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Elastic scattering of one-proton halo nucleus17F on different mass targets using semi microscopic potentials

The elastic scattering of F from different mass targets ( C, N, Ni andPb) at different energies has been studied. We used the double folding optical model potential based on the density-dependent DDM3Y effective nucleon-nucleon interaction without need to renormalize the generated potentials. Two versions of the density distribution of the one-proton halo F nucleus have been taken into account in order to derive the double folding potentials. The measured angular distributions of elastic scattering differential cross section and corresponding reaction cross sections have been successfully reproduced at different energies using the derived potentials. The energy and the target mass number dependences of imaginary volume integrals as well as the total reaction cross sections have been also studied.


Introduction
The nuclei far from β stability line have attracted an immense interest since the first observation of the neutron halo in 11 Li [1]. Further experiments have confirmed the presence of neutron halo in 11 Li and other neutron-rich nuclei [2]. Through the last two decades, experimental data for the proton halo in proton-rich nuclei like 8 B, 17 Ne, and 26,27,28 P have been reported in literature [3][4][5][6][7][8][9][10][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 17 F is one of the candidates for a protonhalo 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 17 F as projectile have been performed in recent years [16][17][18][19][20][21][22][23][24][25][26][27]. It is important to mention that, the 16 O (p, γ) 17 F capture cross section measured in through the energy range E c.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 17 [27][28][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][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37]. In the present work, we have applied the DF model to analyze the elastic scattering of 17 F nucleus on light-mass targets ( 12 C, 14 N), medium-mass target ( 58 Ni), and on 208 Pb 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.

Theoretical Formalism
The real nucleus-nucleus optical potential in the DF model is given by the expression [31] where ρ P (r 1 ) and ρ t (r 2 ) are the nuclear matter density distributions for projectile and target nuclei, respectively, and υ nn (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] v DDM 3Y 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 17 F nucleus, two versions of nuclear matter density are considered. First, it is presumed that the 17 F nucleus consists of an 16 O core and a halo of one proton. The core density distribution is assumed to be of a harmonic oscillator (HO) form as [39] ρ c (r) = ρ 160 (r) = 0.1317(1 + 0.6457r 2 ) where the halo density distribution is described by the Gaussian function as [35]: TABLE I. The best fit imaginary WS potential parameters obtained from the analysis of nine sets of 17 F elastic scattering cross sections. The corresponding real and imaginary volume integrals (JV and JW , respectively), the total reaction cross section (σR), and χ 2 are also tabulated. where R h = 3.78 fm. Therefore, the total matter distribution ρ 17 F (normalized to one nucleon) and the matter radius R m are given as [40]: This density produces a rms radius, r 2 rms 1/2 of 17 F equals to 2.77 fm. We denote this density as version A. In the second approach the density of 17 F nucleus is taken from Ref. [40]. This density produces a rms radius r 2 rms 1/2 of 17 F 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.
For the target nuclei 12 C and 14 N the nuclear matter density has been taken in the harmonic oscillator (HO) form [33,42]: These densities yield rms radii r 2 rms 1/2 equal to 2.407 and 2.48 fm for 12 C and 14 N respectively. For the target nuclei 58 Ni and 208 Pb the nuclear matter density has been taken in the two parameter fermi form [34][35][36][37] ρ 208 Pb (r) = 0.15 1 + exp( r−6.80 0.515 ) .
to analyze the 17 F elastic scattering data. The DF potentials have been developed by considering the procedures given below: (1) The spin-orbit potential has been neglected. It is well known fact that the elastic scattering cross section data in this energy range are not sensitive to this potential [41].
(2) The computer code DOLFIN [45] based on the Fourier-transform technique [32] has been used for this purpose.
(3) The obtained potentials is fed into the code HIOPTM-94 [46] to represent the real part of the optical potential, while the imaginary part of the optical potential is taken in the phenomenological volume Woods-Saxon (WS) shape as MeV, where W 0 , r w and a w are the depth, radius and diffuseness parameters, respectively. Renormalization factors are not considered for the derived real microscopic DF potentials in order to optimize the fits with data.
(4) The resulted potentials are used to analyze nine sets of data of the elastic scattering differential cross section for the following reactions: 1. 17 where σ cal (θ i ) and σ exp (θ i ) are the theoretical and experimental cross sections, respectively, at angle, θ i , ∆σ exp (θ i ) is the experimental error, and N is the number of data points.

Results and Discussion
The derived DF potentials based on the density-dependent DDM3Y effective NN interaction is used to analyze the elastic scattering of 17 F nucleus on 12 C, 14 N, 58 Ni and 208 Pb targets without renormalization factor. The obtained elastic scattering differential cross sections for the considered reactions are listed in Table I 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 17 F + 58 Ni 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 17 F→ 16 O (core)+p (valence) as coupling between (5/2) + ground and (1/2) + states of 17 F are carried out as shown with dot lines in Fig. 3. For this purpose, we used only pure elastic scattering. The obtained parameters of the imaginary phenomenological WS potentials, the corresponding real and imaginary volume integrals per interacting nucleon pair (J V and J W ) in MeVfm 3 and the total reaction cross sections σ R in mb are listed in Table I. We have also studied the elastic scattering of 17 F from the heavy target 208 Pb. The elastic scattering angular distributions of 17 F + 208 Pb 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 17 F+ 208 Pb 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 (θ c.m > 90 • ) at 98 MeV case and the few backward angles data (θ c.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  Table I that fits with data obtained using the version A of the density distribution of the 17 F 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 V B and radius R B can be determined from the following equations: which yield V B = 8.6, 10, 36 and 78 MeV respectively for 12 C, 14 N, 58 Ni and 208 Pb 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][22][23][24][25][26]. The obtained total reaction cross sections, σ R , for 17 F + 208 Pb system are plotted versus the energy E, as shown in Figs. 5, 6. We observe that for all considered energies, σ R increases with increasing the energy E. The target mass-number dependence of the reaction cross-section for all considered systems at E lab = 170 MeV is shown also in this figure. The reaction cross-section increases with increasing the cubic root of the target mass number A 1/3 . To investigate the energy dependence of the imaginary volume integral, we plotted the calculated imaginary volume integral J W , listed in Table I, against the energy E for 17 F + 208 Pb reaction.  The imaginary J W values does not show a clear behavior with energy for the DF potential. Also, we plotted the calculated imaginary volume integral J W , listed in Table I, against the cubic root of the target mass number A 1/3 for all considered systems at 170 MeV as shown in Figs. 7, 8.

Conclusions
In the present work, we have analyzed the elastic scattering data for 17 F nucleus on light-mass targets ( 12 C, 14 N), medium-mass target ( 58 Ni), and heavy-mass target ( 208 Pb) 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 17 F 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 numberdependences have been checked for the resulted reaction cross sections from the derived DF potentials at 170 MeV for the 17 F+ 208 Pb reaction.