The Glauber model has been used for the description of the heavy-ion elastic scattering. In most of applications, the so-called optical limit approximation (OLA), which considered only a leading term in an expansion of the phase shift function, has been employed to evaluate the Glauber elastic S-matrix. A conventional Glauber model [1] is assumed that the projectile moves a straightline path along the collision direction. At relatively intermediate and low energies, the Rutherford trajectory is introduced [2] to account for the Coulomb effect instead of straight-line trajectory. The OLA to the Glauber model modified for the Coulomb effect is called as ”Coulomb-modified Glauber model(CMGM) [3,4]”.

The important constituents characterizing the phase shift in CMGM are the nucleon-nucleon (NN) scattering amplitude and nuclear density distributions. The closed form analytic expression for phase shift can be obtained [2] by taking both the projectile and target density distributions as a Gaussian shape. The adoption of Gaussian density is usually suitable for A < 40. In the case of heavier nucleus having A ≧ 40, realistic nuclear density distribution is employed to describe the surface region where density is small, as the elastic cross section gets generally a predominant contribution from the surface region. In order to use the closed form of phase shift, it is needed to obtain the Gaussian density by using surface matching method which reproduce the experimentally determined nuclear surface texture. Karol [5] obtained the surface-matched Gaussian density (SMGD) parameters by fitting the realistic densities in the surface. On the other hand, Charagi and Gupta [3] computed the profile function for realistic density and matched it to a Gaussian profile one at nuclear surface instead of Karol’s prescription. In our recent paper [6], we calculated the Gaussian density parameters using the Karol’s method and applied it satisfactorily to the analysis of the elastic scattering cross section of α particles from ⁴⁰Ca and ⁵⁸Ni at 240 MeV.

In this paper, we analyze the elastic scattering of 240 MeV α particle from ¹¹⁶Sn and ¹⁹⁷Au targets within the framework of the Coulomb-modified Glauber model. The Gaussian density parameters for target nuclei are calculated from the profile function matching method suggested by Charagi and Gupta [3]. The calculated differential cross sections are compared with the calculations without matching. Further, we calculate the optical potentials from the inversion method, and investigate its behavior at the surface regions around the strong absorption radius. In the following section, we present a theory related with the Coulomb-modified Glauber model and the profile function matching method. Sec. III is devoted to an application of the present model to the elastic α + ¹¹⁶Sn and α + ¹⁹⁷Au scatterings at 240 MeV. Final section presents the concluding remarks.

1. Scattering cross section and optical potential in CMGM

If we assume both the projectile (P) and target (T) densities as a Gaussian shape

and take the NN scattering amplitude as a function of momentum transfer q given by [7]

the nuclear phase shift δ(b) [8] in the OLA to the Glauber model can be written [6]

where

In Eq. (2), k_NN is the wave number of the NN system, α_NN the NN total cross section, α_NN the Re[f_NN(0)]/Im[f_NN(0)], and β_NN the slope parameter.

The CMGM consists of replacing the impact parameter b by the distance of closest approach r_c given by

where k and η are the wave number and the Sommerfeld parameter for α-nucleus system, respectively. Then, we can use the phase shift of Eq. (3) as a function of r_c instead of b.

The differential cross section dσ/dΩ for the α-nucleus elastic scattering is calculated from the square of scattering amplitude f(θ) given by

where f_R(θ) and σ_L are the Rutherford scattering amplitude and the usual Coulomb phase shift, respectively. Nuclear S−matrix element S_L in above equation is expressed as

with the nuclear phase shift δ(r_c) given by Eqs. (3) and (5).

Within the CMGM, one can obtain a complex optical potential U(r) from the inversion formula given by [2,9]

Then, CMGM optical potential by inversion is expressed as

2. Gaussian density parameters

For the light nuclei less than mass number 40, densities are given in the form of Gaussian distribution as Eq. (1), and its density parameters a_i and ρ_i(0) are determined from root-mean-square (RMS) radius R_rms as

But for heavier nuclei (A ≧ 40), realistic density such as two-parameter Fermi (2pF) form is used to describe the surface region where density is small. The 2pF distribution for target density is expressed in terms of radius c_T and diffuseness d_T parameters as

where

Charagi and Gupta [3] reported the profile function matching method to obtain the SMGD parameters. The profile function ρiz (b) is given by

where ρ_i is the nuclear density. Then 2pF profile function for the target nucleus is expressed as

The parameters a_T and ρ_T (0) are adjusted to reproduce the experimentally determined nuclear surface texture by requiring the ρTz (b) at b = c_T and b = c_T + 4d_T in the Gaussian density distribution to be identical to the values calculated from the realistic 2pF density distribution. The Gaussian density parameters a_T and ρ_T (0) obtained by the profile function matching method are given as [3]

and

where cT' = c_T + 4d_T.

The values of the parameters a_T and ρ_T (0) obtained by the profile function matching method are given in Table 1. Fig. 1 shows the profile functions of the 2pF density (solid curves) and SMGD (dotted curves) for two target nuclei (¹¹⁶Sn and ¹⁹⁷Au). We can see in this figure that the agreement between the profile functions for the 2pF densities and the fitted Gaussian densities is satisfactory in the surface region. Since the differential cross section gets a contribution predominantly from the surface region of the colliding nuclei, the discrepancies between those profile functions at the central regions can be neglected.

Table 1 Gaussian density parameters a_P/T and ρ_P/T (0) entering phase shift of Eq. (3). The a_P and ρ_P (0) are obtained from Eq. (10) using the RrmsP. The aT and ρ_T (0) are calculated from Eqs. (15) and (16) given by profile function matching method. RrmsP is taken from Ref. [10] and 2pF density parameters c_T and d_T are taken from Ref. [11].

System	RrmsP (fm)	aP (fm)	ρP (0) (fm-3)	cT (fm)	dT (fm)	RrmsT (fm)	aT (fm)	ρT (0) (fm-3)
α + 116Sn	1.71	1.396	0.06598	5.275	0.539	4.551	2.856	0.02362
α + 197Au	1.71	1.396	0.06598	6.38	0.535	5.327	3.071	0.03439

Figure 1. (Color online) Profile functions for target nuclei (¹¹⁶Sn and ¹⁹⁷Au) plotted versus impact parameter b. The solid and dotted curves represent the profile functions for 2pF and SMGD distributions, respectively. The densities are assumed to be normalized to the mass number of the nucleus.

3. Input parameters

The Gaussian density parameters a_i and ρ_i(0) are determined as follows : (1) The a_P and ρ_P (0) were calculated using Eq. (10) with RMS radius [10]. (2) The calculations of a_T and ρ_T (0) are performed in two ways : one is calculated from the RMS radii [11] using Eq.(10) and the other is from the profile function matching method using Eqs. (15)-(16) where 2pF density parameters c_T and d_T are taken from Ref. [11]. Parameters of the Gaussian and the 2pF densities are collected in Table 1.

The parameter values of α_NN and β_NN related with the NN scattering amplitude are determined as follows : (1) The α_NN values were determined by minimized χ²/N-value obtained from a comparison of the experimental data with the calculations. (2) The slope parameter β_NN was taken to be an average value of β_pp(nn) and β_np(pn) given in Table 6 of Ref. [12]. The remaining input σ_NN in the nuclear phase shift is calculated by the expression

where the values of σ_pp(nn) and σ_np are obtained from a phenomenological formula [13] with ρ = 0.16 fm^-3 [12]. The values of σ_NN, β_NN and α_NN are listed in Table 2.

Table 2 Input parameters (σ_NN, β_NN and α_NN), and physical quantities obtained by phase shift analysis. Cal. 1 and Cal. 2 denote the results using the a_T and ρ_T (0) calculated from the RrmsT (Eq. (10))and the profile function matching method (Eqs. (15) and (16)), respectively. The values in parentheses are the results obtained by the α_NN corresponding to the lowest χ²/N value. 10% error bars are adopted to obtain χ²/N values. All quantities are defined in the text.

	Cal. 1		Cal. 2
Target	116Sn	197Au	116Sn	197Au
σNN(mb)	67.32	67.32	67.32	67.32
βNN(fm2)	1.1433	1.1433	1.1433	1.1433
αNN	1.008 (1.248)	0.987 (0.428)	1.008	0.987
θcross(deg)			12.10	16.40
L1/2	55.64	65.18	53.32	60.70
Rs(fm)	8.885	10.383	8.530	9.709
σRs (mb)	2480	3387	2286	2961
σR(mb)	2424	3220	2200	2754
χ2/N	21.82 (17.28)	29.01 (17.36)	3.48	4.69

We have calculated the elastic differential cross sections for α + ¹¹⁶Sn and α + ¹⁹⁷Au systems at 240 MeV using the CMGM with the Gaussian density parameters of target obtained by the profile function matching method. The calculated results are presented in Fig. 2. The data points (solid circles) are taken from Ref. [14]. The solid curves are calculations using the SMGD given by Eqs. (15)-(16) while the dotted curves are calculations using the non-SMGD given by Eq. (10). As this figure shows, the dotted curves have lower values compared to solid curves and did not adequately reproduced the elastic data as the scattering angle becomes increases, though the dotted curves displayed qualitatively oscillatory structure of the elastic cross section. However, by adopting the SMGD, the dotted curves move upward direction as the angle increases, consequently, leading to fairly good agreements with the observed data. We can see that the calculations with SMGD were found to do better in reproducing the experimental data than those with non-SMGD.

Figure 2. (Color online) CMGM fits to the experimental data for elastic α + ¹¹⁶Sn and α + ¹⁹⁷Au scatterings at E_lab = 240 MeV. The solid and dotted curves are the CMGM calculations using the SMGD and non-SMGD parameters, respectively. The dashed curves are the best results using the non-SMGD parameters. The solid circles are the experimental data [14].

To examine the necessity of adopting the SMGD in the description of elastic cross section, we have also performed a CMGM calculation with Gaussian density parameters determined from RMS radius by varying the free parameter α_NN value. The results of best quality fit are shown by the dashed curves in Fig. 2 and the corresponding α_NN values are also given in Table 2. For α + ¹¹⁶Sn system, the dashed curve is moved to the left as a whole, and provided lower values at the maximum regions, in comparison with the solid curve. In the case of α + ¹⁹⁷Au system, the dashed curve generated decreasing pattern, consequently did not provided the pronounced maxima and minima observed experimentally. From Table 2 and Fig. 2 we can see that the calculations using the SMGD parameters provide more reason able χ²/N−values and give better fits with the observed data than the ones using non-SMGD parameters.

The near- and far-side decompositions of the scattering amplitudes with the SMGD were also performed by following the Fuller’s formalism [15]. The dotted and dashed curves of Fig. 3 represent the contributions of the near-side and the far-side components to the elastic scattering cross sections (solid curves). The near-side contribution due to repulsive Coulomb interaction dominates at small angles and the far-side one due to attractive nuclear interaction at large angles. The magnitudes of the near- and far-side contributions are nearly equal at crossing angles θ_cross = 12.1° and θ_cross = 16.4° for ¹¹⁶Sn and ¹⁹⁷Au targets, respectively. The oscillatory behaviors of the elastic angular distributions are considered to be resulted from the strong interference of the far- and near-side scattering amplitudes. The interference oscillations have maximum amplitudes around the θ_cross. However, the far-side contributions to the cross sections dominate at the regions greater than those angles.

Figure 3. (Color online) Near-side (dotted curves) and far-side (dashed curves) contributions to the differential cross sections (solid curves) following the Fuller’s formalism [15] using the SMGD based on CMGM for elastic α + ¹¹⁶Sn and α + ¹⁹⁷Au scatterings.

Table 2 shows the physical quantities obtained by the phase shift analysis. The L_1/2 is the critical angular momentum for which CMGM yields SL2 = 0.5. We can see that the L_1/2 values obtained from the SMGD are somewhat less than those from the non-SMGD. This feature influences the strong absorption radius R_s given by Eq. (5) at b=L1/2L1/2+1/k. The R_s values obtained from SMGD are slightly lower than the ones from non-SMGD due to a decrease of L_1/2. This situation is also reflected in the reaction cross sections. As shown in Table 2, the strong absorption radius provides a good estimation of the reaction cross section in terms of σ_Rs = πRs2, though the σ_Rs values are somewhat larger than the σ_R ones determined from the partial wave sum formula [1] σ_R = (π/k²) ∑ L=0∞2L+11− SL2.

To further investigate the difference of differential cross sections using the SMGD and non-SMGD parameters, we plotted in Fig. 4 the optical potentials obtained by using the inversion method given in Eq. (9). In this figure, the dashed and dotted curves denote the inverse potentials obtained by using SMGD and non-SMGD parameters , respectively, while the solid curves are the Woods-Saxon ones used in the optical model fit [14] to the same scattering data. It is well known that the elastic scattering cross sections are sensitive to the optical potential at the surface regions around the R_s. This figure shows that the dotted curves produce higher values in the vicinity of R_s than both the solid and dashed curves. Especially, the difference for imaginary potentials around the R_s region is much larger than the one of real potentials. On the other hand, the dashed and solid curves agree fairly well with each other in the vicinity of the R_s. From this figure it is also evident that the potentials determined from using SMGD provided more closer to the Woods-Saxon ones in the vicinity of the R_s in comparison with the potentials from using non-SMGD. We can infer that the difference of two optical potentials (dashed and dotted curves) around R_s produces somewhat different behaviors of elastic angular distribution because the main features of heavy-ion elastic scattering are essentially dominated by the surface regions around the R_s.

Figure 4. (Color online) The optical potentials for elastic α + ¹¹⁶Sn and α + ¹⁹⁷Au scatterings. The dashed and dotted curves denote the optical potentials obtained from Eq. (9) using the SMGD and non-SMGD parameters, respectively, while the solid curves are the Woods-Saxon optical potentials used in the optical model fits [14] to the same scattering data. The arrows indicate the positions of the Rs given in Cal. 2 of Table 2.

1. Scattering cross section and optical potential in CMGM

If we assume both the projectile (P) and target (T) densities as a Gaussian shape

and take the NN scattering amplitude as a function of momentum transfer q given by [7]

the nuclear phase shift δ(b) [8] in the OLA to the Glauber model can be written [6]

where

In Eq. (2), k_NN is the wave number of the NN system, α_NN the NN total cross section, α_NN the Re[f_NN(0)]/Im[f_NN(0)], and β_NN the slope parameter.

The CMGM consists of replacing the impact parameter b by the distance of closest approach r_c given by

where k and η are the wave number and the Sommerfeld parameter for α-nucleus system, respectively. Then, we can use the phase shift of Eq. (3) as a function of r_c instead of b.

The differential cross section dσ/dΩ for the α-nucleus elastic scattering is calculated from the square of scattering amplitude f(θ) given by

where f_R(θ) and σ_L are the Rutherford scattering amplitude and the usual Coulomb phase shift, respectively. Nuclear S−matrix element S_L in above equation is expressed as

with the nuclear phase shift δ(r_c) given by Eqs. (3) and (5).

Within the CMGM, one can obtain a complex optical potential U(r) from the inversion formula given by [2,9]

Then, CMGM optical potential by inversion is expressed as

2. Gaussian density parameters

For the light nuclei less than mass number 40, densities are given in the form of Gaussian distribution as Eq. (1), and its density parameters a_i and ρ_i(0) are determined from root-mean-square (RMS) radius R_rms as

But for heavier nuclei (A ≧ 40), realistic density such as two-parameter Fermi (2pF) form is used to describe the surface region where density is small. The 2pF distribution for target density is expressed in terms of radius c_T and diffuseness d_T parameters as

where

Charagi and Gupta [3] reported the profile function matching method to obtain the SMGD parameters. The profile function ρiz (b) is given by

where ρ_i is the nuclear density. Then 2pF profile function for the target nucleus is expressed as

The parameters a_T and ρ_T (0) are adjusted to reproduce the experimentally determined nuclear surface texture by requiring the ρTz (b) at b = c_T and b = c_T + 4d_T in the Gaussian density distribution to be identical to the values calculated from the realistic 2pF density distribution. The Gaussian density parameters a_T and ρ_T (0) obtained by the profile function matching method are given as [3]

and

where cT' = c_T + 4d_T.

The values of the parameters a_T and ρ_T (0) obtained by the profile function matching method are given in Table 1. Fig. 1 shows the profile functions of the 2pF density (solid curves) and SMGD (dotted curves) for two target nuclei (¹¹⁶Sn and ¹⁹⁷Au). We can see in this figure that the agreement between the profile functions for the 2pF densities and the fitted Gaussian densities is satisfactory in the surface region. Since the differential cross section gets a contribution predominantly from the surface region of the colliding nuclei, the discrepancies between those profile functions at the central regions can be neglected.

Table 1 Gaussian density parameters a_P/T and ρ_P/T (0) entering phase shift of Eq. (3). The a_P and ρ_P (0) are obtained from Eq. (10) using the RrmsP. The aT and ρ_T (0) are calculated from Eqs. (15) and (16) given by profile function matching method. RrmsP is taken from Ref. [10] and 2pF density parameters c_T and d_T are taken from Ref. [11].

System	RrmsP (fm)	aP (fm)	ρP (0) (fm-3)	cT (fm)	dT (fm)	RrmsT (fm)	aT (fm)	ρT (0) (fm-3)
α + 116Sn	1.71	1.396	0.06598	5.275	0.539	4.551	2.856	0.02362
α + 197Au	1.71	1.396	0.06598	6.38	0.535	5.327	3.071	0.03439

Figure 1. (Color online) Profile functions for target nuclei (¹¹⁶Sn and ¹⁹⁷Au) plotted versus impact parameter b. The solid and dotted curves represent the profile functions for 2pF and SMGD distributions, respectively. The densities are assumed to be normalized to the mass number of the nucleus.

3. Input parameters

The Gaussian density parameters a_i and ρ_i(0) are determined as follows : (1) The a_P and ρ_P (0) were calculated using Eq. (10) with RMS radius [10]. (2) The calculations of a_T and ρ_T (0) are performed in two ways : one is calculated from the RMS radii [11] using Eq.(10) and the other is from the profile function matching method using Eqs. (15)-(16) where 2pF density parameters c_T and d_T are taken from Ref. [11]. Parameters of the Gaussian and the 2pF densities are collected in Table 1.

The parameter values of α_NN and β_NN related with the NN scattering amplitude are determined as follows : (1) The α_NN values were determined by minimized χ²/N-value obtained from a comparison of the experimental data with the calculations. (2) The slope parameter β_NN was taken to be an average value of β_pp(nn) and β_np(pn) given in Table 6 of Ref. [12]. The remaining input σ_NN in the nuclear phase shift is calculated by the expression

where the values of σ_pp(nn) and σ_np are obtained from a phenomenological formula [13] with ρ = 0.16 fm^-3 [12]. The values of σ_NN, β_NN and α_NN are listed in Table 2.

Table 2 Input parameters (σ_NN, β_NN and α_NN), and physical quantities obtained by phase shift analysis. Cal. 1 and Cal. 2 denote the results using the a_T and ρ_T (0) calculated from the RrmsT (Eq. (10))and the profile function matching method (Eqs. (15) and (16)), respectively. The values in parentheses are the results obtained by the α_NN corresponding to the lowest χ²/N value. 10% error bars are adopted to obtain χ²/N values. All quantities are defined in the text.

	Cal. 1		Cal. 2
Target	116Sn	197Au	116Sn	197Au
σNN(mb)	67.32	67.32	67.32	67.32
βNN(fm2)	1.1433	1.1433	1.1433	1.1433
αNN	1.008 (1.248)	0.987 (0.428)	1.008	0.987
θcross(deg)			12.10	16.40
L1/2	55.64	65.18	53.32	60.70
Rs(fm)	8.885	10.383	8.530	9.709
σRs (mb)	2480	3387	2286	2961
σR(mb)	2424	3220	2200	2754
χ2/N	21.82 (17.28)	29.01 (17.36)	3.48	4.69

This work was supported by the 2021 education, research and student guidance grant funded by Jeju National university.