Further investigation for the ¹⁸O + ⁴⁰Ca, ⁷⁶Se and ¹¹⁶Sn elastic scattering angular distributions

Abstract

At 275MeV energy, the efficiency of analysing the newly measured angular distributions for ¹⁸O ions elastically scattered from ⁴⁰Ca, ⁷⁶Se and ¹¹⁶Sn targets is examined using double and cluster folding models. Two choices for the imaginary potential part, phenomenological squared Woods Saxon (WS) and (double or cluster) folded forms, were implemented in the calculations. Different ¹⁸O density distributions, including its cluster structure (¹⁴C + α), were put forward for preparing the folding potentials of the considered systems. Furthermore, we investigated the impact of other channels coupling using two distinct methods: incorporating a dynamic polarization surface potential into non-renormalized cluster folded potentials, and  employing the continuum discrete coupled channels (CDCC) approach. To facilitate comparison, we also utilized the phenomenological optical model with real WS and imaginary squared WS potentials. The theoretical results demonstrated significant agreement with the experimental data, providing strong evidence for the efficacy of the methodologies under investigation.

KEYWORDS:

• Elastic scattering
• double folding
• cluster folding
• squared Woods Saxon
• dynamic polarization potential
• continuum discrete coupled channels

PACS:

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

1. Introduction

In heavy-ion physics, a significant drop in the elastic cross section at energies where the nuclei approach each other closely indicates a large absorption out of the elastic channel [1]. This is usually described by means of the imaginary potential in the optical model (OM). Much effort has been undertaken to extract this potential from different approaches [1–5].

Several choices were considered for the imaginary potential part of the optical potential, such as volume Woods Saxon (WS), squared volume WS, surface WS, Fourier Bessel series, etc. in many investigations including heavy ions. Bartnitzky et al. [6] measured and analysed the elastic angular distributions (ADs) for the scattering of ¹⁶O ions from ¹⁶O target at incident energies ranging from 250 to 704 MeV and from ⁴⁰Ca target at 704 MeV. For the imaginary potential part, the phenomenological extracted potentials within the Fourier Bessel series converged better than the squared WS. As well, the former potentials reasonably reproduced the data and compared very favourably with the microscopic calculations using the well-established double folding (DF) potentials. Agarwalla et al. [7] used Lee and Chan [8] imaginary potential and phenomenological WS one to perform a systematic comprehensive analysis for the ¹⁶O + ²⁸Si system at E_lab= 50 and 55 MeV and ¹⁸O + ⁵⁸Ni system at E_lab= 60 MeV. They reported that the latter generates the Fresnel diffraction pattern of the ¹⁸O + ⁵⁸Ni system and that the former shows successfully the oscillatory behaviour presented at the backward angles in the ¹⁶O + ²⁸Si system. Sonika et al. [9] measured the elastic scattering ADs for ¹⁸O + ²⁰⁶Pb system at E_lab= 139 MeV. The measured data was well described within the OM framework utilizing real and imaginary potential parts of a volume WS form. The authors observed that the cluster (CL) structure of ¹⁸O nucleus as a two neutrons orbiting a ¹⁶O core has no significant effects on the OM potential and the reaction cross section of this system when compared with the ¹⁶O + ²⁰⁸Pb system at the same energy. Similar observations were noticed by Jha et al. [10] for the ¹⁸O + ⁹⁰Zr and ¹⁶O + ⁹⁰Zr systems at 90 MeV. They reported that the two neutrons of ¹⁸O do not show an important effect on both the elastic scattering data and OM potential in comparison to the ¹⁶O + ⁹⁰Zr system. Aygun [11] analysed the ADs for ¹⁸O elastically scattered by various targets of masses between 12 and 208 over the energy range of 46–324 MeV. The theoretical analyses were carried out utilizing a real potential part constructed by the double folding model (DFM) in addition to a phenomenological WS imaginary potential. Both the relativistic mean field and the single particle harmonic oscillator density distributions adopted for the ¹⁸O nucleus [12] were quite reasonable in describing the data. Recently, the elastic and inelastic scattering ADs for the ¹⁸O + ⁴⁰Ca, ¹⁸O + ⁷⁶Se and ¹⁸O + ¹¹⁶Sn systems at E_lab= 275 MeV [13–15] were measured, and the measured ADs were analysed within the OM, distorted-wave Born approximation (DWBA) and coupled channel (CC) approaches. In these studies [13–15], three different optical potentials were employed to describe the elastic ADs. The imaginary potential part was taken either in the phenomenological WS form or the DF form as the real one. The three chosen optical potentials failed to describe the elastic scattering experimental data in the full angular range.

The investigation of scattering phenomena in these systems, specifically at the same incident energy, holds significant importance in deducing the potential of interaction between nuclei. This information is crucial for conducting calculations related to single (SCE) and double charge exchange (DCE) nuclear reactions, as they have connections with neutrinoless double beta decay (0νββ) [16,17]. It is worth noting that SCE and DCE reactions are not within the scope of this study.

Elastic scattering [18,19] serves as a fundamental tool for exploring the initial state interactions. Due to its high likelihood of occurrence, describing elastic scattering is an indispensable initial step toward obtaining a precise representation of the interaction between nuclei. This motivated us to reanalyse the newly measured elastic scattering data for these three systems using the widely employed DFM and cluster folding model (CFM).

The DFM is one of the most popular approaches for evaluating the real part of the nucleus (nucleon)–nucleus optical potential for various heavy-ion scattering analyses. The DF calculations rely on the nuclear densities of the colliding nuclei and an effective nucleon–nucleon (NN) interaction. The most widely employed effective NN interactions are the so-called density-independent Michigan 3 Yukawa terms (M3Y) or their density-dependent versions [18,19]. To precisely mimic the features of cold nuclear matter (NM) saturation, many variants of the M3Y density dependency have been presented. When compared to other versions, the CDM3Yn versions (n = 1→6) offer more exact results for nuclear incompressibility. The CDM3Y6 interaction has been extensively studied utilizing the DFM in elastic nucleus–nucleus scattering investigations [20–24]. On the same foot, the velocity dependent Sao Paulo potential (SPP) showed a great accessibility in DFM calculations to describe heavy ion scattering reactions across different energy levels [25–30].

Like the conventional DFM, the CFM computes the real optical potential, but with the effective cluster–cluster (CL–CL) interaction and the CL density distributions. This treatment might highlight the relevance of clusters in nuclei. Once again, SPP showed successful description of elastic scattering reactions for a large number of heavy ion systems in a very wide energy region in CFM calculations [31–35].

Hence, we extend the CL investigation to the case of the ¹⁸O projectile. The ¹⁸O nucleus appears to be a strong contender for α-CL structure, with several theoretical and experimental predictions for probable CL states [36–43]. These studies explored the (¹⁴C + α) and (¹²C + α + 2n) molecular structure states in the spectrum of ¹⁸O. Both the properties of the ¹²C and ¹⁴C nuclei contribute to the likelihood of clustering in ¹⁸O. The ¹⁸O nucleus exhibits a doubly-closed P_3/2 sub-shell, which increases stability and is accentuated by the energy of its first excited state, which is ∼4.4 MeV. The ¹⁴C nucleus is similar in this regard, with a complete neutron and partial proton p-shell closure and a first excited state above 6 MeV [44]. The authors of Ref. [45] demonstrated that whereas monopole and dipole transitions are inhibited for the molecular orbit state, they are facilitated for (¹⁴C + α) CL states.

In keeping with this, our goal is to demonstrate how effective it is to analyse the recently measured experimental ADs for the ¹⁸O ions scattered from ⁴⁰Ca, ⁷⁶Se and ¹¹⁶Sn targets at E_lab = 275 MeV [13–15] using the popular CDM3Y6 interaction in comparison to the SPP one through DFM and CFM calculations, respectively. Various density distributions for the ¹⁸O nucleus in addition to the CL structure (¹⁴C + α) were chosen to calculate the real potential part. On the other hand, the imaginary potential part was taken either as having a phenomenological squared WS shape or microscopically as a factor multiplied by the previously generated real part. Merely for intent of comparability, the phenomenological OM is applied with its parts; WS and squared WS for real and imaginary potential parts, respectively.

This piece of work is divided into four parts. Section 1 begins with a brief introduction, followed by the formalism of Section 2. Section 3 presents the results and discussion, whereas Section 4 has the conclusion.

2. Formalism

The implemented nuclear optical potential in theoretical calculations is separated into two portions: the real $V$ and imaginary $W$ potentials, which are specified as (1) $\begin{array}{r}{U}_N\left(R\right)=V\left(R\right)+\mathrm{iW}\left(R\right)\end{array}$(1) The following phenomenological WS form raised to the power (n) was proposed for both real (n = 1, WS) and imaginary (n = 2, Squared WS) components (2) $\begin{array}{r}X\left(R\right)=-\frac{X_0}{{\left[1+\exp \left(\right(R-R_X\left)/a_X\right)\right]}^n}\text{and (}X=V,W\text{), }\end{array}$(2) where X_o, $a_X$ and $R_X=r_X\left(A_P^{\left(1/3\right)}+A_T^{\left(1/3\right)}\right)$ are potential depth, diffuseness and radius, with A_P and A_T mass numbers of projectile and target nuclei, respectively. Furthermore, the Coulomb potential $U_C\left(R\right)$ for uniformly charge distributions of the colliding nuclei with the radius $R_C=1.3(A_P^{1/3}+A_T^{1/3})$ [46] was added to the nuclear potential of Equation (1) to get the total central potential. This phenomenological OM is named as first model in current analysis.

It is well known that the phenomenological representation does exclude the description of the structure of a projectile or target. Thus we used the second model DFM, in which the phenomenological real potential was supplanted by the DF potential and keeping the imaginary potential part as squared WS. As regards the nucleus–nucleus interaction, we adopt the same DF potential used in Refs. [18,47] (3) $\begin{array}{rl}{V}_{}^{\mathrm{DF}}\left(R\right)& =\int \int {\rho }_1\left(\overrightarrow{r_1}\right){\rho }_2\left(\overrightarrow{r_2}\right)v_{\mathrm{NN}}\left(s\right)d\overrightarrow{r_1}d\overrightarrow{r_2},\\ S& =\left|\overrightarrow{R}+\overrightarrow{r_2}-\overrightarrow{r_1}\right|,\end{array}$(3) where ${\rho }_1$ and ${\rho }_2$ 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 and v_NN (s) stands for the effective NN interaction used in the calculations. In the present work, the well-known density-dependent variant CDM3Y6 is adopted. It consists of two direct Yukawa's terms plus three finite range exchange Yukawa's terms remarked by (D) and (Ex), respectively, with the density dependence function $F\left(\rho \right)$ and energy-dependent factor $g\left(E\right)$ as follows [21, 48]: (4) $\begin{array}{rl}{\upsilon }_{D\left(\mathrm{Ex}\right)}\left(\rho ,R\right)& =g\left(E\right)F\left(\rho \right){\upsilon }_{D\left(\mathrm{Ex}\right)}\left(R\right),\end{array}$(4) (5) $\begin{array}{rl}{\upsilon }_D\left(R\right)& =\left[11062\frac{\text{exp}\left(-4R\right)}{4R}-2538\frac{\text{exp}\left(-2.5R\right)}{2.5R}\right]\\ & \times \text{MeV},\end{array}$(5) (6) $\begin{array}{rl}{\upsilon }_{\mathrm{Ex}}\left(R\right)& =[-1524\frac{\text{exp}\left(-4R\right)}{4R}-518.8\frac{\text{exp}\left(-2.5R\right)}{2.5R}\\ & -7.847\frac{\text{exp}\left(-0.7072R\right)}{0.7072R}]\text{MeV,}\end{array}$(6) (7) $\begin{array}{rl}F\left(\rho \right)& =0.2658\left[1+3.8033\text{exp}\left(-1.41\rho \right)-4.0\rho \right],\end{array}$(7) (8) $\begin{array}{rl}g\left(E\right)& =\left[1-0.003\left(E/A\right)\right],\end{array}$(8) where E/A is the energy per nucleon for projectile mass A.

Because the density distribution of colliding nuclei is basic in folding calculations, three alternative matter density distributions for the nucleus ¹⁸O ground state, namely, Dirac–Hartree–Bogoliubov (DHB) [49], Hartree–Fock–Bogoliubov based on the BSk14 Skyrme force (HFB14) [50] and Harmonic Oscillator (HO) [51] were employed, allowing for a correlation assessment. Figure 1 (linear and logarithmic scale) depicts the three particular densities of the ¹⁸O nucleus: DHB, HFB14 and HO. The DHB form is employed to achieve the considered targets ⁴⁰Ca, ⁷⁶Se and ¹¹⁶Sn density distributions [49]. The resultant real DF potentials are presented in Figure 2.

Figure 1. The densities of ¹⁸O used in the DFM calculations.

Figure 2. The prepared real potentials for ¹⁸O + ⁴⁰Ca, ⁷⁶Se and ¹¹⁶Sn implemented in the DFM calculations.

Additionally, according to the (¹⁴C + α) CL structure of ¹⁸O which appears at an excitation energy of 6.2276 MeV [36–44], it is interesting to analyse the considered systems: ¹⁸O + ⁴⁰Ca, ¹⁸O + ⁷⁶Se and ¹⁸O + ¹¹⁶Sn using real potential constructed based on the CFM as the third employed approach. The main ingredients for generating the ¹⁸O + Target (⁴⁰Ca, ⁷⁶Se and ¹¹⁶Sn) cluster folding (CF) potentials are: ¹⁴C + target and α + target potentials at appropriate energies as expressed in Equation (9): (9) $\begin{array}{rl}{V}^{\mathrm{CF}}\left(\mathbf{\text{R}}\right)& =\int [V_{{}^{14}\text{C}-\text{ Target}}\left(\mathbf{\text{R - }}\frac{\text{2}}{\text{9}}\mathbf{\text{r}}\right)\\ & +V_{\alpha -\text{Target}}\left(\mathbf{\text{R}}+\frac{\text{7}}{\text{9}}\mathbf{\text{r}}\right)]{\left|{\chi }_{{}^{\text{14}}\text{C}-\alpha }\left(\mathbf{\text{r}}\right)\right|}^{2}d\mathbf{\text{r}}\text{,}\end{array}$(9) in addition to the wave function ${\chi }_{{}^{\text{14}}\text{C}-\alpha }\left(\mathbf{\text{r}}\right)$ of the CL which describes the ¹⁴C and α relative motion in the ground state of ¹⁸O. The ¹⁴C + α bound state form factor represents an ⁴S₀ state in a real WS potential of radius $\left(1.25\times {14}^{\left(1/3\right)}\right)$ and diffuseness of 0.65 fm, the potential depth is allowed to be change till the binding energy of the CL (6.2276 MeV) is achieved, in FRESCO code [52]. The CL extracted wave function is displayed in Figure 3.

Figure 3. The extracted wave function ${\chi }_{{}^{\text{14}}\text{C}-\alpha }\left(\mathbf{\text{r}}\right)$ of the cluster which describes the ¹⁴C and α relative motion in the ground state of ¹⁸O.

Of course, the same WS potential parameters for the bound state (¹⁸O → ¹⁴C + α) were used in preparing CF potential for all the considered systems. While, the suitable $V_{{}^{14}\text{C}-\text{ Target}}$ and $V_{\alpha -\text{Target}}$ potentials were prepared and chosen as follows:

1. For ¹⁸O + ⁴⁰Ca system, the considered data is at E (¹⁸O) = 275 MeV. So, the required potentials are ¹⁴C + ⁴⁰Ca at $E=7/9\times 275=213.9$ MeV and α + ⁴⁰Ca at $E=2/9\times 275=61.1$ MeV. As there is no experimental data for the ¹⁴C + ⁴⁰Ca channel, the ¹⁴C + ⁴⁰Ca potential is prepared through SPP potential using REGINA code [26]. The renormalization factor for the real SPP ($N_R^{\mathrm{SPP}}$) was taken by default 1.0. On the other side, the α + ⁴⁰Ca potential at E = 61 MeV [53] was utilized.

2. For ¹⁸O + ⁷⁶Se system, the needed potentials to generate the ¹⁸O + ⁷⁶Se CF potentials are the appropriate potentials for the ¹⁴C + ⁷⁶Se and α + ⁷⁶Se channels at E = 213.9 and 61.1 MeV, respectively. The ¹⁴C + ⁷⁶Se potential is prepared using SPP within the framework of REGINA code. The $N_R^{\mathrm{SPP}}$ was fixed at 1.0. While the α + ⁷⁶Se potential at E = 25 MeV [54] was utilized, as it is the closest existing data to α + ⁷⁶Se at E = 61.1 MeV in the literature.

3. For ¹⁸O + ¹¹⁶Sn system, the needed potentials to generate the ¹⁸O + ¹¹⁶Sn CF potentials are the appropriate potentials for the ¹⁴C + ¹¹⁶Sn and α + ¹¹⁶Sn channels at E = 213.9 and 61.1 MeV, respectively. Real SPP created utilizing REGINA code was implemented for the ¹⁴C + ¹¹⁶Sn channel using $N_R^{\mathrm{SPP}}$ = 1.0. The α + ¹¹⁶Sn potential at E = 65.7 MeV [55] was utilized, as it is the closest existed data in literature to the needed α + ¹¹⁶Sn at E = 61.1 MeV.

Noting that the DHB density distributions employed in REGINA code were taken for all interacting nuclei in preceding three items (i–iii). The generated real CF potentials for the considered systems: ¹⁸O + ⁴⁰Ca, ¹⁸O + ⁷⁶Se and ¹⁸O + ¹¹⁶Sn are shown in Figure 4. The created real CF potential in addition to a phenomenological squared WS imaginary potential, the so-called the CFM was applied to fit the considered data. Within the framework of this model, the potential form of Equation (9) was adopted.

Figure 4. The prepared real CF potential for ¹⁸O + ⁴⁰Ca, ⁷⁶Se and ¹¹⁶Sn systems, implemented in the CFM calculations.

To ensure meaningful comparisons, the imaginary part of the potential is treated using the same folded structure as the real part in both the double folding model (DFM) and the cluster folding model (CFM). Subsequently, Equation (1) is adjusted separately for each case as follows: (10) $\begin{array}{rl}{U}^{\mathrm{DF}\left(\mathrm{CF}\right)}\left(R\right)& =N_R{V}^{\mathrm{DF}\left(\mathrm{CF}\right)}\left(R\right)+\mathrm{iW}\left(R\right)\end{array}$(10) (11) $\begin{array}{rl}{U}^{\mathrm{DF}\left(\mathrm{CF}\right)}\left(R\right)& =N_R{V}^{\mathrm{DF}\left(\mathrm{CF}\right)}\left(R\right)+i{N}_I{V}^{\mathrm{DF}\left(\mathrm{CF}\right)}\left(R\right)\end{array}$(11)

In this context, $N_R$ and $N_I$ are referred to as the renormalization factors for the real and imaginary parts of the DF (or CF) potentials, respectively. These factors are considered as adjustable parameters, and they are allowed to be varied freely until the theoretical calculations achieve the best possible agreement with the experimental data. By employing these renormalization factors, the DFM and CFM can be fine-tuned to accurately reproduce the experimental observations, leading to a more comprehensive understanding of the scattering phenomena in the systems under investigation.

3. Results and discussion

3.1 Comparability of density parameterizations and real nuclear potentials

In Figure 1, the three ¹⁸O projectile density distributions– HO, HFB14 and DHB – are shown in terms of their radial dependency on a linear (upper panel) and logarithmic (lower panel) scales. At radial distance r between 0 and 2.5 fm, the distribution of the nucleons varies in the core, as can be seen in the upper panel. It is noted that at r ∼ 1.5 fm, the DHB density has the highest peak. Also, the DHB and HFB14 density distributions are near to one other and have a lower value in the nucleus's centre than the HO. The lower panel, on the other hand, demonstrates that both HFB14 and DHB densities are nearly comparable at a large distance (r > 5 fm) and have an extended tail compared to HO density.

The selected densities of both the ¹⁸O projectile and the investigated targets, in addition to CDM3Y6-NN interaction potentials, were connected in Equations (3)–(8) to generate real DF potentials. Tables 1 and 2 show the noted parameters related to the determined potentials. Figure 2 depicts how the potential depth (R = 0) decreases as target masses increase. It displays that, when HO and HFB14 densities were utilized, all of the systems studied had near-potential depth. Furthermore, the DHB density correlates to the deepest potential in the case of ¹⁸O + ⁴⁰Ca system. Also, with distances R ranging from 0 to 2.5 fm, a constant core DF potential increases in the following order: ¹⁸O + ⁴⁰Ca <⁷⁶Se < ¹¹⁶Sn.

Table 1. The best fit parameters of different potentials and densities combination for scattering data. The bold numbers represent the double/cluster folded real and imaginary potential case.

Target	Density type	Potential	Vo (MeV) /NR	rV (fm)	aV (fm)	Wo (MeV)/NI	rW (fm)	aW (fm)	JR (MeV.fm^3)	JI (MeV.fm^3)	χ^2	σR (mb)
^40Ca		OM	435.64	0.773	0.910	287.51	1.012	1.554	354.29	273.95	1.74	2933
	DHB	DF	0.58	–	–	319.89	0.935	1.794	339.90	249.04	1.29	3154
			0.27	–	–	0.21	–	–	160.96	123.64	8.20	2251
	HFB14	DF	1.17	–	–	290.02	0.935	1.867	443.31	228.35	1.41	3235
			0.26	–	–	0.15	–	–	96.76	56.71	33.54	1841
	HO	DF	1.20	–	–	256.62	0.935	1.961	449.51	205.22	1.53	3336
			0.26	–	–	0.15	–	–	95.20	56.07	40.97	1810
		CF	0.54	–	–	137.74	0.935	2.169	238.19	114.53	1.36	3231
			0.59	–	–	0.40	–	–	261.78	175.44	26.57	2120
^76Se		OM	448.50	0.806	0.915	52.11	1.209	1.445	294.35	66.01	0.63	3083
	DHB	DF	0.72	–	–	42.44	1.218	1.912	258.81	54.05	0.96	3601
			0.97	–	–	0.44	–	–	345.97	155.71	9.16	2707
	HFB14	DF	0.69	–	–	42.92	1.218	1.911	254.16	54.65	0.96	3607
			0.92	–	–	0.41	–	–	336.23	151.08	9.46	2707
	HO	DF	0.72	–	–	42.23	1.218	1.974	259.96	53.79	1.04	3683
			0.98	–	–	0.42	–	–	352.92	152.55	17.18	2661
		CF	0.59	–	–	40.99	1.218	1.923	244.13	52.11	0.83	3547
			0.78	–	–	0.34	–	–	319.73	139.17	10.29	2683
^116Sn		OM	576.34	0.598	1.127	28.52	1.248	1.372	169.03	34.78	0.83	3126
	DHB	DF	0.95	–	–	63.82	1.213	1.053	334.21	74.17	1.03	2993
			0.87	–	–	0.48	–	–	305.37	168.08	0.99	3027
	HFB14	DF	0.91	–	–	66.79	1.213	1.030	326.08	77.85	1.04	2982
			0.83	–	–	0.45	–	–	297.27	162.32	0.99	3026
	HO	DF	1.06	–	–	67.07	1.213	1.027	374.76	78.23	1.08	2986
			0.99	–	–	0.49	–	–	351.98	176.19	1.04	3004
		CF	1.04	–	–	67.68	1.213	1.028	376.80	78.93	1.04	2983
			0.94	–	–	0.53	–	–	340.48	191.40	0.98	3036

Table 2. Values of some characteristic quantities at the strong absorption radius (R_1/2) in addition to the real and imaginary root mean square radius (R.M.S) for scattering data. The bold numbers represent the double/cluster folded real and imaginary potential case.

Target	Density type	Potential	(R.M.S)R(fm)	(R.M.S)I(fm)	R1/2 (fm)	L1/2 (ħ)	W/V (R 1/2)
^40Ca		OM	4.95	5.44	9.49	93.55	1.33
	DHB	DF	5.14	5.51	9.71	95.86	2.62
			5.14	5.14	8.45	82.46	0.77
	HFB14	DF	4.93	5.62	9.79	96.69	3.43
			4.93	4.93	7.60	73.45	0.59
	HO	DF	4.91	5.34	9.84	97.27	4.81
			4.91	4.91	7.57	73.06	0.59
		CF	4.75	6.06	9.61	94.80	10.88
			4.75	4.75	8.29	80.73	0.67
^76Se		OM	5.47	6.66	10.04	113.0	0.85
	DHB	DF	5.46	7.18	10.53	119.0	3.70
			5.46	5.46	9.61	107.60	0.45
	HFB14	DF	5.48	7.17	10.54	119.10	3.73
			5.48	5.48	9.61	107.60	0.45
	HO	DF	5.45	7.25	10.61	120.00	5.37
			5.45	5.45	9.58	107.20	0.43
		CF	5.47	7.17	10.50	118.70	3.78
			5.47	5.47	9.60	107.40	0.44
^116Sn		OM	5.44	7.31	10.48	121.90	0.86
	DHB	DF	5.91	6.93	10.53	122.60	0.50
			5.91	5.91	10.50	122.20	0.55
	HFB14	DF	5.93	6.92	10.53	122.60	0.49
			5.93	5.93	10.50	122.20	0.55
	HO	DF	5.91	6.92	10.54	122.80	0.47
			5.91	5.91	10.52	122.40	0.50
		CF	5.82	6.92	10.53	122.60	0.49
			5.82	5.82	10.51	122.40	0.56

On the other hand, the computed real potentials are in a good agreement with one another at the surface interaction point, which is specified for each system by the strong absorption radius ($R_{1/2}$) and its associated momentum ($L_{1/2}$). The $R_{1/2}$, at any energy, was computed from the formula of distance of closest approach for the Coulomb trajectories, i.e. (12) $\begin{array}{r}{R}_{1/2}=\frac{\eta }{k}\left[1+{\left(1+{\left(\frac{L_{1/2}}{\eta }\right)}^2\right)}^{1/2}\right]\end{array}$(12) Here, k is the wave number and η is the Sommerfield parameter and $L_{1/2}$ is the partial wave for which the transmission coefficient is 0.5 [56].

Further, real CF potentials were yielded by coupling the extracted (¹⁴C + α) wave function (see Figure 3) with the ¹⁴C, α + ⁴⁰Ca, ⁷⁶Se and ¹¹⁶Sn potentials in Equation (9) based on the CL structure (¹⁴C + α) of ¹⁸O projectile. The densities utilized for all nuclei are all put-up in a form of DHB distribution. Tables 1 and 2 include the estimated CF potential parameters. As seen in Figure 4, the resultant potentials behave similarly to the tested systems and the potential depth (R = 0) increases as target masses increase.

3.2 Fitting quality of elastic scattering models

Three models are used to investigate the very recently measured elastic scattering cross section data of ¹⁸O + ⁴⁰Ca, ⁷⁶Se and ¹¹⁶Sn systems [13–15] at E_lab= 275 MeV. The best fits to the experimental data were acquired via searching on the potential parameters of Equations (1), (10) and (11) by means of the HIOPTIM-94 code [57]. This is achieved by minimizing the χ² value, which is defined as (13) $\begin{array}{r}{\chi }^2=\frac{1}{N}\sum _{i=1}^N{\left[\frac{{\sigma }_{\mathrm{th}}\left({\theta }_i\right)-{\sigma }_{\exp }\left({\theta }_i\right)}{\mathrm{\Delta }{\sigma }_{\exp }\left({\theta }_i\right)}\right]}^2,\end{array}$(13) where ${\sigma }_{\mathrm{th}}\left({\theta }_i\right)$ and ${\sigma }_{\exp }\left({\theta }_i\right)$ are the theoretical and experimental cross sections, respectively, at angle $\left({\theta }_i\right)$, $\mathrm{\Delta }{\sigma }_{\exp }\left({\theta }_i\right)$ is the experimental error, and N is the number of data points. The χ² values are obtained considering 10% uniform errors for all analysed data. The extracted variable parameters are made known in Tables 1 and 2. The predicted angular distributions of the elastic scattering differential cross section are plotted as illustrated in Figures 5–7 compared with the corresponding experimental data.

Figure 5. The experimental ADs for the elastically scattered ¹⁸O + ⁴⁰Ca system at E_lab = 275 MeV versus the theoretical calculations within the OM, DFM and CFM. DHB, HFB14 and HO densities in addition to CL structure of ¹⁸O are used in the DF and CF calculations, separately. Applying the folded forms either for real or both real and imaginary potentials are denoted by (R) and (R + I), respectively. The experimental data are taken from Ref. [13]

Figure 6. The same as Figure 5, but for ⁷⁶Se target. The experimental data are taken from Ref. [14].

Figure 7. The same as Figure 5, but for ¹¹⁶Sn target. The experimental data are taken from Ref. [15].

Table 1 lists the best fit potential (real and imaginary) parameters, the extracted values of the real ($J_R$) and imaginary ($J_I$) volume integrals per pair of interacting nucleons, the reaction cross section (σ_R), along with the minimum χ². Moreover, the corresponding root-mean-square radii of the real (R.M.S)_R and imaginary (R.M.S)_I potentials, besides other quantities at the strong absorption radius $R_{1/2}$ are provided in Table 2.

In the first model, OM, the optimal real WS plus squared WS imaginary potentials’ parameters of Equations (1) and (2) were obtained by performing searches on six parameters. The ones that attain the best fit are listed in Table 1 as OM parameters, while the predicted angular distributions of the elastic scattering differential cross section are shown in Figures 5–7 in correlation with the corresponding experimental data. For the three considered systems, the data are well reproduced over all the measured angular ranges. The obtained potentials give an equally good quality estimation of the scattered data as that using (real and imaginary) WS ones in previous studies [13–15]. This is supported by the nearly close present σ_R value of 3083 to 3327 mb given in Ref. [14] for ¹⁸O + ⁷⁶Se. Unfortunately, the extracted σ_R values from WS fits are not presented in Refs. [13,15] for the other two systems.

In the second model, DFM, the real DF potential determined in Equation (3) was created by consolidating the three density distributions of the ¹⁸O projectile (DHB, HFB14 and HO), the CDM3Y6 designated NN interaction and density distributions for the ⁴⁰Ca, ⁷⁶Se and ¹¹⁶Sn targets. To execute the fitting method, simply one free parameter, $N_R$, for the real potential, and three parameters for the squared WS imaginary potential were used (see Equation 10). We signify this potential as DF (R). In each case, we tried different parameter sets. The imaginary radius parameter (r_W) was discovered to be centred around an average value within each scenario for one system. As a result, this value is kept constant while searching for other parameters to ensure good consistency between the theoretical outcomes and experimental data. Tables 1 and 2 provide the derived parameters, whereas Figures 5–7 show the standout accord between the theoretical calculations and experimental data.

The resultant potentials behave similarly, with fairly near χ² values in each scattering system. The large variation of $N_R$ values from unity for potentials that correspond to the ⁴⁰Ca target (Table 1) suggests coupling of other channels to elastic scattering compared to the ⁷⁶Se and ¹¹⁶Sn targets. Consequently, the flux absorption due to these coupling effects is represented with imaginary parameter values. In contrast to ⁴⁰Ca, the squared WS potential for ⁷⁶Se and ¹¹⁶Sn targets is only somewhat responsive to different types of densities.

_.The same elastic scattering cross-section data have been re-examined. The imaginary potential as well as the real one obtained employing Equation (3) is used to evaluate the DF potential with two separate renormalization factors, $N_I$ and $N_R$, respectively, as seen in Equation (11). This implies that both the real and imaginary parts of the folded potentials are assumed to have the same shape and different strengths. The name for this is DF(R + I) potential. To achieve the maximum compatibility between theoretical predictions and actual results, only two free parameters – $N_R$ and $N_I$ – were unrestrictedly modified, as illustrated in Figures 5–7 and Table 1. As perceived in Figure 5, the observed oscillations in the ¹⁸O + ⁴⁰Ca experimental data are generated with overestimation for the DHB density and a small shift in backward angle between 22^o and 24^o for the HFB14 and HO densities. Similarly, employing SPP interaction potential, this is seen, as in Ref. [13].

The analysis of the ¹⁸O + ⁷⁶Se system utilizing different densities has provided a fruitful representation of the data within the measured angular range, specifically angles less than 16^o. However, it has been observed that there is an overestimation at larger angular distributions. This finding is consistent across all the densities employed in the investigation, as depicted in Figure 6.

It also displays, a larger disparity between $N_R$ and $N_I$ values in the case of ¹⁸O + ⁴⁰Ca than in other examined systems. However, the highest accord with experimental data is shown for all employed densities in the case of ¹⁸O + ¹¹⁶Sn, as is made obvious in Figure 7 and Tables 1 and 2. Oppositely, using the squared WS imaginary potential of the second model appears to give a preferable match to the data over the DF imaginary potential. This is supported by much higher values of χ² values for the latter one in the case of ¹⁸O + ⁴⁰Ca, ⁷⁶Se.

Like the conventional DFM, the CFM computes the real potential part, but with the effective CL–CL interaction and the CL density distributions [31–33] described by Equation (9) to yield two potential versions, CF(R) and CF(R + I). These versions depicted by Equations (11) and (12). Reanalysis is performed using the ¹⁸O-CL elements as (α + ¹⁴C) and the implemented DHB density distributions for all interacting nuclei. The same fitting procedure is followed as in the previous model. Similar to the second model, the results showed a larger overestimation of the experimental data in the case of using the CF (R + I) for ¹⁸O + ⁴⁰Ca than that of using the DF ones. This situation is obvious in Figure 5. Additionally, Tables 1 and 2 show that the same system has a high value of the squared WS imaginary potential depth (137 MeV), which is supported by a high value of its ratio to the real CF potential depth ($W/V_{R_{1/2}}=10.88$) at the interaction surface.

As it is anticipated, the normalized folded real potentials are in decent concurrence with one another at the surface interaction point, which is defined for every nuclear system by the assessed $R_{1/2}$ and its associated $L_{1/2}$ values (see Equation 12) as given in Table 2. By increasing the target mass, $R_{1/2}$ and $L_{1/2}$ values increase.

We have additionally determined the $J_R$ and $J_I$ volume integrals. They assume a significant part in showing the strength of the assessed potential. In this context, the $J_R$ and $J_I$ values are presented in Table 1 for the calculated potentials, and the change in the centre of mass energy ($E_{c.m}$) of these values for examined nuclear systems is revealed in Figure 8. It is obvious from panel (a) in this figure that the obtained $J_R^{\mathrm{OM}}$ and $J_R^{\mathrm{DF}\left(R\right)}$ gradually decrease with increasing centre of mass energy. This behaviour is reversed for the values obtained with the other calculations. It is also clear that the $J_R^{\mathrm{CF}\left(R\right)}$ and $J_R^{\mathrm{CF}\left(R+I\right)}$ are in good agreement with each other. Panel (b) shows the rapid decrease of the $J_I^{\mathrm{OM}}$ and $J_I^{\mathrm{DF}\left(R\right)}$ with increasing centre of mass energy. Besides, as we see from panel (b), the behaviour of the $J_I^{\mathrm{CF}\left(R\right)}$ is different when compared to the $J_I^{\mathrm{CF}\left(R+I\right)}$. On the other hand, as indicated in the two panels, there is a coherency among the $J_R^{\mathrm{DF}\left(R+I\right)}$ and $J_I^{\mathrm{DF}\left(R+I\right)}$ where both reveal similar behaviour.

Figure 8. Change with the centre of mass energy of the $J_R$ (a) and $J_I$ (b) volume integrals obtained for the elastically scattered ¹⁸O + ⁴⁰Ca, ⁷⁶Se and ¹¹⁶Sn systems at E_lab= 275 MeV, in the present work. Applying the folded forms either for real or both real and imaginary potentials are denoted by (R) and (R + I), respectively. The lines are to guide eyes for the energy behaviour.

Another quantity which is frequently investigated to probe the dynamics of ¹⁸O, is the extracted σ_R from the elastic scattering analysis which are listed in Table 1. As suggested by Kolata [58] and Aguilera et al. [59], the extracted reduced total reaction cross section (σ_Red) of different projectiles on several targets as a function of reduced energy can be compared with that given by Wong [60]. The cross section is scaled as ${\sigma }_R/(A_P^{1/3}+A_T^{1/3}{)}^2$ and energy is scaled as $E_{c.m}\left(A_P^{1/3}+A_T^{1/3}\right)/\left(Z_P{Z}_T\right)$. The symbol σ_R denotes total reaction cross section and Z denotes the charge of involved nuclei. The typical geometrical and charge differences between reaction systems were therefore suitably reduced while the dynamical effects of interest were not washed out. The variance in reduced reaction cross sections $\left({\sigma }_{\mathrm{red}.}\right)$ derived in this study from different potentials calculations at different reduced energies $\left(E_{\mathrm{red}.}\right)$ for the ¹⁸O + ⁴⁰Ca, ⁷⁶Se and ¹¹⁶Sn systems, as well as comparisons with previously published values in the literature [14], is shown in Figure 9.

Figure 9. Reduced reaction cross sections for the elastically scattered ¹⁸O + ⁴⁰Ca,⁷⁶Se,¹¹⁶Sn systems at E_lab= 275 MeV obtained in the present work along with that for ^16,17,18O oxygen isotopes on different targets as indicated in Table 3. The red solid curve represents the polynomial fit and the black dashed curve represents the Wong formula fit.

The results for various potentials are close to each other and correspond well with previously published values at the same conditions except for ⁴⁰Ca. Furthermore, the findings presented in Figure 9 incorporate systems of the ¹⁸O, ¹⁷O and ¹⁶O isotope projectiles on various targets at different energies. The σ_R values were derived from References [9–11, 14, 61–64] and summarized in Table 3.

Table 3. Total reaction cross section values at different energies for various systems used in Figure 9.

System	Elab (MeV), σR (mb), [Ref.]
^16O + ^12C, ^16O + ^90Zr, ^16O + ^174Yb, ^16O + ^208Pb	138.65, 2276 [9] 90,1500 [10] – 90,1604.7 [11] 83, 912 [63] 124,1527,1520 [64]
^17O + ^13C, ^17O + ^124Sn	204,1659 [62] – 1899 [11] 340,3952 [11]
^18O + ^12C, ^18O + ^28Si, ^18O + ^76Se, ^18O + ^90Zr, ^18O + ^174Yb, ^18O + ^206Pb	324,1753.8 [11], 216,1695 [62] 352, 2354 −2408 [61] 275,3327-2919-2801 [14] 90,1445 [10]−194.4, 2497.2 [11] 83, 912 [63] 140, 2305 [9]

The energy dependency on the extracted reaction cross sections is depicted in Figure 9 by the black dashed curve of a Wong formula [60]. Starting with the Wong model parameters for ⁴He that Aguilera et al. derived in Ref. [59], yielding ${ϵ}_{\text{o}}=0.175$, $r_o=0.159$ and $V_{\text{red}}=0.913$ as fitting parameters. It is important to note that the fit's parameters could not be entirely reliable given the limited amount of data. Consequently, the trend line of a polynomial fit is also drawn for better fitting as red solid curve in Figure 9 and can be approximated as (${\sigma }_{\text{red}}=0.033{\text{E}}_{\text{red}}^3-1.44{\text{E}}_{\text{red}}^2+18.76{\text{E}}_{\text{red}}-0.90$). Both lines indicate that the ${\sigma }_{\text{red}}$ values increase with increasing energy. With the exception of ¹⁸O + ⁴⁰Ca,⁷⁶Se, the estimated values are in excellent agreement with fitting calculations made on scattering data from several current systems. This might imply that for medium mass targets like ⁴⁰Ca and ⁷⁶Se, the other channel effects on elastic scattering could be very important for ¹⁸O projectile, resulting in greater reaction cross sections. Contrarily, this characteristic is not observed for the lighter or heavier targets, since σ_red for ¹⁸O, ¹⁷O and ¹⁶O follow the same trend, indicating the other channel effects, similarly to what was first noted in Ref. [65].

Regarding the above results from our point of view, employing a combination of a folded real potential and a squared WS (folded) imaginary potential proves to be an appropriate model for describing the measured scattering data of investigated systems. Additionally, the real renormalization factors – $N_R^{\mathrm{DF}}$ and $N_R^{\mathrm{CF}}$ – are calculated to account for the variations among all real folded potentials across different systems. Hence, their deviation from the default value of 1.0 due to the expected other channel coupling effects ($N_R$< 1) can be simulated by introducing an additional dynamic polarization potential (DPP) [66,67]. In the present work, we have adopted a simple model to derive this DPP for the considered systems in a phenomenological form. As shown in Figures 5–7 and Table 1, using renormalized CF(R) potentials, the considered data was well reproduced. The extracted $N_R$ values for the considered targets ⁴⁰Ca, ⁷⁶Se and ¹¹⁶Sn are 0.54, 0.59 and 1.04, respectively. Thus it was intriguing to reproduce the considered ADs using the non-renormalized CF(R) potential $\left(N_R=1\right)$ plus the previously extracted squared WS imaginary potential by including an additional DPP of a repulsive surface form, (14) $\begin{array}{rl}{V}_{\mathrm{DPP}}\left(R\right)& =-V_{\mathrm{pol}}\exp \left(\frac{R-R_{\mathrm{pol}}}{a_{\mathrm{pol}}}\right)\\ & /{\left[1+\exp \left(\frac{R-R_{\mathrm{pol}}}{a_{\mathrm{pol}}}\right)\right]}^2,\\ & R_{\mathrm{pol}}=r_{\mathrm{pol}}(A_P^{1/3}+A_T^{1/3})\end{array}$(14) This potential is named non-renormalized CF(R) + DPP and characterized by three parameters (depth $V_{\mathrm{pol}}$, radius $r_{\mathrm{pol}}$ and diffuseness $a_{\mathrm{pol}}$). The total nuclear potential of Equation (10) is modified for reanalysis the data in this case having the following formula: (15) $\begin{array}{r}U{}^{\mathrm{CF}}\left(R\right)=V{}^{\mathrm{CF}}\left(R\right)+\mathrm{iW}\left(R\right)+V_{\mathrm{DPP}}\left(R\right),(N_R=1)\end{array}$(15) As shown in Figure 10, the comparison between the ¹⁸O + ⁴⁰Ca, ¹⁸O + ⁷⁶Se and ¹⁸O + ¹¹⁶Sn ADs and the theoretical calculation within the framework of the CFM using so-called non-renormalized CF(R) + DPP potential is fairly good using the parameters listed in Table 4. It is worth noting that the CF(R) potential is a case chosen as an example.

Figure 10. The experimental ADs for the elastically scattered ¹⁸O + ⁴⁰Ca, ⁷⁶Se and ¹¹⁶Sn systems at E_lab = 275 MeV versus the theoretical calculations within the CFM using the non-renormalized CF(R)+DPP potential as well as the CDCC calculations. The experimental data are taken from Refs. [13–15].

Table 4. The best fit parameters of the non-renormalized CF(R) + DPP potential used in reproducing the scattering data.

System	$N_R^{\mathrm{CF}}$	W0 (MeV)	rW (fm)	aW (fm)	Vpol (MeV)	rpol (fm)	apol (fm)
^18O + ^40Ca	1.0	137.47	0.935	2.169	−34.32	1.046	0.628
^18O + ^76Se	1.0	40.99	1.218	1.923	−11.79	1.235	0.554
^18O + ^116Sn	1.0	67.68	1.213	1.028	−106.42	1.171	0.95

Finally, we have employed the continuum discretized coupled channels (CDCC) method [68] to investigate the considered ¹⁸O + ⁴⁰Ca, ¹⁸O + ⁷⁶Se and ¹⁸O + ¹¹⁶Sn ADs. The inclusion of couplings to continuum states, which makes it possible to consider effects of projectile breakup that appear mainly at low energies, is an obvious advantage of the CDCC method. The CDCC computations were carried out using FRESCO code, taking into consideration the coupling to the bound non-resonant ¹⁸O states that appear below the α+ ¹⁴C cluster structure at breakup threshold energy of 6.227 MeV. As shown in Figure 10, The CDCC calculations (dashed green curves) as expected failed to reproduce the considered data. The main reason for such failure is the high energy of the incident ¹⁸O projectile, at such high energy; the projectile breakup effects are nearly negligible.

4. Conclusion

In conclusion, this study focused on analysing the angular distributions of ¹⁸O ions elastically scattered from various targets, including ⁴⁰Ca, ⁷⁶Se and ¹¹⁶Sn, at an energy of 275 MeV. The efficiency of this analysis was examined using the DFM and CFM models. Two choices for the imaginary potential part, namely the phenomenological squared WS and DF or CF forms, were employed in the calculations.

To prepare the folding potentials for the considered systems, different distributions of ¹⁸O densities were considered, taking into account its notable (¹⁴C + α) CL structure. For comparison purposes, the phenomenological OM was also utilized, employing real WS potentials and imaginary squared WS potentials.

The results of the theoretical calculations were found to be in reasonable agreement with the experimental data, indicating the effectiveness of the methodologies under investigation. Moreover, the extracted reduced reaction cross section values expect the presence of other channel coupling effects in the field of the target nucleus. Therefore, fairly good calculations are performed using the non-renormalized potentials by adding a DPP surface potential term to simulate these effects. In contrast, the CDCC calculations were unsuccessful in reproducing the analysed data because the projectile breakup effects at the high incident energy were almost negligible.

This provides evidence for the validity and strength of the DFM and CFM in analysing the angular distributions of ¹⁸O ion scattering, and highlights their potential for studying similar nuclear reaction processes.

Acknowledgment

Awad A. Ibraheem express his appreciation to the Deanship of Scientific Research at King Khalid University, Saudi Arabia, for funding this work through research groups program under grant of number R.G.P.2/4/44.

Disclosure statement

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