PRAMANA © Printed in India Vol. 48, No. 5,
__ journal of May 1997
physics pp. 1021-1026
Heavy-ion scattering at low energies: First-order correction to the eikonal phase shift
S K CHARAGI
Nuclear Physics Division, Bhabha Atomic Research Centre, Mumbai 400 085, India MS received 14 November 1996.
Abstract. The first-order non-eikonal correction has been applied to calculate heavy-ion reaction cross sections at low energies. The numerical investigations show that, for many heavy-ion systems, reaction cross sections and elastic scattering angular distributions thus calculated, are in good agreement with the optical model calculations.
Keywords. Heavy-ion scattering; eikonal approximation; Glauber model; heavy-ion reaction cross section.
PACS Nos 25.70; 24.10 1. Introduction
The heavy-ion elastic scattering and reaction cross sections, have been studied in the framework of a microscopic Glauber theory, based on the individual nucleon- nucleon collision in the overlap volume of the colliding nuclei, in the high energy domain [1-6]. In these studies nucleus-nucleus reaction cross section is given as a R = 21tSbdb[1 -- D(b)], where f~(b) = e x p ( - a,nSpp(b,z)pt(b,z)dz), is the overlap inte- gral of the nuclear densities, a.. is the nucleon-nucleon cross section, pp and p, are the projectile and target nucleus densities and b is the impact parameter.
This model has been extended to low energies by taking into account the deviation due to the effect of the Coulomb field, in the straight line trajectory of the colliding nuclei [7-11]. This is achieved by calculating f~(b) at the distance of closest approach, b' as f~(b'). This approach is called the Coulomb-modified Glauber model(CMGM).
A simple, closed-form analytic expression for the heavy-ion reaction cross section, involving nuclear densities of the colliding nuclei and the nucleon-nucleon cross section, has been obtained within the framework of this model [10, 11]. The reaction cross section and the elastic scattering angular distribution of a large number of heavy-ion systems, over a wide energy range, has been obtained [9-11]. Thus, C M G M provides a unified and easy-to-apply microscopic theoretical framework for analyzing heavy-ion reaction cross section and elastic scattering data.
Earlier Waxman et al [12] have shown that the inclusion of the first-order non- eikonal correction in the Glauber model approach considerably improves the agree- ment with data for elastic scattering of 800 MeV protons by nuclei. Recently, Faldt et al [ 13] carried out the numerical investigation of higher order non-eikonal corrections to the Glauber model at intermediate energies. They applied these corrections to elastic
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nuclear scattering of 178 and 800 MeV protons, 180 MeV antiprotons, and 140 MeV ct particles. They calculate the eikonal phase shift function, z(b) in terms of the potential,
V(b,z)
as x(b) = - # / ~V(b,z)dz.
Further, quite recently Cha and Kim [14] applied these corrections for the analysis of the scattering data of 160 + 4°Ca and 160 + 9°Zr at 1503 MeV. In this paper we extend the scope of these corrections to a larger number of heavy-ion systems at low energies by calculating the phase shift function at b' as )~ (b').This is the Coulomb-modified phase shift function [ 14]. This accounts for the deviation in the straight line trajectory due to the Coulomb field.
The plan of the paper is as follows. In § 2 we describe the formalism used in the calculation of the heavy-ion reaction cross section. In § 3 we summarize the results and also draw a few conclusions.
2. F o r m a l i s m
The nucleus-nucleus reaction cross section fiR, is given by
/Z
OR =~-~ ~(21 + 1)(1 --]S/IZ), (1)
where k is the wave number and S z the scattering matrix element given by
Sz = e 2'~1. (2)
Also
tSll 2 = e 4imt~i~, (3)
where Im(61) is the imaginary part of the nuclear phase shift. Wallace [15, 16] has given the following expression for the nuclear phase shift
k[p/(hk)Z]"+1[
[b/nb2 l+bdbJ_ld']]nI~CVn+l[(b2+zE)l/e]dz''
J0 (4) We have considered only first two terms of the nuclear phase shift function i.e. 6 ° and 6], the first order correction to the eikonal phase shifts and evaluated them at b', the distance of the closest approach to account for the deflection effect of the Coulomb field. We write~o _ ~ VN(b',z) dz (5)
h2k
and
where
6~- 2h4k 3
l + b ' d - f fVZ(b',z)dz,
VN(b',z )
= VN[(b '2 + z2)1/2].The distance of closest approach b' is obtained from
kb' =- rl +
[q2 +kZb2]l/2,
(6)
(7)
(8)
H e a v y - i o n scattering at low energies
where q, is the Sommerfeld parameter. The real part of the nuclear potential is written as Re[VN(r)] = - Vo/[1 + e E . . . . (Alia +A~'3)l/av], (9) and the imaginary part of the nuclear potential is
Im[VN(r)] = -- W0/[1 + e E . . . . IAll'3+A~"3)1/a']. (10) The nucleus-nucleus differential cross section da/df~ is deduced from the complex phase shift through
da
d---~ = If(O)lZ' (11)
for nonidentical spinless nuclei, and the scattering amplitude for charged particles is given by
1
f ( O ) =f~(0) +2-~ ~ (2l + 1)e2'~'(e 2'~t - 1)P,(cos0), (12)
where the Coulomb phase shift a~ and the Coulomb amplitudefc(0) are calculated in standard form.
3. Results and conclusions
In the present study, reaction cross sections of heavy-ion systems have been calculated in the framework of the formalism given above. Column 1 of table 1 lists all the heavy-ion systems considered in the present study. The optical potential parameters and the optical model reaction cross section of these systems are well established.
Therefore, these systems are excellent candidates of this study. The optical model parameters are listed in columns 3-6 of table 1. Column 2 gives the energies of these systems and column 7 gives the reference. Here, we have used av = a w and r v = r w.
We have not considered the heavy-ion reaction below 8 MeV. It has been shown [17], that at very low energies, i.e. below 9 MeV/nucleon, even the small values of
Table 1. Optical potential parameters of the heavy-ion systems listed in the table.
Heavy-ion Energy V o W o av rv
system MeV MeV MeV fm fm Ref.
t2C + 28Si 131"5 18'78 52-4 0"9 1'073 [18]
160 "~- 28Si 215'2 24-53 79"0 0"9 1"03 [18]
160 + 4°Ca 2 1 4 . 1 21.23 13.8 1"3 0.54 [183 160 + 2°Spb 192.0 40.00 35"0 1.226 0"634 [19]
312"6 19"32 11-4 1-3 0"6 [18]
a60 + 2°9Bi 164 17.68 46.38 0'58 1-22 [20]
170 53-73 42"48 0-58 1.22 [20]
12C + 2°8pb 96 40-0 25"0 0"56 1-256 [19]
116'4 41-1 26-7t 0-7 1-2 [19]
160 + 2°8pb 129'5 40"0 35"0 0"615 1"249 [19]
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Table 2. This table gives the calculated values of the heavy-ion reaction cross sections. Cal I is the eikonal model result with only 6°(b ') part of the phase shift. Cal II is the result obtained using first two terms ~o (b') and 61(b ') of the phase shift. Cal III includes two terms 3°(b) and/51(b) in the phase shift.
(7 R O" R O" R (7 R
Heavy-ion Energy mb mb mb mb
system MeV Cal I Cal II Optical Cal III
a2C + 28Si 131.5 2236 2267 2273 2645
16 0 + 28 Si 215-2 2463 2487 2484 2839
160 + 4°Ca 214.1 2026 2116 2105 2545
160 -{- 2°8pb 192'0 2810 2894 2930 5013
312"6 3327 3429 3452 4697
x60 + 2°9Bi 164 2405 2442 2486 3065
170 2454 2543 2591 3025
12C + 2°8pb 96 1609 1727 1806 4761
116"4 2159 2276 - - 4787
160 "~ 2°8pb 129"5 1944 2030 2095 5230
l contribute to the scattering matrix in a non-trivial manner and [Sl[ 2 shows oscilla- tions. In such a situation there is a risk of underpredicting the reaction cross sections.
The earlier studies of the non-eikonal corrections [13, 14] have shown that the effect of the second order correction is negligible. Therefore, we have not considered the second order term in our calculations. Cha and Kim have considered the effect of the higher order correction in the case of two heavy-ion systems (160 + 4°Ca and 160 h- 9°Zr) at higher energies (1503 MeV). We extended the scope of the first-order non-eikonal corrections to other heavy-ion systems at low energies.
Table 2 gives the comparison of our calculation of a R, the heavy-ion reaction cross section with the optical model results. These optical model results have been taken from the corresponding references given in the previous table. The calculation labeled as Cal I given in column 3 is the o R calculated by us including only 6 ° part of the phase shift, evaluated at b'. Cal II on the other hand is the o~ calculation including both the terms of the phase shift (6 ° and ~1), evaluated at b'. There is a few per cent enhancement in the prediction of aR by including the first-order term in the phase shift. The agreement between the optical model calculation and the eikonal model with the first-order non-eikonal corrections (Cal II) is good. Also for comparision, we have given in the last column of the table, the prediction of aR, using 6 ° and 6 ~, evaluated at impact parameter b (Cal III). Cal III fails to reproduce the cross sections.
The fit to elastic scattering angular distribution is given in figure 1 for the heavy-ion system ~60 + 2o8 Pb at three different energies. In this figure the solid line is the optical model fit [18] and the dotted line is the fit using Cal II results. Except for a few forward angles, the two fits are identical. Also shown in figure 1 as dot-dash line is the plot using Cal III results. Clearly the ineffectiveness of eikonal approximation without Coulomb correction is demonstrated.
Figure 2 gives similar fits for the heavy-ion system 12C + 2o8 Pb. In this figure, the dotted line is a plot ofCal I result and the solid line is the plot of Cal II results. There is a consistent improvement in the fits by including first order term of the phase shift.
Heavy-ion scattering at low energies
1.6l, I ,&,T, , I, ~,: I I I I ,
1 41- 160-208pb
-f; A
6~-~ok--,~l/ ~ a , , ' \ ~ ~ -
,'~0.Sk312.e~leV
~ 192NeV \ "129.41'4eV
Cn,-~ o ~ 1 - ~ ~, t ~, ... -
o.41--~ ~,co, Iit4 \ -
0.2 \, -
ok, ,
0 5 10 15 20 2520 25 30 35 40 Z,O 50 60 70 ecru (deg)
Figure 1. Elastic angular distribution for the scattering of 160 o n 2°spb at different energies. The solid line is the optical model fit [18]. The dotted line is the plot of Cal II results- Also shown as dot-dash line is Cal III.
1./4 1.2
1.0 0.8 b 0.6
" o
0.4
0.2 0
I I
- 116./4
MeVf %1 t I i J
~ 12 C+ 208pb _
t "- Cal I
40 50 ,\
Col
III I ~ I i
0 20 30 30 40 50 80
ec.m.(deg )
6 0 70
Figure 2. Elastic angular distribution for the scattering of 12C o n 2o8pb at two different energies. The dotted line in Cal I plot. The solid line is Cal II plot.
We conclude that a good description of the heavy-ion elastic scattering and reaction cross section at low energies, can be given by calculating the first two terms of the phase shift at the distance of closest approach. This is a simple and economical procedure as it requires little numerical effort. The predictions of the aR and the elastic scattering angular distribution in the present formalism are as good as the optical model predictions. This good agreement is also an independent verification of the goodness of the optical potential parameters. Since the eikonal calculation with first-order non- eikonal correction is equivalent to the zeroth-order calculation with effective potential given by Ue,(r) = V (r) + 1/(2kv)[2 + r(d/dr)] V(r) 2 where V (r)is the sum of optical and
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Coulomb potentials, we can obtain a new set of effective potentials for these heavy-ion systems at low energies in the framework of the eikonal approximation.
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