The folding potential description of ⁹C + ²⁰⁸Pb elastic scattering at 227 MeV

Abstract

The sensitivity of the ⁹C + ²⁰⁸Pb elastic scattering reaction at E_lab= 227 MeV (three times the barrier energy) towards the yielded potential combinations (⁹C density distributions + effective nucleon–nucleon interactions), bringing about a decent portrayal of the data, was estimated all through the optical double folding model. The merits and demerits of these combinations with regard to experimental data were handled, other than the comparability of results to those acquired by others. For the well-being of culmination, the S-matrix elements modulus, notch perturbation test and alteration of potential parameters are all used to highlight and reiterate some noteworthy aspects of the examined scattering reaction.

KEYWORDS:

• Elastic scattering
• double folding model
• S-matrix and notch test

PACS:

• 13.85.Dz
• 24.10.–i,24.10.Ht
• 25.40.Cm
• 87.15.Cc

1. Introduction

Other than ³He, ⁹C has the most proton-rich nucleus known to be particle bound, with a proton-to-neutron ratio of Z/N = 2. ⁹C's ground state decays to ⁹B through beta decay with a half-life of 126.5 ms. It is a member of the T = 3/2, A = 9 isobaric quartet together with ⁹Li,⁹Be and ⁹B (T = 3/2). It is thought to have two decay channels: one proton (1p) and two protons (2p). Because the 1p separation energy from ⁹C (⁹C → ⁸B + p) is 1.29 MeV and from ⁸B (⁸B → ⁷Be + 2p) being just 0.136 MeV, the 2p channel is all but favoured. Accordingly, recent research has zeroed in on experimental and theoretical studies of its exotic nature [1–7] as it is particularly significant in astrophysics [8–10]. In relation to this, we give a speedy synopsis of the main ones.

The removal reaction cross section of either 1p (σ_1p) or 2p (σ_2p) for ⁸B and ⁹C is estimated by Blank et al. [1], on a set of targets with different mass (A = 12-208) at the energy of 285 MeV/n. Strong evidence supports a 2p configuration for ⁹C given the higher values of deliberated σ_2p compared to σ_1p over two times, which are somewhat dependent upon the various target nuclei masses. Enders et al. [2] and Warner et al.[3] assessed the σ_{2p /} σ_1p ratio and viewed it as deliberately understanding with Blank et al. [1] results.

Furthermore, thorough analyses of the measured and computed scattering reaction (absorption) cross section (σ_R) and the break-up reaction cross section for the angular distributions of our investigated system ⁹C + ²⁰⁸Pb and others were provided [11–15] at various energies. Merely, the only available measurements and analysis of the elastic scattering and break-up processes were done at high incident energies for the ⁹C + ²⁰⁸Pb system. It is worth noting that breakup reactions are not within the scope of this investigation.

Assessing the elastic scattering data for the reactions induced by proton-rich nuclei might reveal some insight into their unusual structures and reaction mechanisms. In this instance, many models may be used to investigate how sensitive the measured elastic scattering data and cross section are to various factors, particularly the nuclear target-projectile potential elements (projectile/target density, effective nucleon–nucleon (NN) interaction) and breakup effects.

For example, utilizing the elastic scattering angular distribution of the ⁹C + p reaction at 277–290 MeV, Matsuda et al. [16] determined the phenomenological two-parameters Fermi (2pF) density distributions and root-mean-square radius of ⁹C nuclear matter by using the relativistic folding model formulated by Murdock and Horowitz [17]. This investigation was unable to provide a trustworthy density distribution in ⁹C or a satisfactory prediction of the reported cross sections. Rafi et al. [18] then reanalysed the same scattered data using the Argonnev-18 NN interaction [19] with the relativistic mean field (RMF) density in the Brueckner–Hartree–Fock model [20]. The derived potential, according to the authors, was able to offer a good level of agreement with experimental results. Additionally, it was necessary to corroborate their findings using experimental data collected over a considerably larger area. Within a novel analytic approach, Bonacorsso et al. [21] evaluated the sensitivity of σ_R for both ⁹C and its mirror nucleus ⁹Be among a series of microscopic densities; Variational Monte Carlo (VMC) [22,23], antisymmetrized molecular dynamics (AMD) [5], relativistic Hatree–Fock (HF) [24] and cluster model (CM) [25] with Jeukenne, Lejeune and Mahaux (JLM) NN interaction [26]. Based on the computed potentials from the optical glauber model, none of those distributions could represent the low energy increase of the σ_R, except when the surface term correction was included.

Further measurement on ⁹C,⁸B,⁷Be + ²⁰⁸Pb elastically scattering at 227,178 and 130 MeV, in that sequence, were reported by Yang et al. [11]. As expected for this incident energy, very small break-up effects were observed. Therefore, the calculated break-up cross sections by using the continuum discrete coupled channel (CDCC) method are viewed as sensitive to the structure of proton-rich nuclei at high incident energies rather than the elastic scattering one as checked by the authors. Hence, for the two ⁹C cluster condurations underlined above, investigations of the measured elastic scattering cross sections on ²⁰⁸Pb are in like manner practically same.

These in-depth measurements and analyses paved the way for more research including proton/neutron-rich unstable nuclei and stable nuclei as well.

Yong Li et al. [12] put upon a systematic global optical model potential (OMP) of ¹²C and ⁹Be to explore the elastic scattering data and the σ_R of ^9–11,13,14C projectiles. The results show that this global potential adequately describes the elastic scattering for the triggered reactions by ^9–11,13C, with disagreement at high angles in the case of ¹⁴C due to break-up effects. What is more, they claimed that this discrepancy could be fine-tuned by varying the potential parameters. Rong et al.[13] examined the responsivity of the Sao-Paulo (SP) [27,28] and Akyüz Winther (AW) [29,30] NN potential along with phenomenological density (2pF) form in ¹⁷F observed elastic scattering on ²⁰⁸Pb at 94.5 MeV (just above the barrier) within the double folded potential (DFP). It is discovered that the extracted potentials exhibit similar behaviour. A fair description of the experimental data was provided by the obtained potentials up to a scattering angle 100^o, however, for large angles, the data were under-predicted. After that, fitting quality enhancement in the intermediate angles was achieved by applying two coupled channel methods. As they stated, the ¹⁷F break up has a negligible influence on the elastic scattering and is independent of target mass. They also summarized other publications on the issue. In the optical DFP calculations based on the density-independent Michigan three-Yukawa (M3Y) NN interaction [31], El-Hammamy et al. [32] examined three density forms, namely Gaussian–Oscillator (GO), Gaussian–Gaussian (GG) and Gaussian (G) [33–36] for the ¹⁷F nucleus scattered by ²⁰⁸Pb at 90.4 MeV (just below the barrier). Once again, by varying the ¹⁷F density forms, the resultant optical DFPs showed almost the same behaviour. However, in their research, the authors attempted to provide a plausible debate of the chosen combination (GO + M3Y). Moreover, the behaviour of the extracted σ_R values is found to be much more similar in comparison with other investigated systems ¹⁷O,¹⁶O + ²⁰⁸Pb than ¹⁹F + ²⁰⁸Pb at nearly the same energies. This proved the weak break-up effect at the energy range around the coulomb barrier.

Anwar et al. [37] examined semi-phenomenological (SP) [38,39], HF, GO and CM [40] density distributions for ⁸B nucleus scattered from ⁵⁸Ni at 20.7–29.3 MeV enhanced by SPP and CDM3Y6 NN interactions [41–43] within the optical DFP. The ability of their established potentials, especially SPP with SP/HF for halo ⁸B nucleus, is found to reproduce successfully the experimental data. Kassem et al. [44] investigated the best ¹¹Li density distribution forms for SP, cluster-orbital shell model approximation (COSMA) [45], and HF in conjunction with several NN interactions (DDM3Y [46], CDM3Y6-RT [47] and SPP) via the optical DFP for ¹¹Li scattered from ¹²C and ²⁸Si at 29, 50 and 60 MeV/n. The major conclusion of this work is that the considered densities and NN interactions predict the best match with experimental data. Similar studies are presented in Refs. [48–55] and authors debated the features of their assorted ingredients of the theoretical models that were used.

Regarding all of the prior research, it is still not clear whether single Yukawa (S1Y) [56] or Knyzakov and Hefter (KH) [57] effective NN interactions with a straightforward density forms for proton-rich nuclei (projectile/target) can be used to produce nuclear potentials in the same way they can for other types of nuclei.

For case, Satchler [56] showed the success of his suggested NN interaction S1Y in analysing 36 sets of heavy ion (HI) elastic scattering at intermediate energies by DFP. Despite his limitations for light HI scattering [56], Farid and Hassanian tested the validity of using four NN interactions, known as S1Y, M3Y, KH (3 terms) and JLM in DFP of ^6,7Li [58,59] projectiles at intermediate energies. They concluded that successful predictions of the data by different NN interactions occurred in the following order: M3Y and S1Y, KH, JLM, with only one chosen published density form for each projectile. In Ref. [60], the KH NN interaction was also used in the potential of several projectiles. Esmael and Allam [61] analysed the elastic scattering of proton from ¹⁶O target at 135 and 200 MeV utilizing a potential built with three forms of KH NN interaction and three densities for ¹⁶O based on optical single folded potential. The analysis confirmed that the potential sets obtained with a single G term (KH1) NN potential give the best description of the data. Anwar [53] reanalysed the elastic scattering cross sections of ⁶Li from various targets at 240 MeV with S1Y and M3Y and three density forms named 2pF, GO and G density distributions for ⁶Li nucleus. The derived potential combination (2pF density + S1Y) predicts an excellent agreement of the data over all measured angular ranges in DF calculations.

Hence, stretching out the assessment of these NN interactions (S1Y and KH) in contrast to the most popular and successful ones (M3Y and CDM3Y6) to the instance of ⁹C isotope seems epochal. Along these lines, the ongoing study centres around gauging the sensitivity of the recently published elastic scattering angular distributions of ⁹C + ²⁰⁸Pb reaction at E_lab= 227 MeV [11] above the coulomb barrier to the constructed potentials by using numerous combinations of ⁹C density distributions and effective NN interactions within the optical DFP which produces a successful description of the data. The outcomes were compared to those obtained for thoroughness. The corresponding techniques for these calculations; S-matrix elements modulus, notch perturbation test, and alteration of potential parameters are also employed to emphasize and restate some remarkable aspects of the examined scattering reaction. So, it should be mentioned that this work supplements our earlier research for different nuclear systems [32,62,63].

This piece of work is broken into four parts. This section 1 started with a concise introduction, followed by the formalism of Section 2. Section 3 renders the results and discussions, while Section 4 provides the conclusion.

2. Formalism

The nuclear potential of the standard OM is separated into two portions for theoretical calculations: the real $V$ and imaginary $W$ potentials, which are specified as (1) $\begin{array}{r}U(R)=V(R)+\mathrm{iW}(R).\end{array}$(1) The following phenomenological Woods–Saxon (WS) form was proposed for both real and imaginary components, (2) $\begin{array}{r}X(R)=-\frac{X_0}{1+\exp ((R-R_X)/a_X)}\text{and}(\text{X}=\text{V,W}),\end{array}$(2) where X_o, a_X and $R_X=r_X(A_p^{1/3}+A_T^{1/3})$ are potential depth, diffuseness and radius with A_P and A_T mass numbers of projectile and target nuclei, individually. Furthermore, the Coulomb potential U_C(R) for uniformly charge distributions of the colliding nuclei with the radius $R_C=1.3(A_p^{1/3}+A_T^{1/3})$ is concluded in Equation (1) [64].

It is well known that the phenomenological representation does exclude the description of the structure of a projectile or target. The present elastic-scattering data were analysed using the program HIOPTIM-94 code [65], in which the phenomenological real potential was supplanted by the DFP. As regards the nucleus–nucleus interaction, we adopt the same DFP used in Refs. [31,66] (3) $\begin{array}{rl}{V}_F^{}(R)& =N\iint {\rho }_1(r_1){\rho }_2(r_2)v_{\mathrm{NN}}^{}(s)d{r}_{1}d{r}_2,\\ & (s=R+r_2-r_1)\end{array}$(3) where ρ₁ and ρ₂ are nucleon densities of the two colliding nuclei, R signifies the distance between the nuclei’s centres of mass, (s) is the relative vector between the interacting nucleon pair, N is renormalization factor and v_NN (s) stands for the effective NN interaction used in the calculations. Several v_NN (s) variants can be implemented for ascertaining the DFPs via the DFPOT code [67]. In this piece of work, four density-independent variants and one density-dependent variant are explored.

There are two variants with a single term, either in Gaussian or Yukawa form. The first is the S1Y interaction, which is indicated by [56] (4) $\begin{array}{r}{v}_{\mathrm{NN}}(s)=-60\frac{\exp (-1.428s)}{1.428s}\left(1-0.0005\frac{E}{A_P}\right),\end{array}$(4) with a depth V₀ of ≈60 MeV, where E and A_P are the incident energy and the projectile mass number, respectively. KH-1 interaction is employed for comparative purposes as Knyzakov and Hefter [57]. This is chosen as a second option, and its configuration is as per the following: (5) $\begin{array}{r}{v}_{\mathrm{NN}}(s)=-20.97\exp (0.68s{)}^2.\end{array}$(5)

The other two variants are comprised of two terms plus Zero range exchange term (3 terms). The third interaction, KH-3 [57] is expressed as (6) $\begin{array}{rl}{v}_{\mathrm{NN}}(s)& =-601.99\exp (1.25s{)}^2+2256.4\exp (2s{)}^2\\ & -276[1-0.005E/A_P]\delta (s)\end{array}$(6) Then again, the most common three Yukawa terms M3Y-Reid effective NN interaction [66] is formed as (7) $\begin{array}{rl}{v}_{\mathrm{NN}}(s)& =7999\frac{\exp (-4s)}{4s}-2134\frac{\exp (-2.5s)}{2.5s}\\ & -276[1-0.005E/A_P]\delta (s)\end{array}$(7)

The last variant consists of two direct terms plus three finite range exchange terms. The fourth interaction, known as CDM3Y6, depends on the M3Y-Paris density-dependent interactions and is described as [42,43] (8) $\begin{array}{rl}{v}_{\mathrm{NN}}(s)& =F(\rho ,E)\ast [(-11602\frac{\exp (-4s)}{4s}\\ & -2538\frac{\exp (-2.5s)}{2.5s})+(-1524\frac{\exp (-4s)}{4s}\\ & -518.8\frac{\exp (-2.5s)}{2.5s}-7.847\frac{\exp (-0.707s)}{0.707s})]\end{array}$(8) $\begin{array}{rl}F(\rho ,E)& =[1-0.003(E/A_p]\\ & \ast [0.2658(1+3.8033\exp (-1.41\rho )-4\rho )]\end{array}$

Because the density distribution of colliding nuclei is basic in folding calculations (see Equation 3), three alternative matter density distributions for the exotic nucleus ⁹C ground state were employed, considering various models of the internal nuclear structure.

The first is phenomenological Gupta1-two parameters Fermi (G1-2pF) [68,69] (9) $\begin{array}{r}\rho (r)=0.237{\left(1+\exp \left(\frac{(r-1.787)}{0.436}\right)\right)}^{-1},\end{array}$(9) with ⁹C rms radius $⟨r^2{⟩}^{1/2}=2.131$ fm, which varies from the value in Ref. [16].

In the second one, we look at the (⁷Be + 2p) cluster structure of the ⁹C nucleus [11]. In this case, we treat ⁹C density as follows: (10) $\begin{array}{r}{\rho }_{{}^{9}C}(r)={\rho }_{{}^7\mathrm{Be}}(r)+{\rho }_{2P}(r).\end{array}$(10) The core (⁷Be) and halo (2p) are presented with different spatial distributions in some phenomenological different forms. The density of the core nucleons is selected to be Gaussian (G), whereas the density of the halo nucleons is characterized by Harmonic Oscillator (HO) [33–35]. The accompanying equations show the density distribution shapes that were used: (11) $\begin{array}{r}{\rho }_{{}^7\mathrm{Be}}(r)=\mathrm{Cexp}\left(-\frac{r^2}{{\alpha }_{{}^7\mathrm{Be}}^2}\right),\end{array}$(11) and (12) $\begin{array}{r}{\rho }_p(r)=\frac{2}{3}C\left(\frac{r^2}{{\alpha }_p^2}\right)\exp \left(-\frac{r^2}{{\alpha }_p^2}\right),\end{array}$(12) where $\begin{array}{rl}{\alpha }_{{}^7\mathrm{Be}}^2& =\frac{2R_{{}^7\mathrm{Be}}^2}{3},{\alpha }_p^2=\frac{2R_p^2}{5},C=N_j{\rho }_{\circ j}\mathrm{and}\\ {\rho }_{\circ j}& ={\left(\frac{1}{\pi {\alpha }_j^2}\right)}^{3/2}\mathrm{with}j={}^7\mathrm{Be},p.\end{array}$where R and N_j are the r.m.s radii and number of the core or halo nucleons, in that order, depending on the density used form. The r.m.s radii for ⁷Be and 2p equal 2.23 fm [70] and 3.542 fm, separately, from this density. For this situation, the resultant Gaussian Oscillator (GO) density has a radius of 2.58 fm (⁹C r.m.s) [5].

Finally, the proton and neutron densities of ⁹C nucleus have been obtained in a microscopic way by the RMF computations as the third density distribution form. Proton and neutron densities are acquired from Ref. [18] and the derived ⁹C r.m.s radius = 2.522 fm.

Figure 1 depicts the three particular densities of the ⁹C nucleus. The (2pF) form is employed to achieve the desired target ²⁰⁸Pb density [71] (13) $\begin{array}{r}\rho (r)=\frac{{\rho }_{\circ }}{1+\exp (\frac{r-6.621}{0.551})},\end{array}$(13) and the deduced r.m.s radius is 5.52 fm.

Figure 1. Panels (a) and (b) show the nuclear matter density distributions of ⁹C as logarithmic and linear scales, respectively.

The elastic-scattering differential cross sections shaped by the five effective interactions S1Y, KH1, KH3, M3Y and CDM3Y6 with three density distributions not entirely set in stone, as shown in Figures 2–4 and compared to the experimental results in Figures 5,6. A computerized search is performed to maximize the fits to data by diminishing the prescribed χ² (14) $\begin{array}{r}{\chi }^2=\frac{1}{N}\sum _{i=1}^N{\left[\frac{{\sigma }_{\mathrm{th}}({\theta }_i)-{\sigma }_{\exp }({\theta }_i)}{\mathrm{\Delta }{\sigma }_{\exp }({\theta }_i)}\right]}^2,\end{array}$(14) where ${\sigma }_{\mathrm{th}}({\theta }_i)$ and ${\sigma }_{\exp }({\theta }_i)$ are the theoretical and experimental cross sections at angle $({\theta }_i)$ respectively, $\mathrm{\Delta }{\sigma }_{\exp }({\theta }_i)$ is the experimental error, and N is the difference number of data points and fit parameters. The χ² values are calculated using a 10% statistical error for all analysed data.

Figure 2. The variation of real folded potential depths by using different density forms of different NN interactions for ⁹C + ²⁰⁸Pb system at 227 MeV.

Figure 3. The real folded potentials by using RMF density form and different NN interactions for ⁹C + ²⁰⁸Pb system at 227 MeV.

Figure 4. The normalized real folded potentials by using different-NN interactions form and different ⁹C densities for ⁹C + ²⁰⁸Pb system at 227 MeV in FM-1 scenario.

Figure 5. The elastic-scattering differential cross sections in comparison with the experimental data of the ⁹C+ ²⁰⁸Pb reaction at 227 MeV [11] by using different combination of densities and NN interactions within DFP of scenario FM-1.Labels indicated on the figures.

Figure 6. The elastic-scattering differential cross sections in comparison with the experimental data of the ⁹C+ ²⁰⁸Pb reaction at 227 MeV [11] by using different combination of densities and NN interactions within DFP of scenario FM-2. Labels indicated on the figures

The real renormalization factor (N_R), as well as the three imaginary WS parameters, are looked for in the first scenario of the double folding model (FM-1). Notwithstanding, there are just two parameters for the second scenario (FM-2): the real (N_R) and imaginary (N_I) renormalization factors, with the real and imaginary potential parts having the same folded shape. Table 1 shows the calculated parameters, the volume integrals per pair of interacting nucleons for the real (J_R) and imaginary (J_I) parts of the potential, the reaction cross-section (σ_R), and the best-fit χ² for every scenario. In the succeeding segments, a thorough discussion is given to exist results.

Table 1. Best-fit parameters of the double folded potentials (DFPs) using the two scenarios; FM-1 and FM-2 for ⁹C + ²⁰⁸Pb scattering data at 227 MeV.

Density	Interaction	Model Scenario	NR	W /NI (MeV)	R (fm)	a (fm)	JR (MeV fm^3)	JI (MeV.fm^3)	σR (mb)	χ^2
G1-2pF	M3Y	FM-1	1.05	23.01	1.404	0.240	406.40	73.48	3140	0.35
		FM-2	1.20	1.49	–	–	464.33	574.67	3233	1.24
	KH1	FM-1	1.03	22.28	1.405	0.226	375.84	71.27	3120	0.34
		FM-2	1.10	1.33	–	–	403.44	488.36	3237	1.38
	S1Y	FM-1	0.96	13.82	1.447	0.187	215.58	48.13	3181	0.36
		FM-2	1.21	1.20	–	–	271.46	269.19	3038	1.51
	CDM3Y6	FM-1	0.81	13.84	1.440	0.138	256.73	47.53	3106	0.34
		FM-2	0.79	0.60	–	–	249.42	190.04	2959	1.76
GO	M3Y	FM-1	0.82	61.62	1.369	0.202	318.94	182.15	3041	0.34
		FM-2	0.75	0.78	–	–	290.19	301.09	3263	2.40
	KH1	FM-1	0.83	54.18	1.384	0.172	304.96	165.34	3037	0.36
		FM-2	0.72	0.73	–	–	264.05	268.03	3270	2.69
	S1Y	FM-1	0.95	29.46	1.394	0.215	212.88	91.86	3082	0.35
		FM-2	1.02	1.07	–	–	228.77	239.00	3244	2.26
	CDM3Y6	FM-1	0.58	28.24	1.406	0.170	205.32	90.25	3069	0.35
		FM-2	0.55	0.59	–	–	195.46	209.39	3262	2.34
RMF	M3Y	FM-1	0.78	70.78	1.365	0.215	302.98	207.33	3062	0.33
		FM-2	0.80	0.78	–	–	309.51	302.42	3259	2.63
	KH1	FM-1	0.75	75.21	1.362	0.202	281.92	219.08	3031	0.34
		FM-2	0.70	0.72	–	–	256.71	265.06	3251	2.34
	S1Y	FM-1	0.91	42.27	1.390	0.214	203.91	130.85	3107	0.35
		FM-2	1.00	1.08	–	–	224.32	242.03	3237	2.01
	CDM3Y6	FM-1	0.78	41.33	1.386	0.140	273.34	126.67	2971	0.42
		FM-2	0.55	0.59	–	–	192.17	205.08	3250	2.18

3. Results and discussions

3.1. Comparability of density parameterizations and real nuclear potentials

The two elements that comprise the optical DFP of Equation (3) have been examined in the field of the elastic scattering angular distributions of ⁹C + ²⁰⁸Pb reaction at energy 227 MeV.

To begin, Figure 1 conveys the radial dependency of the three selected density parameterizations of ⁹C, meant as G1-2pF, GO and RMF. The GO and RMF density distributions are near to one other and have a lower value in the nucleus's centre than the phenomenological G1-2pF, as displayed in Figure 1 panel (a). Simply put, as the radius rises, they tumble to the most elevated. As well, it has been shown that the microscopic density RMF draws out astoundingly farther than the other densities, as seen in Figure 1 panel (b). As an aftereffect, the r.m.s radius of ⁹C is lengthy from 2.13 to 2.58 fm.

Then, with the chosen densities of both the ⁹C projectile and the ²⁰⁸Pb target, Equation (3) was coordinated across S1Y, KH1, KH3, M3Y and CDM3Y6-NN interaction potentials. DFPOT code [67] was used to generate real DFPs. Table 1 and Figure 2 show the noticed parameters related to the determined potentials.

Just a single figure, Figure 3 is shown when the RMF density is in use, because the estimated DFPs with the other densities have about the same depths and ways of behaving with all NN interactions. At the point when the radial dependency of the five built DFPs is examined, it is discovered that S1Y is the shallowest of the five. Nonetheless, the extracted depths of M3Y and KH3 potentials are similar in value and fairly greater than that of KH1 potential. The CDM3Y6 potential is almost in the middle of the preceding potentials. Thus the KH3 potential generated results are eliminated from the analysis.

Additionally, the comparison of the normalized real DFPs for various NN interactions demonstrates the effect of the used densities. Hence, this is shown in Figure 4 in the framework of FM-1 scenario, for instance. It is obvious from four panels (a–d) that the three densities have a slight effect on the S1Y potentials as shown in (c). For M3Y, KH1 and CDM3Y6, the situation is different, the effect of used densities rises in the radial distance region from 0 to 8 fm. These divergences are reflected on the calculated parameters in Table 1.

3.2. Fitting quality of elastic scattering folding models

At 227 MeV, the FM-1 is used to investigate elastic scattering cross-section data of ⁹C +²⁰⁸Pb reaction. In this scenario, the real DFP of Equation (3) was created using the DFPOT code [67] by consolidating the three density distributions of the ⁹C projectile, the five grouped designated NN interactions and 2pF density for the ²⁰⁸Pb target. In addition to the phenomenological WS imaginary potentials, the subsequent potentials were fed into the HIOPTIM-94 [65]. The standout accord between theoretical calculations and experimental data is uncovered in Figure 5, and the resulting parameters are catalogued in Table 1. To execute the fitting method, just one free parameter N_R for the real potential and three parameters for the imaginary potential were used.

The S1Y, KH1, M3Y and CDM3Y6 interaction potentials with G1-2pF, GO and RMF densities demonstrate the comparable way of behaving with each other with approximately identical χ² values. Also, there is a minor underrating of the data at θ∼(12–15)^o for those potentials that correspond to the phenomenological G1-2pF density. As found in Table 1, this is mirrored by the shallowest imaginary potential values. In any case, none of the computations could identify the experimental data in the forward angle θ∼(9–12)^o.

For comparison, the resultant scattering cross section with data is clearly on par with those detailed in Ref. [51] in the framework of density-independent M3Y-NN interaction coupled with RMF density for ^10,11C + ²⁰⁸Pb reactions at 226 and 256 MeV, one by one. Similarly, the experimental data in forward angle θ ∼ (7–11) ^o couldn't be achieved in that analysis.

As far as density distribution sensitivity, any of the ⁹C may be compensated by the N_R and imaginary parameters, regardless of the NN interaction type. It is observed that the strength of the real potential should be reduced by ∼12–28% (see Table 1) to describe the experimental data. the extracted average N_R is 0.88 ± 0.146, 0.87 ± 0.144, 0.94 ± 0.027 and 0.72 ± 0.125 from M3Y, KH1, S1Y and CDM3Y6 calculations. The best match, nonetheless, is obtained for the associated parameters of S1Y interaction potential added to either GO or RMF density. Rather than KH1 and M3Y interaction potentials, this is upheld by a minor deviation of N_R values from unity and a shallow imaginary depth. However, it has almost the same imaginary depths as CDM3Y6 potential. The S1Y potential is marginally sensitive to various kinds of densities, with $N_R^{G1-2\mathrm{pF}},N_R^{\mathrm{GO}},N_R^{\mathrm{RMF}}$ values of 0.96, 0.95 and 0.91, respectively. As an outcome, the values of the real J_R and imaginary J_I volume integrals made by KH1and M3Y potentials are in magnificent agreement and greater than those for S1Y and CDM3Y6 potentials within the same density.

The same elastic scattering cross-section data has been reanalysed employing the FM-2 scenario to look at the strength of the model. In this scenario, the imaginary potential, as well as the real one, were obtained in a DFP type of Equation (2) with two separate renormalization factors. So, just two free parameters; N_R and N_I were uninhibitedly adjusted to get the greatest amicability between theoretical predictions and experimental data, as shown in Figure 6 and Table 1. It is ascertained that the strength of the real potential should be reduced by ∼8–37% to describe the experimental data. the extracted average N_R is 0.92 ± 0.247, 0.84 ± 0.225, 1.08 ± 0.116 and 0.63 ± 0.139 from M3Y, KH1, S1Y and CDM3Y6 calculations.

Applying this scenario, the experimental data in forward angle θ ∼ (7–11)^o couldn't be described once again, as perceived in Figure 6, add-on to under estimation at θ > (17) ^o for all density and potential combinations with the exception of G1-2pF for S1Y and M3Y potentials, which misjudges experimental results at θ ∼ (12–17) ^o. likewise, this is same as indicated in Ref. [11] by using JLM interaction potential and the σ_R value got close to those premeditated for ⁹C (3396 mb).

Similarly, the J_R and J_I volume integrals values made by KH1 and M3Y potentials are higher than those generated by S1Y and CDM3Y6 potentials within the same density, as written in Table 1. It displays also, a small disparity among between N_R and N_I values. Therefore, the highest accord with experimental data is demonstrated by either GO or RMF and S1Y combinations, as it is cleared in Figure 6 and Table 1.

The processed results lay out a high sensitivity to the N_R and W/N_I, which is most likely due to the breakdown of the ⁹C projectile in the field of the heavy target nucleus, ²⁰⁸Pb. All of the σ_R values got are as per each other and close to those distributed in Ref. [11] for ⁹C and [51] for ^10,11C + ²⁰⁸Pb reactions at 226 and 256 MeV, on an individual basis. As a result of this investigation, the FM-1 scenario associated with S1Y-NN interaction and GO or RMF density for ⁹C projectile is preferred.

Sadly, because of an absence of additional energies for the analysed reaction, the overall trend of depths or volume integrals changing with energies still up in the air.

3.3. S-matrix

The modulus of partial wave elastic scattering matrix (S-matrix) elements $|S_L|$ [72–74] as a function of projectile-target orbital angular momentum (L) is introduced in Figure 7. This curve, otherwise called an absorption profile [72], offers a measure of absorption degree and, as a result, the σ_R, the two of which are given by: (15) $\begin{array}{r}{S}_L=e^{2i{\delta }_L}\end{array}$(15) and (16) $\begin{array}{r}{\sigma }_R=\frac{2\pi }{k^2}\sum _L(2L+1)(1-{|S_L|}^2),\end{array}$(16) in terms of the nuclear reflection coefficient ${\eta }_L=|S_L|$ and nuclear phase shift ${\delta }_L$, for complex optical potentials. Where, $(1-|S_L{|}^2)=T_L$ is the nuclear transmission coefficient and k is the wave number. The magnitude $|S_L|$is considered the most important quantity for describing elastic scattering data. It is powerfully related to L, where $|S_L|\approx 0$ which means almost (complete absorption) at small L, increasing $|S_L|$ for intermediate L (partial absorption), whereas $|S_L|\approx 1$ at large L, which explicit (no absorption). Then $T_L\approx 0$is taken as complete transmission while $T_L\approx 1$ is taken as complete reflection. This usual behaviour is shown in Figure 7 and is tantamount to that of the ¹¹Be + ²⁰⁸Pb reaction elastic scattering at 210 MeV, which is roughly 5.2 times the coulomb barrier as portrayed in Ref. [14].

Figure 7. Modulus of the scattering matrix $|S_L|$ to the calculated real folded potential by using RMF density and S1Y-NN interaction within DFP of scenarios; FM-1 and FM-2 versus the orbital angular momentum (L).

The FM-2 potential has the biggest cross section because the $|S_L|$ attains the unitary value at a more prominent L than the FM-1 potential. This is coincidence with what has been stated in the literature [21,75]. But then, the FM-2 potential produces a more keen $|S_L|$, coming about in rather lower critical angular momentum ($L_{1/2}$) and strong absorption radius ($R_{1/2}$) at which both $|S_L{|}^2\mathrm{and}{T}_L=\frac{1}{2}$. The $R_{1/2}$ is commonly described as the distance of the coulomb trajectory's closest approach for the partial waves $L_{1/2}$. Since elastic scattering cross sections are generally impacted by the surface region, changes of $|S_L|$ around L_1/2 (surface collisions) result in fairly differing elastic scattering angular distribution characteristics.

Precisely, these alterations are believed to be brought about by the imaginary ${\delta }_L$ activity in the surface region nearby$L_{1/2}$. This is supported by the way that the zero values of the $|S_L|$at the low angular momentum part mirrors the complete strong absorption effects connected with the higher value of the reduced imaginary potential $(W/V{)}_{R_{1/2}}$ [72] for FM-1 potentials than the FM-2 ones as seen in Figure 7 and Table 2. Thereupon, this confirms the preference for FM-1 over FM-2 scenario in the preceding debate.

Table 2. Values of some characteristic quantities accompanied by the best-fit parameters of the selected bold case in Table 1.

Interaction/density	Model Scenario	R${}_{\frac{1}{2}}$ (fm)	L${}_{\frac{1}{2}}$ ($\hslash$)	W / V${}_{(R\frac{1}{2})}$
S1Y/ RMF	FM-1	11.45	91.26	1.9
	FM-2	11.14	93.68	1.1

3.4. Notch test

Notch perturbation calculations [53,76–78] were used to determine the radial area of the potential that is susceptible to scattering. This technique involves scanning a radial perturbation across the potential. The ${\chi }^2(R)/{\chi }_o^2$ value of the fit remains steady in radial regions where there is no sensitivity, whereas it deteriorates considerably in sensitive parts. ${\chi }^2(R)$ and ${\chi }_o^2$ are the best-fit values obtained for the perturbed and unperturbed real potentials, respectively.

In this work, we use a notch with an amplitude of 1 fm and a width of 0.5 fm to instigate a small perturbation at radius R in the radial distribution, where R fluctuates regularly from 0 to 18 fm in 0.5 fm increments. The results of such a scan for the ⁹C + ²⁰⁸Pb reaction with the best-selected potential within the FM-1scenario are presented in Figure 8. The sensitive zone in ⁹C elastic scattering goes from about (R = 12–14 fm). The ratio W/V = 1.9 (in the nuclear surface) at the radius of greatest sensitivity for the real potential (R_S=12 fm) near the empirical strong absorption radius ($R_{1/2}$ = 11.45 fm) as bestowed in Table 2 attests the domination of the absorptive potential. As well as, this severely restricts the quantity of information that can be obtained about the real one.

Figure 8. Radial sensitivity of elastic-scattering differential cross sections to the calculated real folded potential by using RMF density and S1Y-NN interaction within DFP of scenario FM-1.

Accordingly, this surface localized absorption suggests that the reaction mechanism is controlled by direct reactions, which is predicted at energies over the barrier. What is more, the somewhat larger R_S radius than $R_{1/2}$ radius result is consistent with that reported in Ref. [78] for various weakly bound systems scattered by heavy targets near coulomb energies. By and large, these results point out that ⁹C acts as a heavy ion projectile rather than a light one, which is in agreement with the systematics found in other weakly bound systems such as ⁹Be [79].

3.5. Sensitivity of the real/imaginary potential

To get some insights into the values expected to describe the experimental data well above the coulomb barrier energy, we modify either the N_R or the imaginary potential depth W_o within the FM-1 scenario associated with S1Y-NN interaction potential and projectile RMF density. The adopted procedure is analogous to those delineated in Refs. [13,80,81]. Figures 9 and 10 show a comparison of the experimental data and the computed results using various values of either N_R or W_o.

Figure 9. The elastic scattering calculated with different values of real folded potential renormalization factor (N_R) by using RMF density and S1Y-NN interaction added to WS imaginary potential within the folding model (FM-1) compared with experimental data as linear and logarithmic scales.

Figure 10. The elastic scattering was calculated with different values of imaginary potential depth (W_o) added to real folded potential by using RMF density and S1Y-NN interaction within folding model (FM-1) compared with experimental data as linear and logarithmic scales.

As should be conferred in Figure 9, a comparison of the calculations with the best fit N_R  = 0.91 to those with N_R= 0.5 and 0.25 supplemented with fixed best suitable imaginary parameters reveals that a slight reduction of N_R to 0.91 than its default value ( = 1) is sufficient to reproduce the experimental data. By raising the reduction to 0.25, the theoretical curve goes slightly downward at θ ∼ (12–14)^o and upward at θ > (16^o) without changing its behaviour.

Perversely in Figure 10, a comparability of the results with the foremost acceptable imaginary potentials depth to those with W_o/2, W_o/10 and W_o≈ 0 further with fixed N_R= 0.91 are presented. A search on the radius (r) and diffuseness (a) parameters was performed to evaluate their sensitivity to the quality of the imaginary potential. The diffuseness parameter was found to be more sensitive than the radius parameter at W_o< W_o/2. This agrees with the end in Ref. [80], merely with different behaviour ascribed to lower energies of their inspected system.

Furthermore, using W_o/10 and W_o≈ 0, oscillating curves are existed. Only, as W_o ≈ 0 broad oscillations emerge in the back angle cross sections. These formations are assigned to the manifestation of the real potential’s refractive power. The absorption necessary to match the data dampens these features in the produced angular distributions. This pattern was connected to the use of deep potentials rather than shallow potentials [79,82,83].

Broadly speaking, this implies that a weak real part contrasted with its relating strong imaginary part in the surface region provokes reasonable agreement with the experimental data. Accordingly, no apparent impacts of associated phenomena (rainbow scattering), which is typical of well-bound heavy nuclei scattering can be detected [11,84]. Homologous investigations are conferred for various systems with miscellaneous conclusions, such as ⁶Li + ²⁰⁹Bi [80], ¹⁷F + ²⁰⁸Pb [13], ¹⁷F + ¹⁴N [83] and ⁷li + ¹³⁸Ba [85].

4. Conclusion

Throughout this study, we have compared our results to the current elastic scattering cross sections of the ⁹C + ²⁰⁸Pb reaction at 227 MeV (3 times over the barrier energy) using two basic scenarios of the optical DFM (FM-1 and FM-2) of the elastic scattering process. Various DFPs in light of the combination of ⁹C density distribution (G1-2pF,GO and RMF), ²⁰⁸Pb (2pF) density distribution and the effective NN interaction (density-independent S1Y, KH1, KH3, M3Y and density-dependent CDM3Y6) were examined. Principally, this analysis glares the undermentioned four significant aspects:

1. All resultant potentials give a similar outstanding prediction of the data except in forward angle θ ∼ (9–12)^o. The strength of the real potential should be reduced by ∼12–28% and ∼8-37% within the framework of FM-1 and FM-2 scenarios, in that order, to reproduce the data.

2. The ascertained scattering matrix elements $|S_L|$ from the data analysis agree with one another using the concerned scenarios. It verifies preferring of the FM-1 scenario over the other one and focuses on an absorptive and a peripheral nature of the investigated scattering reaction.

3. The calculated notch test shows the radius of maximum sensitivity (R_S  = 12 fm) which is likewise near the empirical strong absorption radius ($R_{1/2}$ = 11.45 fm). This is joined by a higher reduced potential value $(W/V{)}_{R_{1/2}}=1.9$ than unity. In our opinion, the explanation may be owing to the weak binding of ⁹C nucleus.

4. An examination of the favoured real deep potential bespeaks that there are no apparent impacts of associated phenomena (rainbow scattering). As per our assessment, there is sufficient absorption to match the experimental data and eliminate the recognized oscillations in the computed angular distributions.

Disclosure statement

No potential conflict of interest was reported by the author(s).