The parametrized phase shift model (PPSM)[1-3] is known to be convenient tool for the analysis of measured data of heavy-ion elastic scattering. The McIntyre PPSM[1] has been used[4-12] for a long time to interpret the elastic scattering data. In this model, the nuclear scattering matrix element (S_L) is analytically expressed in terms of angular momentum L, and its parameters are determined by the χ2/N−calculation. The early work of Cha and Kim[6] and Mermaz et al.,[4, 5] reported a phase shift analysis of ¹²C + ¹²C elastic scattering using the McIntyre PPSM. The experimental data of the ⁶Li elastic scattering are analyzed[11] using both the Frahn-Venter PPSM and McIntyre PPSM. A few years ago, we performed[12] a systematic analysis of 240 MeV α−particle elastic scattering on various targets from ¹⁶O to ¹⁹⁷Au using the McIntyre model.

The elastic scatterings between alpha particle and various target nuclei have been extensively studied[13-16]. Li et al.[15] measured the angular distributions for α−particle beam elastically scattered from tin isotopes (¹¹²Sn, ¹²⁰Sn and ¹²⁴Sn) at 386 MeV, and analyzed this elastic data using the optical model. It is interesting to utilize the McIntyre PPSM for the description of α + ^112,120,124Sn elastic scattering at 386 MeV and for the investigation of the trend of physical quantities such as strong absorption radius, nuclear rainbow angle and reaction cross section as the target mass number increases. We name McIntyre parametrized phase shift model simply as a “McIntyre model” in this paper.

This work presents the numerical analysis of the experimental data of 386 MeV alpha particle elastic scattering on three tin isotopes using the McIntyre model. The best input parameter values are determined from the minimized χ2/N−condition and its values are used to analyze the α−particle elastic scattering. The near- and the far-side decompositions of scattering amplitude are also performed to qualitatively understand the features of angular distributions that oscillates strongly at small scattering angle regions and oscillates weakly while decreasing at relatively large angle regions. Another approach using the average of best input parameter values is made to examine whether the average inputs can reproduce the α + ^112,120,124Sn elastic data because three tin isotopes have very similar mass. Strong absorption radius, reaction cross section and nuclear rainbow are studied. We also investigate the effect of reduced radius on differential and reaction cross sections.

The elastic scattering amplitude f(θ) of a spin-zero alpha particle incident on a spin-zero target nucleus can be expressed in the form:

where first and second terms are the Rutherford and nuclear scattering amplitudes, respectively. In this equation, k the wave number, σL the Coulomb phase shift, PL(cosθ) the Legendre polynomial and S_L the nuclear scattering matrix element related with the nuclear phase shift δL with a complex form (δL=δLR+iδLI) through the relation :

where |SL| (modulus of S_L) is obtained by the imaginary phase shift δLI from the relation |SL|=exp[−2δLI] (which is real). The elastic differential cross section is the absolute square of the scattering amplitude, f(θ), given by

In the McIntyre model[1, 4, 5], the modulus of S_L is

and the real nuclear phase shift is

The grazing angular momentum L_g and its width Δ are related to the reduced radius r_1/2 and the diffusivity d through the two following semiclassical formula[4] :

and

where R1/2=r1/2(AP1/3+AT1/3) and η the Sommerfeld parameter. Similarly, the phase grazing angular momentum Lg′ and its width Δ′ in Eq. (5) are given in the same form as L_g and Δ except that r_1/2 and d are replaced by the phase radius r_ph and the phase diffusivity d_ph.

The deflection function is defined as twice the angular momentum derivative of the Coulomb phase shift (σL) plus the real nuclear phase shift (δLR) given by the analytical formula[4] :

The function θL is defined as positive for net repulsion and as negative for net attraction. When the deflection function displays a deep minimum in the region of negative values, the angle at which this occurs is known as the nuclear rainbow angle.

1. Angular distributions using best input parameter values

The McIntyre model is used to describe the differential cross section of 386 MeV alpha particle elastic scattering on three tin isotopes. Each of five input parameter values (r1/2, d, rph, d_ph and μ) is carefully changed in small steps, and the best fit to elastic data is reached when χ2/N−value is minimized. In this study, the χ2/N−values are obtained under the assumption that all experimental data have a 10% error. The determined best input values for three scattering system are listed in Table 1 along with the χ2/N−values. We can see that the McIntyre model provides a very reasonable χ2/N−value for each scattering system. The experimental data (solid circles)[15] and calculated results (solid curves) for angular distributions are presented in Fig. 1. As shown in this figure, experimental angular distribution of each α + ^112,120,124Sn system has two different structures: one is strong oscillations at small angle regions, and the other is decreasing patterns with weak oscillations at relatively large angle regions. It is seen that present results are in good agreement with the experimental measurement[15]. In particular, the solid curves satisfactorily reproduce both the number and the structure of the oscillations present in the experimental data.

Figure 1. (Color online) Comparisons between the experimental data (solid circles) and McIntyre model predictions (solid curves) using the best input parameter values given in Table 1 for the elastic scattering angular distributions of α + ^112,120,124Sn systems at 386 MeV. The experimental data are taken from Ref. 15.

Table 1 . Best input parameter values and analysis results of the McIntyre model for 386 MeV alpha particle elastic scattering on three tin isotopes (¹¹²Sn, ¹²⁰Sn and ¹²⁴Sn). The χ2/N−values were obtained by assuming that all experimental data have a 10% error.

Target	112Sn	120Sn	124Sn
r1/2 (fm)	1.154	1.159	1.163
d (fm)	0.691	0.664	0.685
μ	4.037	4.098	4.115
rph (fm)	0.892	0.895	0.901
dph (fm)	0.951	0.926	0.920
Lg	59.7387	61.2281	62.0359
Δ	5.7364	5.5248	5.7054
Lg′	45.8004	46.9051	47.6885
Δ′	7.8968	7.7067	7.6646
L1/2	64.7964	66.0976	67.0646
Rs (fm)	8.06352	8.20159	8.30917
θcross (deg)	5.55	5.70	5.65
θn.r. (deg)	-10.636	-11.318	-11.530
σRs (mb)	2043	2113	2169
σR (mb)	2020	2086	2144
χ2/N	2.71	2.59	2.33

The decomposition of elastic scattering amplitude into near-side and far-side (N/F) components is known to be useful tool for understanding qualitatively the characteristic behavior of the differential cross sections. The N/F cross sections following the Fuller formalism[17] were calculated within the framework of McIntyre model using the best input parameter values given in Table 1. In Fig. 2, the solid curves are the differential cross sections drawn in Fig. 1 while the dotted and dashed curves are the N/F cross sections for α + ^112,120,124Sn systems. A near-side contribution (dotted curves) dominates over the far-side one (dashed curves) at angle regions less than the θcross, where θcross (given in Table 1) is the crossing angle at which the N/F cross sections have equal magnitude. The oscillatory structures appearing near θcross regions in the angular distributions (solid curves) of three scattering systems are resulted from the strong interference between the N/F components of the scattering amplitude. As shown in Fig. 2, it can be seen that the maximum amplitude in oscillation appears near θcross. On the other hand, the far-side component of each system provides the major contribution to the elastic angular distribution for the angle regions larger than θcross, where the near-side component has small magnitude.

Figure 2. (Color online) Differential cross sections (solid curves) calculated with the McIntyre model using the best input parameter values given in Table 1, its far- (dashed curves) and near-side (dotted curves) components following the Fuller's formalism [17] for the elastic scattering of α−particle from ¹¹²Sn, ¹²⁰Sn and ¹²⁴Sn at 386 MeV.

2. Angular distributions using average input parameter values of best fits

In this subsection, we use the average input parameter values (hereinafter named as ``AIPV") to examine whether AIPV can describe the α−particle elastic scattering on three tin isotopes. The AIPV is obtained by taking average of best input parameter values of three scattering systems given Table 1 and its values are

The calculated results using AIPV are shown in Fig. 3 as dotted curves, together with the best fits (solid curves). As expected, AIPV provides successful agreement (χ2/N=2.63) with the experimental data of ¹²⁰Sn target because AIPV has similar values to the ones given in Table 1. In the case of α + ^112,124Sn systems, the calculations (dotted curves) provided acceptable agreements (χ2/N−% value is 4.18 for ¹¹²Sn and 3.40 for ¹²⁴Sn) with the experimental data[15], although they displayed somewhat higher (lower) values for ¹¹²Sn (¹²⁴Sn) target at θc.m.> 15⁰. In particular, the calculations showed that the structures shown in experimental angular distribution of the α + ^112,124Sn systems were satisfactorily reproduced.

Figure 3. (Color online) Angular distributions of the elastic scattering of α−particle from ¹¹²Sn, ¹²⁰Sn and ¹²⁴Sn at 386 MeV. The solid curves are the calculated results obtained from the McIntyre model using the best input parameter values given in Table 1 while the dotted curves are the results using AIPV. The dashed curves are the results from McIntyre model with μ=4.003, 4.082 and 4.165 for ¹¹²Sn, ¹²⁰Sn and ¹²⁴Sn targets, respectively, where other input values except for μ in AIPV are fixed. AIPV is average input parameter values of best fits defined in the text. The experimental data are taken from Ref. 15.

In order to improve the quality of fit, we calculated the elastic angular distributions of each scattering systems by varying only the parameter μ related with the strength of δLR , where other input parameters (r1/2, rph, d, and d_ph) in AIPV are fixed. The determined μ−values (corresponding χ2/N−value) from χ2/N−calculation are the μ=4.003 (χ2/N=2.85), 4.082 (2.63) and 4.165 (2.44) for ¹¹²Sn, ¹²⁰Sn and ¹²⁴Sn targets, respectively. The calculated results are shown as dashed curves of Fig. 3 and comparable to the best fit results (solid curves). In particular, the dashed curves provided successful fits to the experimental angular distributions, especially at large angles of α + ^112,124Sn systems. We can see that for the angle regions exceeding about 15⁰, the elastic differential cross section curve moves up (down) as the μ−value increases (decreases).

3. Transmission and deflection Functions

The transmission functions T_L (TL=1−|SL|2) using the best input values are plotted versus the orbital momentum quantum number L, along with the deflection functions. As Fig. 4(a) shows, the T_L functions are rapidly varying from unity to zero over the transition regions centered on the critical angular momenta L_1/2 corresponding to TL=1/2. The rapid transition regions of T_L are slightly shifted to the right as the target mass number (A_T) increases. This movement of T_L function gives a slightly larger L_1/2 value related with the strong absorption radius R_s obtained from the kRs=η+η2+L1/2(L1/2+1) expression. The L_1/2 and R_s values extracted from the phase shift analysis using the best input values are collected in Table 1. The R_s has an effect on the geometrical reaction cross section σRs=πRs2, and the obtained σRs values are listed in Table 1. On the other hand, σR in this table is the reaction cross section obtained from the partial wave sum given by the formula: σR=πk2∑ L=0∞(2L+1)TL. As Table 1 shows, the σRs value is comparable to the value of σR for each system. It can be seen that the R_s value allows us to predict the approximate magnitude of the σR. In addition, both the σR and σRs values show an increasing trend as A_T increases.

Figure 4. (Color online) (a) Transmission functions and (b) deflection functions obtained from the McIntyre model using the best input parameter values given in Table 1 for the elastic scattering of α−particles from ¹¹²Sn, ¹²⁰Sn and ¹²⁴Sn at 386 MeV plotted versus the orbital angular momentum quantum number L.

Figure 4(b) displays the behavior of the deflection functions θL(deg) as a function of the orbital angular momentum quantum number L for α + ^112,120,124Sn system. It is known that the nuclear rainbow angle θn.r. occurs at the negative maximum of the θL(deg) (so to speak, deep minimum on θL(deg) graph) in consequence of a nuclear attraction. As Fig. 4(b) shows, each deflection function displays apparently a negative maximum, which is typical for the nuclear rainbow. Such negative angles tell evidently us the existence of the nuclear rainbow for each scattering system. As shown in Table 1 and Fig. 4(b), the absolute magnitude of θn.r. increases slightly as the mass number of tin isotopes increases.

4. Reduced radius effect on differential and reaction cross sections

In this subsection, the effect of reduced radius r_1/2 on differential and reaction cross sections of α + ¹²⁰Sn system is investigated. As described in Eqs. (6) and (7), the r_1/2 is an important factor in determining the grazing angular momentum L_g and its width Δ included in |SL| expressed by Eq. (4). In Fig. 5, we present a study of the r_1/2 effect on the angular distributions when other four input parameter values (d, rph, d_ph and μ) given in Table 1 are fixed. The solid curve denotes the calculated elastic cross section (best fit) with r1/2=1.159 fm, while the other four curves are the calculated results with r1/2=0.159±0.01 fm and r1/2=0.159±0.02 fm. We can see that for r1/2>1.159 fm, the calculated curves are moved downward with an increase in r_1/2 relative to the solid one, and for r1/2<1.159 fm the calculated curves are moved upward as the r_1/2 decreases, though the five curves display reasonable agreement with each other in the small angle regions. Also, the r_1/2 affects the reaction cross section because r_1/2 is related with the |SL| important for σR calculation. The calculated σR values are 2020 mb for r1/2=1.139 fm, 2053 mb for r1/2=1.149 fm, 2086 mb for r1/2=1.159 fm, 2119 mb for r1/2=1.169 fm and 2152 mb for r1/2=1.179 fm, respectively. This fact tells us that the σR value increases as the r_1/2 value increases. Calculations of α + ^112,124Sn (not shown) give similar results.

Figure 5. (Color online) Elastic scattering angular distributions for α + ¹²⁰Sn system at 386 MeV calculated using different reduced radius (r_1/2) values. The other input parameter values except for r_1/2 are fixed and its values are given in Table 1.

In this study, an analysis of α−particle elastic scattering on three tin isotopes (¹¹²Sn, ¹²⁰Sn and ¹²⁴Sn) at 386 MeV is carried out using the McIntyre model. The calculated results reproduced satisfactorily the pattern of experimental angular distribution that oscillate strongly at small angles and decrease with weak oscillations at relatively large angles, and showed good agreement with the corresponding measured cross sections. The oscillatory patterns of elastic cross section observed near the crossing angle are thought to be related with the strong interference between far- and near-side scattering amplitudes. The behaviors of differential cross section at relatively large angle regions are mainly determined by the far-side part of the scattering amplitude.

Another approach using the average of best input parameter values has been made to examine whether this input values can describe the experimental data of α + ^112,120,124Sn systems. The results reproduced fairly well the pattern of the angular distributions and provided acceptable fits to the elastic data, although the results displayed somewhat higher (lower) values for ¹¹²Sn (¹²⁴Sn) target at scattering angles beyond about 15⁰. In order to improve the quality of fit, we calculated the elastic cross sections by varying only the parameter μ related with the strength of real nuclear phase shift δLR, where other input parameter values in AIPV are fixed. The calculated results removed considerably the discrepancies at large angle and reproduced successfully the experimental data.

The rapidly decreasing region of the transmission function is moved toward slightly larger L values as the A_T increases. This movement produced slightly larger the critical angular momentum L_1/2 and the strong absorption radius R_s. The geometrically obtained reaction cross section (σRs=πRs2) for each α + ^112,120,124Sn system has a magnitude similar to that of the reaction cross section using partial wave sum, indicating that the magnitude of σR may be estimated by R_s. The existence of a nuclear rainbow in each scattering system was confirmed by the fact that the deflection function has a negative maximum. The magnitudes of reaction cross sections and nuclear rainbow angles are slightly increasing with the increase in A_T. An increase (decrease) in reduced radius value r_1/2 moved the calculated differential cross section curve downward (upward) and provided a larger (smaller) reaction cross section when other four input parameters (d, rph, d_ph and μ) are fixed.

This research was supported by the 2023 scientific promotion program funded by Jeju National University.