density at the surface region. The method is extended to the 12C⫹12C,208Pb systems at intermediate energies, and in this case information about the 12C density at much inner distances is obtained. The paper is organized as follows. In Sec. II, we present the experimental results and the determination of the bare potential strengths from optical model data analyses for the 12C⫹58Ni,208Pb systems. A brief summary of the nonlocal model, and a comparison between the present results and the early systematics for the nuclear potential are contained in Sec. III. The extraction of the 12C density and the limitations of the method are discussed in details in Sec. IV. Section V contains the main conclusions. II. EXPERIMENTAL RESULTS AND DATA ANALYSIS The measurements were made at the Sa ˜ o Paulo 8UD Pelletron Accelerator, Brazil. The detecting system has already been described in Ref. 关1兴. The thickness of the targets were about 60 ␮ g/cm2. Figure 1 exhibits the elastic scattering cross sections for the 12C⫹58Ni,208Pb systems in several sub-barrier energies. In the optical model calculations, we have adopted a procedure similar to that described in the analysis of the subbarrier data for the 16O⫹58,60,62,64Ni, 88Sr, 90,92Zr, 92Mo, PHYSICAL REVIEW C, VOLUME 65, 044314 0556-2813/2002/65共4兲/044314共7兲/$20.00 ©2002 The American Physical Society65 044314-1 120Sn, 138Ba, 208Pb systems 关1–4兴. We have adopted a Woods-Saxon shape for the optical potential, with an inner imaginary part which takes into account the rather small internal absorption from barrier penetration. The values assumed for the parameters of the imaginary part of the potential result in very small strengths at the surface region. This procedure must be adopted in the sub-barrier data analysis due to the negligible cross sections of peripheral reaction channels. Concerning depth variations of this absorptive potential, no sensitivity in the elastic scattering cross section predictions has been detected. The radius parameters of the real part of the optical potential were fixed at r0(A1 1/3 ⫹A2 1/3), with r0⫽1.06 fm, and the depth and diffuseness parameters were searched for the best data fits. For each angular distribution, we have found a family of potentials which give equivalent fits. These potentials cross 共see Fig. 2兲 at a particular distance RS, hereafter referred to as the sensitivity radius. The heavy-ion elastic scattering is sensitive to averages of the potential over distances comparable to the wavelength of the relative motion; therefore the determination of a sharply defined sensitivity radius presents some dependence on the shape assumed for the real part of the optical potential 关11兴. In most cases, including those of the present work, the scattering of heavy ions is sensitive only to the potential for a very restricted range of surface distances around the sensitivity radius. In this region, a realistic potential, such as the double folding, should present an approximately exponential shape with diffuseness values around 0.6 fm 关12兴. Thus, in order to avoid ambiguities in the potential determination, we have assumed a realistic shape for the potential in the surface region. Indeed, such an exponential behavior with realistic diffuseness values is presented by the Woods-Saxon potential adopted in our analyses 共see Fig. 2兲. The sensitivity radius is energy dependent 共see Fig. 2兲, because at sub-barrier energies it is related to the classical turning point. We have used this fact to characterize the nuclear potential in the surface region 共see Fig. 3兲. For such large interaction distances, the shape of the potential is nearly an exponential 共solid lines in Fig. 3兲, with a diffuseness value about 0.64 fm. Similar behavior has been observed for systems with 16O as the projectile 关1–4兴. The nuclear potentials equal 1 MeV at R⫽9.7 fm and R ⫽12.5 fm 共see Fig. 3兲for the 12C⫹58Ni and 12C⫹208Pb systems, respectively. As discussed in the next section, this difference between the nuclear potentials is directly connected with the corresponding different densities of the 58Ni and 208Pb nuclei. FIG. 1. Elastic scattering angular distributions for the 12C ⫹58Ni, 208Pb systems at several sub-barrier energies. The solid lines represent optical model predictions, in which the nonlocal model is assumed for the real part of the interaction. Note that the cross section is represented in a linear scale. FIG. 2. The real part of the optical potential for the 12C⫹58Ni system at two different energies, as obtained by optical model data fits for several values of the diffuseness parameter. The potentials cross at the sensitivity radii RS, where the corresponding potential strengths are determined without ambiguities. FIG. 3. The nuclear potential strength as a function of the sensitivity radius for the 12C⫹58Ni, 208Pb systems. The bombarding energies of the elastic scattering angular distributions in which the sensitivity radii have been determined are indicated in the figure. The solid lines represent exponentials with diffuseness value 0.64 fm. The radii at which the potentials equal 1 MeV are indicated in the figure. L. R. GASQUES et al. PHYSICAL REVIEW C 65 044314 044314-2 III. COMPARISON WITH THE NONLOCAL MODEL The elastic scattering data analyses for different systems in a very large energy range have resulted in phenomenological optical potentials with significant dependence on the bombarding energies 关13兴. Several theoretical models have been developed to account for this energy dependence; one of them associates this dependence with nonlocal quantum effects related to the exchange of nucleons between target and projectile 关6–9兴. Within this model, the bare interaction VNis connected with the folding potential VFthrough 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 speed between the nuclei v2共R,E兲⫽2 ␮ 关E⫺VC共R兲⫺VN共R,E兲兴.共2兲 For the Coulomb interaction VCwe have used the expression for the double sharp cutoff potential 关14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3兲 With the aim of providing a global description of the nuclear interaction, in Ref. 关6兴a systematization of nuclear densities has been proposed, based on an extensive study involving charge distributions extracted from electron scattering experiments and theoretical densities calculated through the Dirac-Hartree-Bogoliubov model. This study has indicated that the two-parameter Fermi 共2PF兲distribution can be adopted to describe the nuclear densities, and an useful distinction between nucleon and matter distributions has been made. The radii of the 2PF distributions are well described by Ri⫽1.31Ai 1/3⫺0.84 fm, 共4兲 where Ais the number of nucleons of the nucleus. The nucleon and matter densities present average diffuseness values aN⫽0.50 fm and aM⫽0.56 fm, respectively. Due to effects of the structure of the nuclei, along the table of stable nuclides the Riand aparameters vary around the corresponding average values . However, concerning the nuclear potential, the effects of the structure of the nuclei are mostly present at the surface and mainly related only to the diffuseness parameter 关6兴. Within this context, an extensive systematization of optical potential strengths extracted from heavy-ion elastic scattering data analyses at low and intermediate energies was performed 关6兴. The experimental potential strengths have been described within 25% precision, by combining Eqs. 共1兲 and 共3兲through two different and equivalent methods. In the first alternative, the double-folding potential is treated in the usual interpretation 关12兴: the nucleon densities and an effective nucleon-nucleon interaction for u0(r ជ ) are adopted in Eq. 共3兲. The standard M3Y interaction ‘‘frozen’’ at 10 MeV/ nucleon 关6,8兴has been assumed for the effective nucleonnucleon interaction. In the other alternative, the matter densities are adopted in Eq. 共3兲, with a zero-range delta function assumed for u0(r ជ ): u0共r ជ 兲⫽V0 ␦ 共r ជ 兲,共5兲 with V0⫽⫺456 MeV fm3. This zero-range alternative is interesting because it results in approximate analytical expressions for the folding potential 关6兴. For example, in the surface region VF共R⭓R1⫹R2兲⬇V0 ␳ 01 ␳ 02 ␲ aM 2Rg共 ␶ 兲共1⫹s/aM兲e⫺s/aM, 共6兲 with s⫽R⫺(R1⫹R2), R⫽2R1R2/(R1⫹R2), ␶ ⫽s/R. The ␳ 0iare obtained from the normalization of the densities 4 ␲ 冕 0 ⬁ ␳ 0i 1⫹e(r⫺Ri)/aMr2dr⫽Ai,共7兲 and the function gis defined by g共 ␶ 兲⫽1⫹ ␶ ⫹ ␶ 2 ␨ /3⫹aM/R⫹共aM/R⫹1/2兲e⫺s/aM 1⫹ ␨ ␶ , 共8兲 ␨ ⫽R/(R1⫹R2). Expression 共1兲has accounted for the energy dependence of experimentally extracted potential strengths for a large number of different systems in a very wide energy range 关6–9兴. At sub-barrier energies and for radii close to the barrier radius, Eq. 共1兲indicates that VN⬇VF. In order to compare potentials from different systems, we have defined a reduced quantity Vred , which removes the dependence of the sub-barrier potential strengths on the radii of the nuclei: Vred⫽VN V0 ␳ 01 ␳ 02Rg共 ␶ 兲.共9兲 Taking into account Eqs. 共6兲and 共9兲, the reduced potential should be a universal function of s Vred共s⭓0兲⬇ ␲ aM 2共1⫹s/aM兲e⫺s/aM,共10兲 with the matter diffuseness approximately system independent and close to the average value aM⬇0.56 fm 关6兴. The experimental reduced potential strengths for the 12C ⫹58Ni,208Pb systems, calculated through Eq. 共9兲, are in good agreement 共see Fig. 4兲with the predictions 关Eq. 共10兲with aM⫽0.56 fm兴of the early systematics for the bare potential. Assuming the nonlocal model with the densities proposed in such a systematics, good predictions for the elastic scattering cross sections are obtained 共see the solid lines in Fig. 1兲. In order to extend the analysis to higher energies, experimental elastic scattering angular distributions 共from Refs. 关15–17兴兲 at several intermediate energies for the 12C ⫹12C,208Pb systems have been included in our study. In the sub-barrier case the imaginary potential used in the optical DETERMINATION OF THE 12C NUCLEAR DENSITY . . . PHYSICAL REVIEW C 65 044314 044314-3 model calculations is based on very fundamental ground: the lack of surface absorption. Provided this condition is assumed, the results of the analyses at sub-barrier energies are independent of the parameters adopted for the imaginary potential. With the purpose of treating the absorptive part of the potential within a fundamental context also for intermediate energies, we have assumed 关Eq. 共11兲兴 the Lax-type interaction with Pauli blocking 关18兴, which is the single-scattering term of the Glauber multiple scattering theory 关19兴 W共R,E兲⫽⫺ E kN ␴ T NN共E兲 冕 ␳ 1共 兩 R ជ ⫺r ជ 兩 兲 ␳ 2共r兲dr ជ ,共11兲 where ␴ T NN(E) is the average nucleon-nucleon total cross section. In Fig. 5, a comparison between data and theoretical predictions 共solid lines兲for the angular distributions at intermediate energies is presented. Again we have assumed the nonlocal model for the real part of the interaction with the densities proposed in the systematics of Ref. 关6兴. Although the calculated cross section shows stronger oscillatory behavior, the magnitude, however, is in reasonable agreement with the data. Part of the theoretical oscillatory pattern could be damped in the data due to the angular aperture of the collimation system used in the experiments. We point out that no adjustable parameter has been used in either of the real and imaginary parts of the potential. It must be also remembered that only the single-scattering term of the multiple scattering series of Glauber has been included in the absorptive part of the potential, although higher order terms are quite likely to contribute significantly to the cross section. As we showed before 关8兴, better fits can be obtained using a Woods-Saxon shape for the imaginary potential with three adjustable parameters. However, we regard the present approach as more fundamental. IV. DETERMINATION OF THE 12C NUCLEAR DENSITY As we have discussed in Sec. III, the experimentally extracted potential strengths for the 12C⫹58Ni,208Pb systems are compatible with the systematics for the nuclear potential of Ref. 关6兴. That systematics is based on densities with the shape of Fermi distributions, radii obtained from Eq. 共4兲, and average diffuseness values aM⫽0.56 fm and aN⫽0.50 fm for the matter and nucleon distributions, respectively. In this sense, the analysis presented in Sec. III provides information about the nuclear densities of the partners in the collision. In this section, we present another form of analyzing the same set of cross section data, which determines the densities in a more direct manner than that of Sec. III. If the nonlocal model is assumed for the heavy-ion interaction and the density of one nucleus is known, an unfolding method can be used to extract the density of the other nucleus from the elastic scattering data analyses. In Fig. 6, we compare the data 共from Refs. 关20–22兴兲 with predictions for electron scattering cross sections on several nuclei. In the theoretical calculations, we have used charge distributions derived from the Dirac-Hartree-Bogoliubov 共DHB兲model 关23兴. The predictions are in good agreement with the data for the heavier nuclei, but discrepancies are observed for the 12C and 16O. This fact is an indication that the heavier the nucleus is, the more realistic is the theoretical density calculated through the DHB model. Therefore, we consider that the 58Ni and 208Pb densities are well described by the DHB calculations, and we have used the unfolding method to determine the 12C nuclear density. In this section, the double-folding potential is considered in the usual interpretation: the nucleon densities and the M3Y effective nucleon-nucleon interaction are adopted in Eq. 共3兲. The 12C density is extracted from data analyses within a procedure similar to that used in the determination of potential strengths at the sensitivity radii. We have assumed the Fermi distribution to describe the 12C nucleon density, with diffuseness (aN) and radius (Ri) searched for the best data fits, and with the ␳ 0iparameter determined by FIG. 4. The experimental reduced potential strength as a function of the reduced distance sfor the 12C⫹58Ni, 208Pb systems. The solid line represents the theoretical prediction, Eq. 共10兲with aM ⫽0.56 fm. FIG. 5. Elastic scattering angular distributions for the 12C ⫹12C, 208Pb systems at several intermediate energies. The solid lines represent optical model predictions, in which the freeparameter nonlocal model and the Lax-type interaction are assumed for the real and imaginary parts of the potential, respectively. L. R. GASQUES et al. PHYSICAL REVIEW C 65 044314 044314-4 the normalization condition 关Eq. 共7兲兴. The real part of the optical potential is obtained from Eqs. 共1兲and 共3兲, and has no adjustable parameters except those (aNand Ri) connected only with the quantity to be determined: the 12C nucleon density. For each angular distribution, we have found a family of densities which give equivalent data fits. These densities cross at two particular radii 共see Fig. 7, top兲, and we associate only one of these radii to the sensitivity radius (rS) for the density. To choose rS, we have used the notch test 共Fig. 7, bottom兲, in which a spline with a Gaussian shape is included in the 12C density, and the variation of the chisquare is studied as a function of the position of this perturbation. The crossing chosen as the sensitivity radius is that closest to the center of the region which affects significantly the elastic scattering data fit. The determination of the error bar for the density at the sensitivity radius is illustrated in Fig. 8 for a particular elastic scattering angular distribution. At rSthe data extracted density value does not depend on the diffuseness assumed for the distribution. Thus, the dependence of the total chi-square on Riis studied for a fixed aNvalue, and the parameters that correspond to the minimum value ␹ min 2and to ␹ min 2⫹ ␹ min 2/n are found 关see the determination of R0min,R0⫺, and R0⫹in Fig. 8共a兲兴;nis the number of experimental data points of the angular distribution. Figure 8共b兲presents the Fermi distributions for the R0min,R0⫺, and R0⫹values, and the respective determination of the error bar for the density at rS. Similar to the case of the potential determination, the sensitivity radius for the density is energy dependent and this fact allows the characterization of the density over a large range of distances. Figure 9共a兲contains the 12C experimental nucleon density values at the corresponding sensitivity radii obtained from data analyses of several angular distributions. The sub-barrier elastic scattering data gives information about the density at the surface region, while inner distances are probed through the data at intermediate energies. We point out that data analyses for different systems provide consistent similar results for the 12C density. The theoretical prediction from the DHB model 共see Fig. 9兲does not match the experimental results at the surface region. Taking into account the discussion about the nuclear potential of Sec. III, FIG. 6. Experimental electron scattering cross sections for the 12C, 16O, 58Ni, and 208Pb nuclei as a function of the effective momentum transferred. The solid lines represent theoretical predictions using charge distributions derived from Dirac-Hartree-Bogoliubov calculations. The dashed line in the 12C case represents calculations based on a 2PF distribution 共see text for details兲. FIG. 7. 共Top兲Example of the determination of the sensitivity radius rSand the corresponding experimental value for the 12C nucleon density, using two-parameter Fermi distributions which give equivalent data fits for the angular distribution of the 12C ⫹58Ni system at Elab⫽28.5 MeV. 共Bottom兲The sensitivity region for the 12C nucleon density characterized by the notch test. DETERMINATION OF THE 12C NUCLEAR DENSITY . . . PHYSICAL REVIEW C 65 044314 044314-5 another consistent result of our analysis is the agreement 共see Fig. 9兲between the Fermi distribution that has been assumed in the potential systematics of Ref. 关6兴and the experimental density values from sub-barrier data analyses. We mention that other experimental data for the 12C density in the region 2⭐r⭐4 fm could be found through the analyses of other angular distributions in an energy region in between the subbarrier and intermediate energies analyzed in the present work, but in this case the imaginary potential would have adjustable parameters and the reliability of the results for the density should be studied much more carefully. Now we evaluate the effects of two possible sources of systematical errors in the density determination: the polarization potential and the shape of the density distribution. As discussed in Sec. I, coupled-channel calculations have indicated that the polarization represents about 10% of the real part of the optical potential at sub-barrier energies. In the data analysis, we have neglected the polarization and associated the real part of the optical potential only with the bare interaction, which is directly proportional to the nuclear densities. Thus, a systematical error of about 10% is expected in our results for the density in the surface region, due to the procedure of neglecting the polarization potential for subbarrier energies. The threshold anomaly 关24兴indicates that the contribution of the polarization to the real part of the optical potential should be more significant in the region of the Coulomb barrier, so we estimate this contribution at intermediate energies 共inner density distances兲to be even less than 10%. Another source of systematical errors is the shape assumed for the density distribution. Similar to the case of the potential determination, the notch test indicates that the data fit is sensitive to a density region of width ( ␴ ⫽2.5 fm, see Fig. 7兲comparable to the wavelength of the relative motion. However, in contrast with the potential case, in such a region (2.5⭐r⭐5.0 fm) a realistic shape for the density may present a significant deviation from a pure exponential form 共see Fig. 7兲. Therefore, one could expect some dependence of the results for the sensitivity radius on the shape adopted for the density distribution, particularly for intermediate energies in which inner distances are probed. Thus, in order to investigate the dependence of the method on the shape assumed for the 12C distribution, we have also performed data analyses using the harmonic oscillator 共HO兲 shape 关Eq. 共12兲兴, with two adjustable parameters (wand ␣ ): ␳ 共r兲⫽ ␳ 0 冉 1⫹ ␣ r2 w2 冊 e⫺r2/w2.共12兲 The corresponding results for the densities at the sensitivity radii are presented in Fig. 9共b兲. At the surface region an average difference of 22% between the HO and 2PF results has been found. Therefore, within this precision, our studies indicate that the results for the 12C density at the surface region are rather independent of the model assumed for the shape of the distribution. The results from intermediate energies are more sensitive to the shape, but even in this case the different models 共HO and 2PF兲provide similar overall trends for the density. As a further test of the consistency of our results for the 12C density, in Fig. 6 we compare the data with predictions for electron scattering cross sections. In the theoretical calculations, we have obtained the 12C charge distribution by folding the proton density of the nucleus ( ␳ p) with the intrinsic charge distribution of the proton in free space ( ␳ chp) ␳ ch共r兲⫽ 冕 ␳ p共r ជ ⬘兲 ␳ chp共r ជ ⫺r ជ ⬘兲dr ជ ⬘,共13兲 FIG. 8. The figure presents an example of the determination of the error bar for the 12C density at rSfor the angular distribution of the 12C⫹58Ni system at Elab⫽27 MeV. 共a兲The total chi-square as a function of the radius of the Fermi distribution for the fixed diffuseness parameter a⫽0.5 fm, and the determination of the R0min , R0⫺, and R0⫹values. 共b兲The Fermi distributions which correspond to the R0min ,R0⫺, and R0⫹values, and the determination of the error bar for ␳ at rS. FIG. 9. Experimental nucleon density values for the 12C nucleus, as obtained from elastic scattering data analyses for different heavy-ion systems at sub-barrier 共open symbols兲and intermediate 共closed symbols兲energies. Parts 共a兲and 共b兲of the figure concern the results of analyses considering the Fermi or harmonic oscillator shapes for the 12C density, respectively. The lines correspond to theoretical Dirac-Hartree-Bogoliubov 共DHB兲calculations, and to the Fermi distribution 共2PF兲proposed in Ref. 关6兴. L. R. GASQUES et al. PHYSICAL REVIEW C 65 044314 044314-6 where ␳ chp is an exponential with diffuseness achp ⫽0.235 fm. We have estimated the 12C proton distribution as one half of the total 共proton ⫹neutron兲nucleon distribution. The dashed line in Fig. 6 represents the results for the cross sections obtained by considering the 2PF distribution of Ref. 关6兴for the total 12C density 共solid lines in Fig. 9兲. Such results are much closer to the data than the DHB predictions. V. CONCLUSION In this work, we have presented elastic scattering data at sub-barrier energies for systems involving 12C as projectile, and we have extended our studies to other data sets earlier obtained at intermediate energies. In our optical model data analysis, the imaginary part of the potential is based only on very fundamental grounds and has no adjustable parameters. The nonlocal model is assumed to describe the energy dependence of the real part of the interaction, which is connected to the folding potential through the very simple Eq. 共1兲. Within this context and assuming the systematics of Ref. 关6兴for the nuclear densities, a reasonable prediction of the elastic scattering cross sections is obtained for the whole data set without the use of any adjustable parameter. If the target densities are known, we have shown that the density of the projectile can be extracted from the data analysis in a direct procedure. The sub-barrier elastic scattering data gives information about the density at the surface region, while inner distances are probed through the data at intermediate energies. The results for the 12C nuclear density are consistently independent of the target nucleus, and in reasonable agreement with the Fermi distribution resulting from the systematics of Ref. 关6兴. 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