Elastic scattering between heavy-ions (HI) has been one of research topics in nuclear physics over the past decades. The main observation quantity in HI elastic scattering experiment is differential scattering cross section. This physical quantity is theoretically obtained from the square of the scattering amplitude. Meanwhile, the scattering matrix element is a factor that constitutes the scattering amplitude, and is closely related to the nuclear phase shift. Therefore, an important ingredient in calculating the differential scattering cross section is the nuclear phase shift. The eikonal model is a convenient and useful approach to describe the HI elastic scattering cross section in the incident energy range from several tens of MeV/nucleon to about 100 MeV/nucleon. The nuclear phase shift in this model is obtained by integrating the nuclear potential. There have been many studies[1-8] attempting to describe HI elastic scattering using the eikonal model. Trajectory corrections using the effective impact parameters in HI elastic scattering are discussed[5]. In our work[6] conducted a long time ago, analysis of ¹⁶O + ¹⁶O elastic angular distribution data at incident energies of 480 and 704 MeV was performed using the second-order eikonal model. Elastic scattering of 240 MeV ⁶Li ions by ²⁴Mg and ²⁸Si was analyzed[7] using an eikonal model that takes into account the tangential velocity at the distance of closest approach.

Elastic scattering between α-particle and various target nuclei has been actively studied[9-13]. Nayak et al.[11] measured the α + ⁵⁸Ni elastic scattering cross section at 386 MeV, and used a folding model to describe these data. The measured data of α + ³²S system were also obtained[12] at the same incident energy, and analyzed using the DWBA. It is interesting to analyze the measured data of 386 MeV α-particles elastic scattering on ³²S and ⁵⁸Ni using the eikonal model. In this paper, we analyze these experimental data using the eikonal model considering the tangential velocity at the distance of closest approach. It is also investigated how the use of tangential velocity affects the results of differential and reaction cross sections calculations. Furthermore, the structure of angular distributions, the Argand diagram of scattering matrix element, the strong absorption radius, the density overlap of two colliding nuclei determined by strong absorption radius, and the effect of variation in potential radius parameters on the differential and reaction cross sections are investigated.

The elastic scattering amplitude f(θ) is expressed as the sum of Rutherford scattering amplitude (fR(θ)) and nuclear scattering amplitude (second term in the equation below), given as follows:

where k=2μE/ℏ, and θ is the scattering angle. Here Coulomb phase shift σL, nuclear scattering matrix elements SL, and Legnedre polynomial PL(cosθ) depend on the orbital angular momentum quantum number L. The SL has a relationship given by the following equation with the nuclear phase shift δL

where δL has a complex form. Then, the elastic differential cross section, called angular distribution, is calculated as

In the eikonal model based on the Coulomb trajectory of colliding nuclei, the nuclear phase shift δL(rc) is expressed as an integral over the nuclear potential U(r) as follows[3]

Here v=ℏk/μ is the asymptotic velocity, r=rc2+z2, and the rc is the distance of closest approach for a Coulomb trajectory given by[14]

where ac=Z1Z2e2/2E is the Coulomb length parameter and b=(L+1/2)/k is the impact parameter. In the Coulomb trajectory, the straight-line trajectory is deflected around rc due to the Coulomb field between the two colliding nuclei. Thus, we will use the tangential velocity considering the deflected trajectory at rc, and the tangential velocity vc is given as follows[5]

If v is replaced by vc in Eq. (4), the nuclear phase shift δL(rc) can be written as follows

In this study, the nuclear phase shift given by Eq. (7) is used instead of Eq. (4). As the U(r), we use six-parameters Woods-Saxon (WS) potential given as follows:

where V0 (W0) is the real (imaginary) nuclear potential depth, and the form factor f(r,rx,ax) is usually taken as

In this equation rx (ax) denotes the radius (diffuseness) parameter, where x=v (w) represents the real (imaginary) part, while A is the target mass number. The potential parameters given in Eqs. (8) and (9) are determined by the minimum χ2/N-fit to the measured data.

1. Angular distributions

The elastic scattering cross sections of α + ³²S and α + ⁵⁸Ni systems at 386 MeV were calculated using the eikonal model considering the vc at rc. The WS potential parameters that best describe the measured data[11, 12] when using Eq. (7) are listed in Table 1 and the corresponding angular distributions are shown as solid curves in Fig. 1. The dotted curves in this figure are the results of using the asymptotic velocity (v) instead of the tangential velocity (vc) in Eq. (7). Although the two curves show relatively good agreement in the small θc.m. regions, it can be seen that there is a substantial difference between the solid curve and the dotted curve in the large θc.m. regions. The solid curves reproduce successfully the complex behavior of experimental angular distribution and give good agreement with the elastic data[11, 12] in the entire measured angle regions. The χ2/N-values obtained from the calculations using v and vc at rc, respectively, are shown in Table 1. As this Table shows, the χ2/N-values using tangential velocity at rc are reasonable for each scattering system.

Figure 1. (Color online) Angular distributions for the elastic scattering of 386 MeV α-particles from ³²S and ⁵⁸Ni. The solid curves are the eikonal model calculations using tangential velocity whereas the dotted curves are the calculations using asymptotic velocity, where both results were calculated using the WS potential parameters given in Table 1. The dashed curves are the calculation results that best fits the experimental data when using the asymptotic velocity, and the potential parameters used are shown in Table 2. Experimental data are taken from Refs. 11, 12.

Table 1

Potential parameters extracted from the minimum χ2/N-fit to the measured data and corresponding χ2/N-value in the eikonal model analysis using the tangential velocity at rc for 386 MeV α-particle elastic scatterings from ³²S and ⁵⁸Ni. σR in the last column is the reaction cross section obtained using the partial wave sum given by Eq. (10). Values in parentheses are the results obtained using the asymptotic velocity. χ2/N-values were obtained under the assumption that all experimental data have a 10% error.

Target	V0	rv	av	W0	rw	aw	χ2/N	σR
	(MeV)	(fm)	(fm)	(MeV)	(fm)	(fm)		(mb)
32S	63.7	0.883	0.843	29.6	0.991	0.671	1.33 (4.84)	1052 (1048)
58Ni	74.8	0.916	0.834	33.5	1.022	0.756	1.83 (3.30)	1539 (1532)

In order to compare the degree of agreement with the experimental data[11, 12] in the calculation results of the eikonal model considering the v and vc, the optimal potential parameters that best fit the measured data were found when using the asymptotic velocity, and the values found are shown in Table 2. The eikonal model calculation results using asymptotic velocity and potential parameters given in Table 2 are shown as dashed curves in Fig. 1. As shown in Tables 1 and 2, the χ2/N-values of the calculation results using the tangential velocity are smaller than the results using the asymptotic velocity. This means that the solid curves of Fig. 1 match the measured data better than the dashed curves.

Table 2

Same as Table 1, but potential parameters that best fit the measured data when using the asymptotic velocity.
χ2/N-values were obtained under the assumption that all experimental data have a 10% error.

Target	V0	rv	av	W0	rw	aw	χ2/N	σR
	(MeV)	(fm)	(fm)	(MeV)	(fm)	(fm)		(mb)
32S	75.4	0.836	0.909	34.2	1.003	0.622	2.17	1067
58Ni	85.3	0.883	0.883	40.1	1.014	0.700	2.11	1510

The technique of decomposing the f(θ) into near- and far-sides using the Fuller formalism[15] helps to qualitatively understand the structure of the angular distribution. The contributions of the near-side and far-side components to the dσ/dσRuth obtained from the eikonal model using vc at rc are shown as dotted and dashed curves, respectively, in Fig. 2 along with the dσ/dσRuth shown as solid curves. The near-side cross sections dominate at small scattering angles, while the far-side cross sections contribute little. However, the far-side contributions increase in magnitude as the θc.m. increases and become equal to the near-side contributions at crossing angle θcross (θcross=3.30∘ for α + ³²S and θcross=4.50∘ for α + ⁵⁸Ni). At scattering angles larger than the θcross, the near-side amplitude does not contribute significantly, while the far-side amplitude makes an important contribution in determining the behavior of the elastic cross section. The strong interference between the near- and far-side scattering amplitudes can be thought of as giving rise to an oscillating structure observed around the θcross.

Figure 2. (Color online) Differential cross sections (solid curves) drawn as solid curves in Fig. 1 for the elastic scattering of 386 MeV α-particles from ³²S and ⁵⁸Ni, their near-side (dotted curves) and far-side (dashed curves) components calculated from the Fuller's formalism[15].

2. Scattering Matrix Elements and Reaction Cross Sections

Argand diagrams of the scattering matrix elements SL using eikonal phase shift given by Eq. (7) are plotted in Fig. 3. The Arabic numerals along the curve in this figure represent the values of orbital angular momentum quantum number L. The Argand diagrams of two scattering systems show similar shapes. This graphical images of SL shows two distinct characteristics. The first one is that they starts at (Re(SL), Im(SL))=(0.0, 0.0) and finally end at (1.0, 0.0). The other is that the graphical image depicts the shape of a nautilus shell spiral, as shown in Fig. 3.

Figure 3. (Color online) Argand diagram of the scattering matrix element for α + ³²S and α + ⁵⁸Ni elastic scatterings at 386 MeV. The Arabic numerals along the curve denote the L-values.

The transmission functions TL=1-|SL|2 using Eqs. (2) and (7) are plotted in Fig. 4(a). The TL functions have a value of 1 for small L and 0 for large L, and change rapidly from 1 to 0 around the critical angular momenta L1/2 (42.44 for α + ³²S and 54.17 for α + ⁵⁸Ni) corresponding to TL=0.5. The L1/2 is closely related to the strong absorption radius Rs defined as Rs=ac+ac2+(L1/2+12)2/k2. The calculated Rs values are 5.688 fm for α + ³²S and 6.912 fm for α + ⁵⁸Ni systems, respectively. Furthermore, using the Rs, the so-called geometric reaction cross section can be calculated from the relationship σRs=πRs2, and the obtained values are 1017 mb and 1501 mb for ³²S and ⁵⁸Ni targets, respectively. Meanwhile, the reaction cross section σR given in Tables 1 and 2 is obtained using the partial wave sum given by

Figure 4. (Color online) (a) Transmission functions, and (b) density overlaps of two colliding nuclei when the distance between two nuclear centers is equal to Rs for α + ³²S and α + ⁵⁸Ni systems at 386 MeV.

where TL is the transmission function mentioned just before. From the σR values given in Table 1, the following three facts can be known. First, the σR obtained using tangential velocity is slightly larger than that obtained using asymptotic velocity. Second, the σRs value for each scattering system is similar to the σR ones, which indicate that Rs provides quite a satisfactory account of the σR obtained using partial wave sum. Third, as the target mass increases, the Rs increases, and thus the σR (or σRs) increases.

The overlaps of the projectile and target densities are shown in Fig. 4(b) when the distance between the centers of the two colliding nuclei is equal to the Rs. In this figure, it is assumed that both the projectile and the target have a Gaussian-shaped density distribution given by :

where root-mean-square radii (Rrms) were taken as 1.71 fm for ⁴He, 3.243 fm for ³²S and 3.764 fm for ⁵⁸Ni taken from Ref. 16. The overlap provides an interpenetration of the two colliding nuclei with a fixed absorption rate (TL=1/2) for two scattering systems. As Fig. 4(b) shows, the density overlap of two colliding nuclei for α + ³²S system is larger than the one for α + ⁵⁸Ni system, indicating weak absorption. This weakening absorption allows colliding nuclei to interpenetrate more deeply without being absorbed. Consequently, the σR of α + ³²S system is smaller than the one of α + ⁵⁸Ni system.

3. Potential radius parameter effect on differential and reaction cross sections

To investigate how variations in real (rv) and imaginary (rw) radius parameters of WS potential affect the differential cross section, we plotted the dσ/dσRuth of α + ³²S system by varying the rv (or rw) value, where the other potential parameters except rv (or rw) were fixed. The radius parameters are taken ±0.02 fm from rv (or rw) value given in Table 1, and the calculation results using the tangential velocity at rc are plotted in Fig. 5. As this figure shows, in the relatively large angle region, the dσ/dσRuth move upward (downward) as the value of rv increases (decreases). On the other hand, an increase (decrease) in the value of rw shifts the dσ/dσRuth downward (upward). In both Figs. 5(a) and 5(b), the three curves show that the difference becomes larger as the θc.m. increases.

Figure 5. (Color online) Angular distributions for α + ³²S elastic scattering at 386 MeV calculated using different real (or imaginary) radius parameters rv (or rw). Other input parameters in the angular distribution calculation were fixed, and these values are given in Table 1.

The TL function can be expressed as TL=1-exp[-4Im(δL)] with the help of Eq. (2) because the phase shift δL is given in complex form as δL=Re(δL)+iIm(δL). Meanwhile, as given in Eq. (10), the reaction cross section σR is related to the TL function obtained from the Im(δL). Therefore, the σR is directly related to the Im(δL). As given in Eqs. (7)–(9), the Im(δL) is obtained from the imaginary part of WS potential (Im(U(r))), so the σR is ultimately affected by the Im(U(r)). Three different rw values produce somewhat different σR values (σR =1016 mb,1052 mb and 1089 mb for rw= 0.971 fm, 0.991 fm and 1.011 fm, respectively) in α + ³²S system. It can be seen that the σR value increases as the imaginary radius parameter rw increases.

This paper presented the results of eikonal model analysis for the measured data of 386 MeV α-particle elastic scattering from ³²S and ⁵⁸Ni targets. For model calculation, the WS potential with six-parameters and the tangential velocity at rc are used. The calculation results successfully reproduced the structure of experimental angular distribution and were in good agreement with the measured data. Strong interference between near- and far-side scattering amplitudes is thought to give rise to the oscillatory structures that appears around the crossing angle in the angular distribution. In the regions of scattering angle greater than crossing angle, the far-side amplitude made an important contribution in determining the behavior of the elastic cross section.

The graphical images of scattering matrix elements SL of two systems show similar shapes and display two distinct characteristics: the first one is that they begin at (Re(SL), Im(SL))=(0.0, 0.0) and end at (1.0, 0.0) ; the other is that the graphical images look like a nautilus shell spiral. The critical angular momentum L1/2 and the strong absorption radius Rs for ³²S target are smaller than those for ⁵⁸Ni target. The geometric reaction cross section σRs can be extracted from the Rs in terms of σRs=πRs2, and its values are comparable to the reaction cross sections σR obtained using partial wave sum. The fact that the σRs and σR values are similar indicates that Rs can be usefully used to estimate the magnitude of σR. When two colliding nuclei are separated by Rs, projectile and target densities are found to overlap more in α + ³²S system than α + ⁵⁸Ni case, indicating weak absorption. As a result, the σR value of α + ³²S system is smaller than that of α + ⁵⁸Ni system.

As the value of real radius parameter rv of WS potential increases (decreases), the elastic cross section moves upward (downward), especially in the relatively large angle regions. On the other hand, increasing (decreasing) the imaginary radius parameter rw value shifts the elastic cross section downward (upward). Additionally, as rw increased, the reaction cross section also increased.

This research was supported by the 2024 scientific promotion program funded by Jeju National University.