1. Introduction

The nuclei far from β stability line have attracted an immense interest since the first observation of the neutron halo in ¹¹Li^[1]. Further experiments have confirmed the presence of neutron halo in ​11Li and other neutron-rich nuclei^[2]. Through the last two decades, experimental data for the proton halo in proton-rich nuclei like ⁸B, ¹⁷Ne, and ^26,27,28P have been reported in literature ^[3-^11]. However, the amount of experimental data on the proton halo is relatively small compared to those on the neutron halo. The short-lived radioactive nucleus ¹⁷F is one of the candidates for a proton-halo nucleus due to its small (600 keV) proton separation energy. But, it cannot be studied with usual spectroscopic techniques due to its short lifetime. Hence, one must resort to indirect methods to deduce information about its structure. Reactions are the most used tools to study halo nuclei. Elastic scattering ^[12,^13] and breakup ^[14,^15] provide interesting information about the structure of the projectile. Therefore, many experiments with ¹⁷F as projectile have been performed in recent years ^[16-^27]. It is important to mention that, th¹⁶e O (p, γ) ¹⁷F capture cross section measured in through the energy range Ec.m.=200-3750 keV covers five orders of magnitude of cross sections. Some data give striking different energy dependences of the branching ratio between the transition to the 5/2​+ ground state and to the 1/2​+ first excited state of ​17F which is bound by only 105 keV. On the other hand, the ¹⁶O( ³He,d) ¹⁷F reaction was used to determine asymptotic normalization coefficients for transitions to the ground and first excited states of ¹⁷F. Full coupled channels calculations were performed to study the ¹⁶O(d,p) ¹⁷F and the ¹⁶O(d,n) ¹⁷F transfer reactions at sub-Coulomb energies (Ed=2-3 MeV) ^[27-^29].

On the other side, through the past three decades, nucleus-nucleus optical model (OM) potentials have been studied extensively through elastic scattering measurements. The observed data were interpreted in the framework of both phenomenological and microscopic double folding (DF) potential models with the adjustment with none and with only a few parameters ^[20-^37]. In the present work, we have applied the DF model to analyze the elastic scattering o¹⁷f F nucleus on light-mass targets (¹²C, ¹⁴N), medium-mass target (⁵⁸Ni), and on ²⁰⁸Pb which is an example of a heavy target. The paper is organized as follows: In Sec. 2, we give a brief account of the adopted formalism. The procedure is discussed in Sec. 3. Results and discussion, and their comparison with the available experimental data are listed in Sec. 4, while concluding remarks are finally presented in Sec. 5.

2. Theoretical Formalism

The real nucleus-nucleus optical potential in the DF model is given by the expression ^[31]

where ρP(r1) and ρt(r2) are the nuclear matter density distributions for projectile and target nuclei, respectively, and υnn(s) is the effective nucleon-nucleon (NN)

interaction with s=∣R→-r→1+r→2∣ the distance between the two nucleons. In the present calculation, we use the most popular density-dependent DDM3Y, effective NN interaction of Bertsch et al.^[38] which has the following form ^[31]

The functional form of this density and energy dependent factor, f(ρ,E) is chosen as

The M3Y interaction is taken in the form

where the third energy-dependent term is a zero range pseudo-potential to account for the single nucleon exchange effect and E is the laboratory energy per nucleon. For the projectile ¹⁷F nucleus, two versions of nuclear matter density are considered. First, it is presumed that the ¹⁷F nucleus consists of an ¹⁶O core and a halo of one proton. The core density distribution is assumed to be of a harmonic oscillator (HO) form as ^[39]

where the halo density distribution is described by the Gaussian function as ^[35]:

where Rh=3.78 fm. Therefore, the total matter distribution ρ17F (normalized to one nucleon) and the matter radius Rm are given as ^[40]:

This density produces a rms radius, ⟨rrms2⟩1/2 of ¹⁷F equals to 2.77 fm. We denote this density as version A. In the second approach the density of ¹⁷F nucleus is taken from Ref. ^[40]. This density produces a rms radius ⟨rrms2⟩1/2 of ¹⁷F which equals to 2.74 fm. We denote this density as a version B. Figure 1 shows a comparison between the two considered A and B densities. It is evident that both densities have identical radial distributions over the range r=0-5 fm as shown in the inset layer in Fig. 1. In the surface region (r>5 fm) the density A has more extended tail than that of the B form. This indicates that the halo structure is more pronounced for the density A than the density B.

Figure 1 Comparison between the two densities A and B of ​17F. 

For the target nuclei ¹²C and ¹⁴N the nuclear matter density has been taken in the harmonic oscillator (HO) form ^[33,^42]:

These densities yield rms radii ⟨rrms2⟩1/2 equal to 2.407 and 2.48 fm for ​12C and ​14N respectively. For the target nuclei ​58Ni and ​208Pb the nuclear matter density has been taken in the two parameter fermi form ^[34-^37]

These densities yield rms radii ⟨rrms2⟩1/2 equals to 3.745 and 5.482 fm, respectively.

3. Procedure

The DF optical potentials generated from Eq. (1) using the density-dependent DDM3Y effective NN interaction are used to analyze the ​17F elastic scattering data. The DF potentials have been developed by considering the procedures given below:

4. Results and Discussion

The derived DF potentials based on the density-dependent DDM3Y effective NN interaction is used to analyze the elastic scattering of ¹⁷F nucleus on ¹²C, ¹⁴N, ⁵⁸Ni and ²⁰⁸Pb targets without renormalization factor. The obtained elastic scattering differential cross sections for the considered reactions are listed in Table I. These results are plotted as shown in Figs. 2-4 compared with the corresponding experimental data. The solid and dashed curves represent the results using, respectively, the density distribution A and B of the one-proton halo ¹⁷F nucleus. We used the calculated DF potential to analyze the elastic scattering experimental data of ¹⁷F + ¹²C, ¹⁴N reactions as examples of the light-mass tar- gets for only one energy value of Elab=170 MeV. As shown in Fig. 2, our results using the WS and microscopic DF potentials based on the JLM effective NN interaction are in good agreement with the corresponding measured data by Blackmon et al. ^[23]. For the medium mass targets, the elastic scattering data of ¹⁷F +²⁸Ni at E=51.94 and 170 MeV measured by Liang et al. ^[24] Mazzocco et al. ^[26], as shown in Fig. 3, are well reproduced by our calculated potentials. It should be mentioned that, the effect of the quasi-elastic scattering is very weak. However, in our calculations, the coupled channel calculations using the cluster form factor for ¹⁷F→​16O (core)+p (valence) as coupling between (5/2)​+ ground and (1/2)​+ states of ​17F are carried out as shown with dot lines in Fig. 3. For this purpose, we used only pure elastic scattering.

Figure 2 Angular distributions of the elastic scattering differential cross section relative to the Rutherford one for the ¹⁷F + ¹²C and ¹⁷F + ¹⁴N reactions using the DDM3Y potentials at Elab=170MeV. Experimental data are taken from Ref. ^[23]. 

Figure 3 Same as Fig. 2, but for ​17F + ​58Ni reaction at E=51.94 and 170 MeV. Experimental data are taken from Ref.^[24,^26]. 

Figure 4 A: Same as Fig. 2, but for ¹⁷F + ²⁰⁸Pb reaction at E=86.0, 90.4 and 98.0 MeV. Experimental data are taken from Refs. ^[25,²², ^21], respectively. B: Same as Fig. 2, but for ​17F + ²⁰⁸Pb reaction at E=120 and 170 MeV. Experimental data are taken from Ref. ^[21,^24]. 

The obtained parameters of the imaginary phenomenological WS potentials, the corresponding real and imaginary volume integrals per interacting nucleon pair (JV and JW) in MeVfm​3 and the total reaction cross sections σR in mb are listed in Table I. We have also studied the elastic scattering of ¹⁷F from the heavy target ²⁰⁸Pb. The elastic scattering angular distributions of ¹⁷F + ​208Pb were measured at 98 MeV and 120 MeV ^[21], 90.4 MeV ^[22], 170 MeV ^[24], and 86 MeV ^[25]. We used the calculated DF potentials for this system to describe the experimental data at these energies. The results of our calculations for the ¹⁷F+²⁰⁸Pb system are presented in Fig. 4 compared with the experimental data. At energies of 86, 90.4 and 98 MeV, the calculated angular distributions of the differential cross section produced reasonable agreements with experimental data for the three considered energies as shown by the solid and dashed lines in Fig. 4, except the few forward angles data (θc.m>90∘) at 98 MeV case and the few backward angles data (θc.m>150∘) at 86 and 90.4 MeV. For energies 120 and 170 MeV, successful reproduction of data is obtained. Also, it is evident from these figures that the present predictions for the 120 and 170 MeV data are more successful than those predicted for the 86, 90.4 and 98 MeV. It is noticed from these figures and the values of χ2 shown in Table I that fits with data obtained using the version A of the density distribution of the ¹⁷F nucleus is better than those resulted using version B. An additional important piece of information that can be deduced from the elastic scattering analysis, is the total reaction cross section as well as the determination of the Coulomb barrier ^[46,47]. The Coulomb barrier strength VB and radius RB can be determined from the following equations:

which yield VB=8.6, 10, 36 and 78 MeV respectively for ¹²C, ¹⁴N, ⁵⁸Ni and ²⁰⁸Pb target nuclei. This information is useful to investigate the role of breakup (or other reaction mechanisms) for weakly bound exotic nuclei.

The obtained reaction cross section values for all considered reactions are given in Table I. The extracted values are quite consistent with those reported in recent studies ^[21-^26]. The obtained total reaction cross sections, σR, for ¹⁷F +²⁰⁸Pb system are plotted versus the energy E, as shown in Figs. 5, 6. We observe that for all considered energies, σR increases with increasing the energy E. The target mass-number dependence of the reaction cross-section for all considered systems at Elab=170 MeV is shown also in this figure. The reaction cross-section increases with increasing the cubic root of the target mass number A1/3. To investigate the energy dependence of the imaginary volume integral, we plotted the calculated imaginary volume integral JW, listed in Table I, against the energy E for ¹⁷F +²⁰⁸Pb reaction.

Figure 5 The energy dependence of the reaction cross-section for ¹⁷F +²⁰⁸Pb system.  

Figure 6 The target mass-number dependence of the reaction cross-section for 17¹⁷F + 12¹⁷C, ¹⁷F + ¹⁴N, ¹⁷F + ⁵⁸Ni and ¹⁷F + ²⁰⁸Pb systems at Elab=170 MeV. 

The imaginary JW values does not show a clear behavior with energy for the DF potential. Also, we plotted the calculated imaginary volume integral JW, listed in Table I, against the cubic root of the target mass number A1/3 for all considered systems at 170 MeV as shown in Figs. 7, 8.

Figure 7 The energy dependence of the imaginary volume integral for ¹⁷F + ²⁰⁸Pb reaction.  

Figure 8 The target mass-number dependence of the imaginary volume integral for ¹⁷F + ¹²C, ¹⁷F + ¹⁴N, ¹⁷F + ⁵⁸Ni and ¹⁷F + ²⁰⁸Pb systems at Elab=170 MeV. 

5. Conclusions

In the present work, we have analyzed the elastic scattering data for ¹⁷F nucleus on light-mass targets (¹²C,¹⁴N), medium-mass target (⁵⁸Ni), and heavy-mass target (²⁰⁸Pb) at different energies in the framework of the OM. The DF model is employed to generate the real part of the optical potentials, by folding the density-dependent DDM3Y effective NN interaction over two versions of the density distribution of the one-proton halo ¹⁷F nucleus besides the density of targets, while the imaginary part is treated phenomenologically through the WS form. Successful predictions of different sets of data at above the Coulomb barrier energies are obtained all over the measured angular ranges without needing to renormalize the generated potentials. Furthermore, the total absorption (reaction) cross sections for the four reactions are investigated. It is found that, the extracted values from the present elastic scattering calculations agree well with the measured data. The energy- and target mass number-dependences have been checked for the resulted reaction cross sections from the derived DF potentials at 170 MeV for the ¹⁷F+²⁰⁸Pb reaction.