Optical Model Analysis of Elastic Scattering of 16 O + 12 C

This paper presents the optical model analysis of the elastic scattering of O + C at the incident energies of 608 and 1503 MeV using optical potentials derived from B3Y-Fetal effective interaction. Optical model (OM) analysis of the elastic data of this system at these incident energies has shown two of the four optical potentials, the DDB3Y1-Fetal (K=176 MeV) and BDB3Y1-Fetal (K=235 MeV), to give a better description of the elastic data than the BDB3Y2-and BDB3Y3-Fetal potentials, making them the best-fit folded potentials, in agreement with previous work done with the M3Y-Reid effective interaction in both identical and non-identical heavy ions. This is a conclusive and convincing confirmation of the concensus among Nuclear Physics scholars, based on OM analyses of heavy ions, that nuclear matter has an underlying soft equation of state. In addition, results of calculations herein have also shown the best-fit folded potentials, the DDB3Y1-and BDB3Y1-Fetal with-227.8 and-220.6 MeV at 608 MeV and-124.3 MeV and-120.5 MeV at 1503 MeV, respectively as the largest values at smaller inter-nuclear distances, to be in good agreement with their counterparts, the DDM3Y1-Reid and BDM3Y1-Reid, whose largest values at smaller inter-nuclear distances are-231.6 and-223.8 MeV at 608 MeV and-138.8 MeV and-134.2 MeV at 1503 MeV respectively, in terms of magnitude, shape and trend. This is a further validation of the viability of the B3Y-Fetal, corroborating the findings of previous studies carried out with it. In the final analysis, the findings of this study have not shown the occurrence of distinctive features of refractive scattering such as Airy minima in the calculated cross sections at both 608 MeV and 1503 MeV in agreement with previous work.


Introduction
Optical potential is a potential that represents the interaction between a nucleon or a group of nucleons and a target nucleus [1]. It is a fundamental tool, used for many decades for the analysis of nuclear reactions, that enables elastic cross-sections over a wide range of ions, ion beam energies and scattering angles to be calculated [2,3]. In many optical-model studies, the optical potential is constructed by theoretical and phenomenological methods. The former uses the fundamental theory of nucleon-nucleon interaction to solve the many-body problem of nucleon-nucleon or nucleusnucleus interaction, whereas the latter starts from a physically reasonable form for the potential and adjusts its parameters systematically to optimize the fit to the experimental data. The theoretical method has the disadvantage of formidable mathematical difficulties, but is successful in giving an overall quantitative picture of the optical potential through a number of approximations. The phenomenological method has the advantage of giving a very good fit to experimental data by varying parameter sets, but is unsatisfactory on grounds that many different potentials can be found which give acceptable fits to the data. Therefore, to have a satisfactory potential, the two approaches are combined so that the fundamental theories give the overall form of the potential and the phenomenological method determines the optimum parameters of the potential. The parameters of the composite potential usually vary smoothly with energy and are similar for neighbouring nuclei. The origin of an erratic variation of the parameters for different nuclei can be traced to the differences in their structure, especially in collective states [4].
Generally, this optical potential is complex, having real and imaginary components. The real component accounts for elastic scattering while the imaginary component, arising from phenomenological method, represents non-elastic processes which cause the absorption of incident particles in nuclear reactions [1,2,5]. The convenient parameterization of the imaginary component is the Fermi distribution, which, in the context of the optical model, is referred to as the Woods-Saxon form, whereas the real component is represented by a folding-model potential in many theoretical optical model (OM) studies [4,6,7]. Since this study is a theoretical analysis of heavy-ion nucleus-nucleus elastic scattering, the real component is represented by a double folding potential in this work [8]. Such a folded potential is more frequently used for the study of α-particle as well as heavy-ion (HI) scattering because it gives a better representation of the surface part of the optical potential than the phenomenological potentials [1].
For the analysis of the elastic scattering of light heavy ions, the double folding model, with an effective interaction appropriately adjusted to nuclear matter properties, enters the OM analyses as a mean-field effect which is, indeed, very deep. It is well-known that heavy-ion collisions are usually accompanied by strong absorption, making the elastic scattering data sensitive only to the surface part of the nucleus-nucleus potential. However, lately, ever-increasing understanding of the interaction between heavy ions arising from studies of elastic scattering has shown certain combinations of heavy ions, for which absorption is relatively weak, to exhibit refractive scattering at incident energies of a few MeV per nucleon [7,9,10,11]. One key advantage of refractive scattering is that it makes the determination of HI optical potential up to small internuclear distances possible, giving room for the possibility of testing different theoretical models for HI optical potentials by analyzing these systems (heavy ions) [12,13]. This refractive scattering has been a subject matter of increased attention in the last two decades; and it has been established that only deep potentials, such as those generated by double folding model, are needed to describe the systematics of heavy-ion scattering.
So far, experimental evidence of refractive scattering has been found mainly in 12 C + 12 C, 16 O + 16 O and 16 O + 12 C systems [9,14]. Their elastic scattering angular distributions reveal unmistakable refractive features, such as rainbow scattering patterns and broad interference minima called "Airy minima" [10]. It is now known that these refractive features can be described consistently by optical potentials with deep (several hundreds MeV) real parts. Therefore, the clear observation of rainbow scattering features in 12 C + 12 C, 16 O + 16 O and 16 O + 12 C elastic scattering data has definitely and particularly established that the real part of the light heavyion optical potential is strongly attractive and deep; and the imaginary part of the potential is weak enough to minimize absorption. Undoubtedly, a combination of these two features-deep real optical potential and incomplete absorption-makes possible the observation in the elastic angular distribution of distinctive refractive effects such as Airy minima superimposed on more classical diffractive features [7,9].
This refractive behaviour has been conspicuously observed in the systematic analysis of the 16 O + 16 O system carried out by Nicoli et al. [15] at incident energies between 75 and 124 MeV and by Khoa et al. [9,13] between 124 and 1120 MeV; and of the 16 O + 12 C system by Nicoli et al. [16] between 62 and 124 MeV and by Oglobin et al. [17] at 132 MeV [12]. In the systematic analysis of the 12 C + 12 C data at 14 energies between 71 and 127 MeV, refractive effects were not observed in the elastic data, so they concluded that only deep real potentials could describe the data [18]. Khoa et al. [9] did an extensive analysis of the elastic data for the same system at incident energies between 112 and 1016 MeV with the deep real potential generated by the double folding model and observed distinctive refractive features in the elastic scattering angular distribution. Attempts have been made to extend the methods and approaches of previous researches to heavier projectile-target combinations, but absorption has consistently been observed to increase rapidly with the masses of the colliding nuclei, damping the nuclear reaction unfavourably. Thus, for the 20 Ne + 12 C system, only pure diffraction was observed and the system was dubbed "strongly absorbing" [19]. Also, previous experiments aimed at observing rainbow or at least refractive scattering in the 16 O + 20 Ca system at incident energies of 281 and 704 MeV failed to see any signatures of such effects [20]. Only at the energy of 214 MeV was what seemed to be a semblance or signature of nuclear rainbow observed at 45° in the angular distribution of the elastic cross-section as a "minimum" [20].
While efforts continue to be made to establish clear signatures of rainbow scattering in heavier systems, it is still, no doubt, a seriously contemplated idea among nuclear Physicists to find the possibility of observing refractive scattering at higher incident energies in those systems that are known to exhibit weak absorption so far. New theoretical models which might be used for better studies of elastic scattering may be developed in the future. However, within the confines of this study, the newly developed B3Y-Fetal is to be applied to the folding analysis of 16 O + 12 C at the incident energies of 608 and 1503 MeV within the framework of optical model in anticipation that the expected results will show signs of refractive scattering as well give a good insight into distinctive qualities of the new effective interaction among numerous theoretical models for heavy-ion optical potential. For comparative purposes, the new effective interaction is used alongside the M3Y-Reid effective interaction in this work.
The organization of this paper is such that Section 2.0 discusses briefly the double folding procedure, which is followed by presentation and discussion of results of the study in Section 3.0; and Section 4.0 is devoted to concluding remarks.

Double Folding Procedure
In a bid to study the elastic scattering of 16 O + 12 C nuclear system at the incident energies of 608 and 1503 MeV, a nucleus-nucleus optical potential is constructed which is composed of the double folding model as the real part while the imaginary part is parameterized phenomenologically as a two-parameter Woods-Saxon shape.
The basic inputs to the double folding model calculation are the nuclear densities of the colliding nuclei and a realistic effective nucleon-nucleon (NN) interaction which reproduces the basic nuclear matter properties as well as the density and energy dependence of the nucleon optical potential. Here, the effective interaction of particular interest is the B3Y-Fetal interaction, derived on the basis of the lowest order constrained variational principle, whose radial form in terms of three Yukawa functions is [21]: The well-known M3Y-Reid interaction is also used along with the B3Y-Fetal interaction as a standard with which the latter is compared. For this reason, the explicit functional form of the M3Y-Reid NN interaction is expressed in terms of three Yukawas as [6,7,22]: For the optical model analysis of the elastic scattering of the chosen nuclear systems, the B3Y-Fetal interaction is to be used in its density-and energy-dependent form: where E is the incident nucleon energy, the energy dependence of the optical potential, g(E) ~ 1-0.002E for M3Y-Reid interaction [7,9,23] as well as B3Y-Fetal interaction [24] and F(ρ) is the density-dependent factor whose explicit forms are and representing the DDM3Yn (n=1) and BDM3Yn (n=0, 1, 2, 3) interactions respectively. Herein, the B3Y-Fetal effective interaction is used in the DDM3Y1, BDM3Y1, BDM3Y2 and BDM3Y3 density-dependent versions so that the resulting density-dependent B3Y-Fetal interactions are called the DDB3Y1-Fetal, BDB3Y1-Fetal, BDB3Y2-Fetal and BDB3Y3-Fetal interactions respectively. In a test of viability, this new effective interaction was applied to the study of symmetric nuclear matter in an earlier work [25] in the named density dependences and was found to reproduce the saturation properties of this nuclear matter with results that were in excellent agreement with previous work done with M3Y-Reid and M3Y-Paris interactions [6,7,9]. Shown in Table 1 are the parameters of density dependence and incompressibility obtained with the B3Y-Fetal interaction in that paper. In this model, the direct and exchange parts of the nucleusnucleus potential are obtained by folding the densities of the projectile and target nuclei with the direct and exchange components of the effective nucleon-nucleon (NN) interaction as presented in equations (6) and (7) respectively , = ! , , " # # with where and are the densities of the interacting nuclei. The exchange term is non-local due to the effect of antisymmetrization occasioned by the single-nucleon knock-on exchange. To solve the non-locality problem, an exact, consistent microscopic approximation to the exchange potential, instead of the semi-phenomenological zero-range approach, developed by Khoa et al. [6,7,9,26] is adopted in this work. Their approach produces an accurate local approximation, resulting from treating the relative motion locally as a plane wave so that k(R) in equation (7) is the local momentum of relative motion represented by: where is the reduced mass, 6.4 the centre-of-mass (c.m) energy, E the incident laboratory energy per nucleon and m the bare nucleon mass. , = , + &' , is the total nuclear potential and V C (R) is the Coulomb potential.
In this work, the exchange potential is evaluated by exact iterative method; and the energy dependence of the total folded potential, V (E, R) on the exchange term which is much stronger than the energy dependence denoted by g (E), is obtained from the real folded optical potential through accurate evaluation of the knock-on exchange effects [6,11,27]. A very important parameter which gives insight into the energy dependence of the optical potential is the volume integral of the folded potential per interacting nucleon pair expressed as [7,9,27]: As a measure of the potential strength, it can be used to distinguish a family of real folded potentials [14].
Essentially, the double-folding procedure generates the real part of the nucleus-nucleus optical potential, supplemented by an imaginary potential which accounts for absorption into non-elastic channels. This hybrid approach, composed of the folding model as the real component and a standard Woods-Saxon (WS) potential as the imaginary component of the optical potential, is used for the opticalmodel (OM) analysis of elastic scattering in this study. In totality, the optical potential is: where the real and imaginary parts of the optical potential are: , = B / , , where N R is the normalization factor for the real (folded) potential adjusted appropriately to fit the elastic data alongside the parameters W v , R W and a W which are the strength, radius and diffuseness of the Woods-Saxon function [28]. All the nuclear densities in the folding model are taken as a Fermi distribution with parameters that can reproduce the shell-model densities for the nuclei under consideration; and all computational analyses of the optical model are carried out in the present work using the ECIS-97 code.
To study the energy dependence of the real heavy ion potential, efforts were made in previous work to plot all N R and J R values obtained by using one of the best-fit real folded potentials, the DDB3Y1-Fetal, in the analyses of 16 O and 12 C [24,30] scattering data against the incident energy. The plots are reproduced in the upper and lower regions, respectively, of Figure 1. The folding model used for the nucleus-nucleus potential therein was adjudged satisfactorily realistic as the plot in the upper region has shown N R to depend weakly on energy and vary around a value of 0.9 [7,9,24,30]. The folding results have additionally established the intrinsic energy dependence arising from the exchange effects and the weak explicit energy dependence g(E) factor introduced into the effective NN interaction (Equation (3)) as the main sources of the total energy dependence of the real HI optical potential [9,24,30].
In the lower region of Figure 1, the volume integral of the real folded potential per interacting nucleon pair, J R is seen to decrease smoothly from about 340 MeV/m 3 to 150 MeV/m 3 as the incident energy increases from about 10 to 120 MeV/nucleon. This trend is known to agree well with previous work [9], validating the prediction of the energy dependence of the real HI potential by the folding model in that work to be reliably satisfactory. It is with this confidence that the same folding model is used in this work for the analysis of the elastic data of the non-identical 16 O + 12 C system.

Results and Discussion
This work essentially represents the application of the new M3Y-type effective interaction, the B3Y-Fetal, to optical model analyses of the elastic scattering of 16 O + 12 C at E lab =608 and 1503 MeV. The results of calculations of the real optical potential for 16 O + 12 C scattering at E lab =608 and 1503 MeV alongside the calculated cross sections are shown in Table 2 and Figures 2-7. The heavy-ion optical potential derived from the B3Y-Fetal has been found to demonstrate weak absorption in the chosen nuclear system. The results show the optical potentials derived from the B3Y-Fetal effective interaction to be in good agreement with the optical potentials based on the M3Y-Reid effective interaction.

Figure 2. Real Optical Potentials for 16 O + 12 C Based on B3Y-Fetal (Upper Part) and M3Y-Reid (Lower Part) Obtained at the Incident Energy of 608
MeV.

The Real Folded Potential
The four real folded potentials, derived from the B3Y-Fetal effective interaction and used in this work-the DDB3Y1-Fetal, BDB3Y1-Fetal, BDB3Y2-Fetal and BDB3Y3-Fetal-are shown in the upper regions of Figures 2  and 3 along with the M3Y-Reid-based DDM3Y1-Reid, BDM3Y1-Reid, BDM3Y2-Reid and BDM3Y3-Reid potentials in the lower regions. lt is quite informative to note that the first two optical potentials (DDM3Y1-Reid and BDM3Y1-Reid) in the lower regions of Figures 2 and  3, being close in magnitude, are widely separated from the last two optical potentials (BDM3Y2-Reid and BDM3Y3-Reid), which are also shown to be close in magnitude. This wide separation between the two sets of optical potentials appears to be a sharp demarcation that puts them in two classes that are manifested in their description of the elastic data Here, the differences between the DDB3Y1-Fetal and BDB3Y1-Fetal at 608 and 1503 MeV are-7.2 MeV and-3.8 MeV respectively while the differences between DDM3Y1-Reid and BDM3Y1-Reid at these energies are-7.8 MeV and 4.6 MeV respectively. This shows that the B3Y-Fetal is in excellent agreement with the M3Y-Reid effective interaction. The BDB3Y2-Fetal potential is seen in this nonidentical ( 16 O + 12 C) system to have a much larger value than the BDB3Y3-Fetal potential; this unusual behaviour is not exhibited in identical systems such as 12 C + 12 C [29] and 16 O + 16 O [30]. As regards trend, each real folded potential based on the B3Y-Fetal or M3Y-Reid becomes repulsive as the incident energy increases from 608 MeV to 1503 MeV in Figures 2 and 3 in conformity with the well-established trend in [9].

Fits to 16 O + 12 C Elastic Data
In the present work, the elastic scattering of 16 O + 12 C system is studied at the energies of 608 MeV and 1503 MeV respectively. The graphical representation of the elastic cross sections obtained with the four folded potentials is given in Figures 4-7 while the optical parameters are presented in Table 2. The fit to the data at 608 MeV was done using the weak WSI potential, but for the fit to the data at 1503 MeV, two sets of the imaginary potential were used: a weak one (WSI) with a depth of about 16 MeV and a strong one (WSII) about 50 MeV deep. Figure 4 shows clearly that the DDB3Y1-Fetal and BDB3Y1-Fetal potentials give a better description of the data with the weak WSI potential (minimal absorption) at both 608 MeV and 1503 MeV than the BDB3Y2-Fetal and BDB3Y3-Fetal potentials. This is in agreement with previous work [9] and the results for 12 C + 12 C [29] and 16 O + 16 O [30] systems in which DDM3Y1-Reid and BDM3Y1-Reid as well as DDB3Y1-Fetal and BDB3Y1-Fetal were described as the best-fit folded potentials. Figure 5 has been included to show that the optical potentials based on B3Y-Fetal and M3Y-Reid are in agreement. Herein, the distinctive features of refractive scattering such as Airy minima are not observed in the calculated cross sections obtained with the M3Y-Reid and B3Y-Fetal optical potentials at both incident energies; the sensitivity of the elastic data at either energy to the real optical potential at small inter-nuclear distances is not pronounced.
Folding results at 1503 MeV show that the real folded potentials have better fits to the data with WSII imaginary potential than with the WSI potential (Tables 2b and 2c). For the data at 1503 MeV in heavy absorption conditions, the DDB3Y1-Fetal and BDB3Y1-Fetal have exactly the same fit while the BDB3Y2-Fetal and BDB3Y3-Fetal have about the same fit as shown in Table 2c and Figures 6 and 7. Furthermore, the renormalization factors, N R for the best-fit folded potentials, DDB3Y1-Fetal and BDB3Y1-Fetal as well as DDM3Y1-Reid and BDM3Y1-Reid in the heavyabsorption case are found in Table 2c to be greater than unity contrary to the systematic behaviour of the best-fit real folded potentials seen so far in the present work and in the elastic scattering of 12 C + 12 C and 16 O + 16 O systems [9,29,30]; and this suggests strongly that the results for this system at 1503 MeV for the weak-absorption imaginary potential (WSI) might be more relevant.
In summary, this version of heavy-ion optical potential, based on B3Y-Fetal, applied to the study of the elastic scattering of 16 O + 16 C system has demonstrated weak absorption in the cases considered in this work. Finally, for the sake of emphasis, it has to be said that the c, which nsistent agreement of DDB3Y1-Fetal and BDB3Y1-Fetal [24,25,29,30] or DDM3Y1-Reid and BDM3Y1-Reid [9] potentials, as best-fit folded potentials, in the description of the elastic data of 12 C + 12 C and 16 O + 16 O systems, which is also observed in the present work is a convincing pointer to the concensus among Nuclear Physics scholars that the cold nuclear matter is governed by a soft equation of state [9]. Figures 2 and 3 clearly separate the best-fit folded potentials, being very close in magnitude, from the BDM3Y2-and BDM3Y3-Reid or BDB3Y2-and BDB3Y3-Fetal, which appear to be in a separate group of potentials with poor description of elastic data, known to represent "stiff" equation of state [9,31].

Conclusion
In this study, the folding analysis of the elastic scattering of 16 O + 12 C has been carried out within the framework of optical model at the incident energies of 608 and 1503 MeV.
Some density dependences have been introduced into the B3Y-Fetal effective interaction to reproduce the saturation properties of normal nuclear matter within a Hartree-Fock scheme; and these have predicted the nuclear incompressibility K to have values ranging from 176 to 467 MeV. The B3Y-Fetal-based density dependent interactions have been used to calculate the real part of the heavy-ion optical potentials for the non-identical system studied herein. In this way, the optical model analyses have been used to probe the sensitivity of the scattering data to different forms of density dependence as well as to the various values of the associated incompressibility K of the cold nuclear matter. The results of this study have shown the DDB3Y1-Fetal and BDB3Y1-Fetal optical potentials to be the best-fit folded potentials; this confines a realistic K value to a range of about 176 to 235 MeV, which corresponds to a "soft" nuclear EOS. This study allows one to suggest the most realistic density dependent interaction that is currently available for the folding model calculations of nucleus-nucleus potentials. In addition, the results obtained in this work show clearly and particularly the importance of refractive nucleus-nucleus scattering experiments as a means to provide accurate data to test an effective interaction.
Clear signatures of refractive scattering such as Airy minima have not been observed in the calculated cross sections at the energies of interest in this work, but the B3Y-Fetal-based optical potentials have been found to demonstrate good agreement with the optical potentials derived from the M3Y-Reid effective interaction in the description of the elastic data of this system.
Finally, the findings of this study could find useful applications in Nuclear Astrophysics where nuclear EOS is a major ingredient in calculations of reaction rates for understanding primordial nucleosynthesis and hydrostatic burning in stars.