npsm 새물리 New Physics : Sae Mulli

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New Phys.: Sae Mulli 2023; 73: 934-940

Published online November 30, 2023 https://doi.org/10.3938/NPSM.73.934

Copyright © New Physics: Sae Mulli.

Phase Shift Analysis of 386 MeV Alpha Particle Elastic Scattering on Tin Isotopes

Yong Joo Kim*

Department of Physics, Jeju National University, Jeju 63243, Korea

Correspondence to:*yjkim@jejunu.ac.kr

Received: September 20, 2023; Accepted: October 20, 2023

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License(http://creativecommons.org/licenses/by-nc/3.0) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

A phase shift analysis of the elastic scattering angular distributions for α + 112,120,124Sn systems at 386 MeV is performed using the McIntyre parametrized phase shift model. The calculated results not only reproduced well the structure of measured angular distribution of three scattering systems, but also showed very good agreements with the experimental data. Average of input parameter values extracted from the best fit to each α + 112,120,124Sn elastic data also provided a fairly well reproduction of experimental results. The oscillatory structure of angular distributions observed near the crossing angle is explained in terms of the strong interference between the near-side and the far-side scattering amplitudes. The behaviors of elastic cross section in the region of relatively large angles are mainly governed by the far-side contribution. As the target mass number increases, the magnitude of strong absorption radius, reaction cross section and nuclear rainbow angle tended to increase slightly.

Keywords: Phase shift analysis, McIntyre parametrized phase shift model, Elastic scattering, &alpha, + 112,120,124Sn

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 (SL) is analytically expressed in terms of angular momentum L, and its parameters are determined by the χ2/Ncalculation. 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 αparticle 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 αparticle 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/Ncondition and its values are used to analyze the αparticle 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:

f(θ)=fR(θ)+1ik L=0(L+12%)exp(2iσL)(SL1)PL(cosθ),

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 SL the nuclear scattering matrix element related with the nuclear phase shift δL with a complex form (δL=δLR+iδLI) through the relation :

SL=exp[2iδL]=|SL|exp[2iδLR],

where |SL| (modulus of SL) 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

dσdΩ=|f(θ)|2.

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

|SL|=1+exp(LgLΔ)1,

and the real nuclear phase shift is

δLR=μ1+exp(LLgΔ%)1.

The grazing angular momentum Lg and its width Δ are related to the reduced radius r1/2 and the diffusivity d through the two following semiclassical formula[4] :

Lg=kR1/2(12ηkR1/2 )1/2,

and

Δ=kd(1ηkR1/2)(12ηkR1/2)1/2,

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 Lg and Δ except that r1/2 and d are replaced by the phase radius rph and the phase diffusivity dph.

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] :

θL=2ddL(σL+δLR)=2tan1(%ηL)+2μddL1+exp(LLgΔ )1.

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, dph and μ) is carefully changed in small steps, and the best fit to elastic data is reached when χ2/Nvalue is minimized. In this study, the χ2/Nvalues 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/Nvalues. We can see that the McIntyre model provides a very reasonable χ2/Nvalue 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/Nvalues were obtained by assuming that all experimental data have a 10% error.

Target112Sn120Sn124Sn
r1/2 (fm)1.1541.1591.163
d (fm)0.6910.6640.685
μ4.0374.0984.115
rph (fm)0.8920.8950.901
dph (fm)0.9510.9260.920
Lg59.738761.228162.0359
Δ5.73645.52485.7054
Lg45.800446.905147.6885
Δ′7.89687.70677.6646
L1/264.796466.097667.0646
Rs (fm)8.063528.201598.30917
θcross (deg)5.555.705.65
θn.r. (deg)-10.636-11.318-11.530
σRs (mb)204321132169
σR (mb)202020862144
χ2/N2.712.592.33


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 αparticle 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 αparticle 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

AIPV:r1/2=1.159 fm, d=0.680 fm, %rph=0.896 fm,dph=0.932 fm and μ=4.083

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 αparticle 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 dph) in AIPV are fixed. The determined μvalues (corresponding χ2/Nvalue) from χ2/Ncalculation 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 μvalue increases (decreases).

3. Transmission and deflection Functions

The transmission functions TL (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 TL functions are rapidly varying from unity to zero over the transition regions centered on the critical angular momenta L1/2 corresponding to TL=1/2. The rapid transition regions of TL are slightly shifted to the right as the target mass number (AT) increases. This movement of TL function gives a slightly larger L1/2 value related with the strong absorption radius Rs obtained from the kRs=η+η2+L1/2(L1/2+1) expression. The L1/2 and Rs values extracted from the phase shift analysis using the best input values are collected in Table 1. The Rs 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 Rs value allows us to predict the approximate magnitude of the σR. In addition, both the σR and σRs values show an increasing trend as AT 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 αparticles 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 r1/2 on differential and reaction cross sections of α + 120Sn system is investigated. As described in Eqs. (6) and (7), the r1/2 is an important factor in determining the grazing angular momentum Lg and its width Δ included in |SL| expressed by Eq. (4). In Fig. 5, we present a study of the r1/2 effect on the angular distributions when other four input parameter values (d, rph, dph 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 r1/2 relative to the solid one, and for r1/2<1.159 fm the calculated curves are moved upward as the r1/2 decreases, though the five curves display reasonable agreement with each other in the small angle regions. Also, the r1/2 affects the reaction cross section because r1/2 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 r1/2 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 (r1/2) values. The other input parameter values except for r1/2 are fixed and its values are given in Table 1.

In this study, an analysis of αparticle 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 AT increases. This movement produced slightly larger the critical angular momentum L1/2 and the strong absorption radius Rs. 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 Rs. 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 AT. An increase (decrease) in reduced radius value r1/2 moved the calculated differential cross section curve downward (upward) and provided a larger (smaller) reaction cross section when other four input parameters (d, rph, dph and μ) are fixed.

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

  1. J. A. McIntyre, K. H. Wang and L. C. Becker, Phys. Rev. 117, 1337 (1960).
    CrossRef
  2. D. M. Brink, Semi-Classical Methods for Nucleus-Nucleus Scattering (Cambridge University Press, Cambridge, 1985).
  3. S. K. Charagi, et al., Phys. Rev. C 48, 1152 (1993).
    Pubmed CrossRef
  4. M. C. Mermaz, Z. Phys. A 321, 613 (1985).
    CrossRef
  5. M. C. Mermaz, B. Bonin, M. Buenerd and J. Y. Hostachy, Phys. Rev. C 34, 1988 (1986).
    Pubmed CrossRef
  6. M. H. Cha and Y. J. Kim, J. Phys. G: Nucl. Part. Phys. 16, 281 (1990).
    CrossRef
  7. H. M. Fayyad, T. H. Rihan and A. M. Awin, Phys. Rev. C 53, 2334 (1996).
    Pubmed CrossRef
  8. Y. J. Kim and M. H. Cha, Int. J. Mod. Phys. E 9, 299 (2000).
    CrossRef
  9. M. H. Cha and J. Korean, Phys. Soc. 49, 1389 (2006).
    CrossRef
  10. R. I. Badran, H. Badahdah, M. Arafah and R. Khalidi, Int. J. Mod. Phys. E 19, 2199 (2010).
    CrossRef
  11. R. I. Badran and D. Al-Masri, Can. J. Phys. 91, 355 (2013).
    CrossRef
  12. Y. J. Kim, New Phys.: Sae Mulli 67, 162 (2017).
    CrossRef
  13. M. El-Azab Farid, Z. M. M. Mahmoud and G. S. Hassan, Phys. Rev. C 64, 014310 (2001).
    CrossRef
  14. M. A. Alvi, J. H. Madani and A. M. Hakmi, Phys. Rev. C 75, 064609 (2007).
    CrossRef
  15. T. Li, et al., Phys. Rev. C 81, 034309 (2010).
    CrossRef
  16. Y. K. Gupta, et al., Phys. Lett. B 748, 343 (2015).
    CrossRef
  17. R. C. Fuller, Phys. Rev. C 12, 1561 (1975).
    CrossRef

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