npsm 새물리 New Physics : Sae Mulli

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Research Paper

New Phys.: Sae Mulli 2022; 72: 893-899

Published online December 31, 2022 https://doi.org/10.3938/NPSM.72.893

Copyright © New Physics: Sae Mulli.

Analysis of 6Li Elastic Scattering at 240 MeV : Eikonal Model Approach

Yong Joo Kim*

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

Correspondence to:*E-mail: yjkim@jejunu.ac.kr

Received: September 3, 2022; Accepted: October 24, 2022

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.

The differential cross sections of 240 MeV 6Li elastic scattering on 24Mg, 28Si, 40Ca, 58Ni, 90Zr, and 116Sn targets are systematically analyzed using the eikonal model based on the Coulomb trajectories of colliding nuclei. The model calculations successfully reproduced the structure of the elastic scattering cross sections, and provided fairly good description of the experimental measurements. The oscillatory patterns shown in the experimental elastic angular distributions are well explained in terms of the strong interference between the near- and far-side scattering amplitudes. The reaction cross sections obtained from the analysis of 6Li elastic scattering on various targets from 24Mg to 116Sn are found to have linear dependence on the (AP1/3+AT1/3)2. The partial reaction cross sections increased linearly up to some peak values and thereafter decreased rapidly to zero.

Keywords: Elastic scattering, Eikonal model, Differential cross section, Reaction cross section, 6Li + nucleus

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/Nmethod. 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/Nmethod (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 :

dσdΩ=|f(θ)|2.

The elastic scattering amplitude f(θ) has the expression

f(θ)=fR(θ)i2k L=0(2L+1)exp(2iσL)(SL1)PL(cosθ),

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 SL is the nuclear scattering matrix element obtained from the nuclear phase shift δL through the relation

SL=exp[2iδL].

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

fN(θ)=fRN(θ)i2k L=0(2L+1)exp(2iσL)(SL1)QL()(cosθ)

and

fF(θ)=fRF(θ)i2k L=0(2L+1)exp(2iσL)(SL1)QL(+)(cosθ),

where fRN and fRF 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

QL()(cosθ)=12PL(cosθ)±i2%πQL(cosθ).

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]

δL(rc)=m2k0U(%rc2+z2 )dz,

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

rc=1kη+η2+(L+12)2,

with η=mZ1Z2e2/(2k) being the Sommerfeld parameter. The potential U(r) used in this work is

U(r)=V0 fv(r)iW0 fw(r),

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

fr,w(r)=1+exp(rrv,w(AP1/3+AT1/3)av,w)%2.

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. AP and AT 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 SL 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/Nmethod, in which the χ2/Nvalue 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/Ncalculation. Determined values of the potential parameters (V0, rv, av, W0, rw, and aw) are presented in Table 1, together with the χ2/Nvalue. 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/Nvalues 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/Nvalues for the 6% Li + 24Mg, 28Si, 40Ca, 58Ni, 90Zr, and 116Sn elastic scatterings at 240 MeV. The χ2/Nvalues were obtained by considering uniform 10% errors for all analyzed data.}

TargetV0 (MeV)rv (fm)av(fm)W0 (MeV)rw (fm)aw (fm)χ2/N
24Mg1540.8961.39144.81.1401.6813.61
28Si1990.8301.55533.11.2281.5183.09
40Ca2220.8491.52139.71.2161.5233.77
58Ni2390.8851.50341.11.2421.4563.14
90Zr2570.9321.45541.81.2801.3673.63
116Sn2710.9371.52649.01.2591.3016.24


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.

Targetθcross
(deg)
θn.r.
(deg)
L1/2Rs
(fm)
Lpeakσpeak
(mb)
σRs
(mb)
σR
(mb)
24Mg8.127.846.57.213747.216331689
28Si7.5-30.348.27.293846.716681695
40Ca10.3-34.855.17.924449.519691971
58Ni13.3-39.762.08.595152.823182270
90Zr18.4-47.870.59.525958.328482718
116Sn23.0-49.174.19.946360.931052906


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]

θL=2ddL(σL+ReδL).

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 L1/2 is the critical angular momentum determined by the condition of |SL|2=1/2, where SL is the scattering matrix element expressed by Eq. (3). As this Table shows, the L1/2 value is increasing from 46.5 to 74.1 with the increase in target mass from 24Mg to 116Sn. The L1/2 value and its trend are also reflected in the strong absorption radius Rs obtained from the formula : Rs=η+η2+(L1/2+12)2%/k. The obtained Rs 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 :

σR=πk2 L=0(2L+1)(1|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 Rs 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 Rs 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 L1/2 and the strong absorption radius Rs 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 Rs 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.

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