Most measurements in heavy-ion elastic scattering is the differential cross section. This physical quantity is calculated from the elastic scattering amplitude. The phase shift is an important factor in determining the scattering amplitude. The eikonal model[1-3] has been used for describing the high energy elastic scattering between heavy-ions. The basic assumption of this model is that its classical trajectory is little deflected from a straight line because the incident energy is sufficiently high. The eikonal phase shift is obtained from the integral equation by further approximating the Wentzel-Kramers-Brillouin (WKB)results. The only input to calculate the eikonal phase shift is the optical potential, and its parameters are usually determined by the χ2/N−method. Several kinds of eikonal model, including the higher-order eikonal corrections or using the effective impact parameter, were reported[4, 5] to extend eikonal model to the regime of relatively low bombarding energies. Numerous studies[6-11] for describing the elastic scattering between heavy-ions were performed using the eikonal model. An approximated analytic expression of the eikonal phase for the symmetrized Fermi type potential has been derived[8] and compared with that obtained by exact calculations. Various corrections to the eikonal approximations have been studied[11] for two- and three-body nuclear collisions to extend the range of validity of this approximation to beam energies of 10 MeV/nucleon. The early work of Cha and Kim[6] reported the 1st- and 2nd-order corrections to the eikonal phase shift taken into account the deflection effect due to the Coulomb field.

In the recent past, the elastic differential cross sections of ⁶Li projectile at 240 MeV have been measured[12-15] on ²⁴Mg, ²⁸Si, ⁴⁰Ca, ⁵⁸Ni, ⁹⁰Zr, and ¹¹⁶Sn targets. Most theoretical studies employed optical and folding models for analyzing these elastic data. It is interesting to analyze the experimental measurements for ⁶Li + ²⁴Mg, ²⁸Si, ⁴⁰Ca, ⁵⁸Ni, ⁹⁰Zr, and ¹¹⁶Sn elastic scatterings at 240 MeV by using the eikonal model based on the Coulomb trajectory of colliding nuclei. The purpose of this paper aims at a systematic analysis of 240 MeV ⁶Li elastic scattering from six different targets having various mass from ²⁴Mg to ¹¹⁶Sn. The main points which will be of concern are the followings : (1) the best fit to the experimental data using the χ2/N−method (2) the far- and near-side contributions to the differential cross section (3) the deflection function which provides an information about the existence of nuclear rainbow, (4) the reaction cross sections obtained from the strong absorption radius and the partial wave sum, respectively, (5) a linear dependence of reaction cross section on (AP1/3+AT1/3)2, and (6) the partial reaction cross sections.

This paper is organized as follows : In the next section, we briefly outline the theory related with the scattering amplitude and eikonal model. The results obtained from eikonal model calculation and their discussion are presented in section III. The concluding remarks are given in section IV.

The differential cross section for elastic scattering between two non-identical spin zero nuclei is given by the following equation :

The elastic scattering amplitude f(θ) has the expression

where fR(θ), k, σL, and PL(cosθ) are the Rutherford scattering amplitude, wave number, Coulomb phase shift, and Legendre polynomials, respectively. Second term on the right side of the above equation denotes the nuclear scattering amplitude, and S_L is the nuclear scattering matrix element obtained from the nuclear phase shift δL through the relation

Following the formalism of Fuller[16], the elastic scattering amplitude is decomposed in the near-side (fN(θ)) and far-side (fF(θ)) components through

and

where f_RN and f_RF correspond to the near- and the far-side components of the Rutherford amplitudes[16], and QL(∓) are linear combinations of Legendre functions of the first and the second kinds according to

In the eikonal model, the phase shift function δL(rc), taking into account the deflection effect due to Coulomb field, is given by[4-6]

where m is the reduced mass, U(r) is the nuclear potential, and r_c is the distance of closest approach given by

with η=mZ1Z2e2/(ℏ2k) being the Sommerfeld parameter. The potential U(r) used in this work is

where fv,w(r) is the Woods-Saxon squared shape given by

In the above equation, rv,w (av,w) represents the radius (diffuseness) parameter for the nuclear potential U(r), in which subscript v and w denote the real and imaginary parts, respectively. A_P and A_T are the mass numbers for projectile and target nuclei, respectively. Then the eikonal phase shift δL(rc) can be obtained by inserting this potential U(r) into Eq. (7). Thus one can calculate the nuclear scattering matrix element S_L with the help of δL(rc) from Eq. (3).

In this section, the eikonal model mentioned in previous section is applied to the systematic analysis of 240 MeV ⁶Li elastic scattering from six targets (²⁴Mg, ²⁸Si, ⁴⁰Ca, ⁵⁸Ni, ⁹⁰Zr, and ¹¹⁶Sn). The only input required for the eikonal model calculation is the nuclear potential. We use the optical Woods-Saxon squared potential given by Eqs. (9) and (10) as the nuclear potential. The parameter values of the nuclear potential are determined from the χ2/N−method, in which the χ2/N−value shows the quality of fit between the experimental data and theoretical results. In this work, uniform 10% errors of all analyzing experimental data are considered for the χ2/N−calculation. Determined values of the potential parameters (V0, rv, av, W0, r_w, and aw) are presented in Table 1, together with the χ2/N−value. The differential cross sections (normalization to the Rutherford cross section) of ⁶Li elastic scattering are displayed by the solid curves in Fig. 1 (for ²⁴Mg, ²⁸Si and ⁴⁰Ca targets) and Fig. 2 (for ⁵⁸Ni, ⁹⁰Zr and ¹¹⁶Sn targets). In these figures, the solid circles with error bars are the experimental data taken from Ref.[13] (for ⁶Li + ²⁴Mg and ⁶Li + ²⁸Si), Ref.[15] (for ⁶Li + ⁴⁰Ca), Ref.[14] (for ⁶Li + ⁵⁸Ni and ⁶Li + ⁹⁰Zr), and Ref.[12] (for ⁶Li + ¹¹⁶Sn). The calculated angular distributions show fairly good quality of fit to the experimental ones over the whole angles covered by the data. Also, the calculations provide a satisfactory reproduction of the oscillatory pattern shown in the experimental angular distribution of all ⁶Li + six target systems. As Table 1 shows, the obtained χ2/N−values were reasonable. We can see that it is possible to give a satisfactory account of 240 MeV ⁶Li elastic scattering from various targets in the mass number range AT=24 to 116 by using the eikonal model.

Table 1 Best-fit Woods-Saxon squared potential parameters extracted from the eikonal model analysis, and corresponding χ2/N−values for the ⁶% Li + ²⁴Mg, ²⁸Si, ⁴⁰Ca, ⁵⁸Ni, ⁹⁰Zr, and ¹¹⁶Sn elastic scatterings at 240 MeV. The χ2/N−values were obtained by considering uniform 10% errors for all analyzed data.}

Target	V0 (MeV)	rv (fm)	av(fm)	W0 (MeV)	rw (fm)	aw (fm)	χ2/N
24Mg	154	0.896	1.391	44.8	1.140	1.681	3.61
28Si	199	0.830	1.555	33.1	1.228	1.518	3.09
40Ca	222	0.849	1.521	39.7	1.216	1.523	3.77
58Ni	239	0.885	1.503	41.1	1.242	1.456	3.14
90Zr	257	0.932	1.455	41.8	1.280	1.367	3.63
116Sn	271	0.937	1.526	49.0	1.259	1.301	6.24

Figure 1. (Color online) Eikonal model results (solid curves) of the elastic scattering angular distributions using the parameters of Table 1 for ⁶Li + ²⁴Mg, ⁶Li + ²⁸Si and ⁶Li + ⁴⁰Ca systems at 240 MeV. Observed data (solid circles with error bars) are taken from Refs. [13, 15].

Figure 2. (Color online) Same as Fig. 1 but for the ⁶Li + ⁵⁸Ni, ⁶Li + ⁹⁰Zr and ⁶Li + ¹¹⁶Sn systems at 240 MeV. Observed data (solid circles with error bars) are taken from Refs. [12, 14].

Qualitative features of the dσ/dσRuth of the ⁶Li elastic scattering from ²⁴Mg, ²⁸Si, ⁴⁰Ca, ⁵⁸Ni, ⁹⁰Zr, and ¹¹⁶Sn targets can be understood using the near- (fN(θ)) and far-side (fF(θ)) amplitudes obtained from the decompositions of elastic scattering amplitude suggested by Fuller[16]. Figure 3 shows the results of total (solid curves), near- (dotted curves) and far-side (dashed curves) cross sections, in which the total cross sections denote the differential cross sections shown by the solid curves of Figs. 1 and 2. As Fig. 3 and Table 2 show, the dotted and dashed curves have equal magnitudes at the crossing angle θcross=8.1o 23.0o for ²⁴Mg ¹¹⁶Sn targets. As the target mass increases, the crossing angle is found to increase except for ²⁸Si target. Figure 3 further shows the near-side (far-side) dominance for scattering angle regions less (greater) than the crossing angle θcross. The oscillatory patterns appeared in the dσ/dσRuth (solid curves of Fig. 3) are considered to be resulted from the strong interference between the fN(θ) and the fF(θ) given by Eqs. (4) and (5). Maximum amplitude of interference oscillation is occurred around the crossing angle θcross. The far-side cross section plays a significant role in determining the total differential cross sections at scattering angle regions greater than the θcross.

Table 2 Physical quantities obtained from the eikonal model analysis for the ⁶Li + ²⁴Mg, ²⁸Si, ⁴⁰Ca, ⁵⁸Ni, ⁹⁰Zr, and ¹¹⁶Sn elastic scatterings at 240 MeV. The meaning of the quantities is explained in the text.

Target	θcross (deg)	θn.r. (deg)	L1/2	Rs (fm)	Lpeak	σpeak (mb)	σRs (mb)	σR (mb)
24Mg	8.1	−27.8	46.5	7.21	37	47.2	1633	1689
28Si	7.5	-30.3	48.2	7.29	38	46.7	1668	1695
40Ca	10.3	-34.8	55.1	7.92	44	49.5	1969	1971
58Ni	13.3	-39.7	62.0	8.59	51	52.8	2318	2270
90Zr	18.4	-47.8	70.5	9.52	59	58.3	2848	2718
116Sn	23.0	-49.1	74.1	9.94	63	60.9	3105	2906

Figure 3. (Color online) Elastic differential cross sections (solid curves) given in Figs. 1 and 2, near-side contributions (dotted curves) and far-side contributions (dashed curves) following the formalism of Fuller[16] by using the eikonal model for 240 MeV ⁶Li scattering from ²⁴Mg, ²⁸Si, ⁴⁰Ca, ⁵⁸Ni, ⁹⁰Zr, and ¹¹⁶Sn targets.

In order to examine the presence of nuclear rainbow, we plotted the deflection function θL(deg) as a function of the orbital angular momentum quantum number L. This function can be obtained from the variations of the Coulomb phase shift and real nuclear phase shift over L given by the formula[17]

In a rainbow situation, the strong nuclear force attracts projectiles toward the scattering center and deflects them to negative scattering angles. As shown in Fig. 4, the θL(deg) functions have negative maximum value corresponding to the nuclear rainbow angle θn.r.. This fact indicates that a nuclear rainbow is clearly appeared in all ⁶Li + six target systems at 240 MeV. As Table 2 and Fig. 4 show, the magnitude of θn.r. displayed a gradual increase from −27.8o to −49.1o as the target mass became heavier from ²⁴Mg to ¹¹⁶Sn.

Figure 4. (Color online) Deflection functions for the ⁶Li + ²⁴Mg, ²⁸Si, ⁴⁰Ca, ⁵⁸Ni, ⁹⁰Zr, and ¹¹⁶Sn elastic scatterings at 240 MeV plotted versus the orbital angular momentum L.

In Table 2, the physical quantities extracted from eikonal model analysis are collected. The L_1/2 is the critical angular momentum determined by the condition of |SL|2=1/2, where S_L is the scattering matrix element expressed by Eq. (3). As this Table shows, the L_1/2 value is increasing from 46.5 to 74.1 with the increase in target mass from ²⁴Mg to ¹¹⁶Sn. The L_1/2 value and its trend are also reflected in the strong absorption radius R_s obtained from the formula : Rs=η+η2+(L1/2+12)2%/k. The obtained R_s value has a tendency of slightly increasing from 7.21 fm to 9.94 fm when the target mass is increasing from ²⁴Mg to ¹¹⁶Sn.

The reaction cross section σR is expressed[18] using the |SL|2 :

Within the framework of eikonal model, |SL|2 is influenced by the imaginary part of nuclear potential U(r) given by Eqs. (9) and (10) because |SL|2 is calculated from the imaginary phase shift. In Table 2, we listed σR values calculated in this work. As expected, the values of σR increased when the target nucleus became heavier. On the other hand, the so-called geometrical reaction cross section σRs are obtained from the strong absorption radius R_s from the formula : σRs=πRs2. As shown in Table 2, this σRs value is comparable to σR value obtained from Eq. (12) for each scattering system. The percentage errors of σRs relative to σR are 3.3, 1.6, 0.1, 2.1, 4.8, and 6.8% for ²⁴Mg, ²⁸Si, ⁴⁰Ca, ⁵⁸Ni, ⁹⁰Zr, and ¹¹⁶Sn target nuclei, respectively. This fact tells us that the R_s may be used to estimate the magnitude of σR value.

In Fig. 5(a), the σR values obtained from Eq. (12) are plotted as a function of (AP1/3+AT1/3)2 because the strong absorption radius is generally proportional to (AP1/3+AT1/3) and provides a good estimation of the σR value. This figure shows that σR values have a nearly linear dependence on (AP1/3+AT1/3)2, where the solid curve can be described by the formula : σR(b)=0.0689(AP1/3+AT1/3)2. Figure 5(b) shows the partial reaction cross sections σL=πk2(2L+1)(1−|SL|2) of ⁶Li elastic scattering from six different target nuclei versus the orbital angular momentum quantum number L. As Table 2 and Fig. 5(b) show, the partial reaction cross sections increased linearly up to some peak values σpeak=47.2, ⋯, 60.9 mb for ²⁴Mg, ⋯, ¹¹⁶Sn targets, respectively, and thereafter decreased rapidly to zero. These σpeak values became larger as the target mass increased : the only exception is the case of ²⁸Si target. In Table 2, the Lpeak is the orbital angular momentum quantum number corresponding to σpeak. The Lpeak values displayed a gradual increase from 37 to 63 when the target mass is increasing from ²⁴Mg to ¹¹⁶Sn.

Figure 5. (Color online) (a) Reaction cross section versus the (AP1/3+AT1/3)2, and (b) partial reaction cross sections versus the orbital angular momentum L for ⁶Li + ²⁴Mg, ²⁸Si, ⁴⁰Ca, ⁵⁸Ni, ⁹⁰Zr, and ¹¹⁶Sn elastic scatterings at 240 MeV.

In this paper, we systematically analyzed the differential cross sections for 240 MeV ⁶Li elastic scattering from six targets (²⁴Mg, ²⁸Si, ⁴⁰Ca, ⁵⁸Ni, ⁹⁰Zr, and ¹¹⁶Sn) using the eikonal model based on the Coulomb trajectories of colliding nuclei. We have shown that the eikonal model provides a satisfactory account of elastically measured data of all ⁶Li + six targets scattering systems at 240 MeV. In this study, the calculated results yielded a successful reproduction of the elastic angular distribution structure, and gave fairly good quality of fit to the experimental measurements over the whole angles covered by the data for a wide mass range of target nuclei. Qualitative understanding of the elastic cross section can be gained by the near- and far-side decompositions of the elastic scattering amplitude. The strong interference between these two components of the scattering amplitude resulted in the oscillatory pattern appeared in the dσ/dσRuth of all ⁶Li + six target systems. Maximum amplitude of interference oscillation is observed around the crossing angle. No oscillatory behavior of the dσ/dσRuth at large θc.m. regions is dominantly influenced by the far-side scattering.

The deflection function has a negative maximum value for each scattering systems, which indicates evidently a presence of an nuclear rainbow. The magnitude of nuclear rainbow angle became larger as the target mass increased. The critical angular momentum L_1/2 and the strong absorption radius R_s displayed an increasing trend as the target mass increased. The geometrical reaction cross section σRs values obtained from σRs=πRs2 are similar to the σR values from partial wave sum. This fact tells us that the R_s provides a good estimation of the σR value. Linear dependence of σR on (AP1/3+AT1/3)2 is observed with the formula : σR(b)=0.0689(AP1/3+AT1/3)2. The partial reaction cross-sections increased linearly up to some peak values (σpeak) and thereafter decreased rapidly to zero. The orbital angular momentum quantum number Lpeak corresponding to the σpeak increased as the target mass increased.

Finally, our results confirm that the eikonal model calculation successfully reproduce the elastically measured scattering data of ⁶Li projectile on six targets having various mass from ²⁴Mg to ¹¹⁶Sn at 240 MeV. Therefore, we can see that the eikonal model is a useful tool to be used in the analysis of the nucleus-nucleus elastic scattering data

This work was supported by the research grant of Jeju National University in 2022.