PHYSICAL REVIEW C VOLUME 48, NUMBER 3 SEPTEMBER 1993

Elastic scattering of Si from Al at 70, 80, 90, and 100 Mev

S. K. Charagi, S. K. Gupta, M. G. Betigeri, C. V. Fernandes, and KuldeepNuclear Physics Division, Bhabha Atomic Research Centre, Bombay $00-085, India

(Received 12 May 1993)

Differential cross sections for the elastic scattering of Si from Al have been measured at 70,80, 90, and 100 MeV bombarding energies. The data have been analyzed in the framework of theoptical model, the parametrized phase shift model, the generalized Fresnel model, and the Coulomb-

modified Glauber model. The scattering matrix elements, determined from the analysis of the dataat each energy using these models, are in agreement with each other. The strong absorption radius

over the range of energies for the measurement is energy independent. The total reaction cross

section at these energies are deduced. The merits and demerits of various models in the context ofthe experimental data are discussed.

PACS number(s): 25.70.Bc, 25.70.—z, 24.10.Ht

I. INTRODUCTION

The study of elastic scattering is a basic ingredientto understand more complicated reaction processes inhadronic collisions. The analysis of the differential crosssection for elastic scattering provides a simple handle todetermine the interaction between nuclei. Phenomenlog-ically, this interaction is modeled in terms of the opti-cal potential with a Woods-Saxon geometry [1]. Othermodels, such as the parametrized phase shift model [2],have been successfully used to intrepret the elastic scat-tering data. A special case of this model is the strongabsorption model referred to as the generalized Fresnelmodel [3]. Recently the scope of the Glauber model [4] inthe interpretation of the elastic scattering data has beenextended to low energies [5]. The so called Coulomb-modified Glauber model has been shown to Gt elasticscattering angular distributions for several heavy-ion sys-tems at low energies.

The present work aims at a comprehensive study of thescattering process for the Si + Al system at diferentbombarding energies. We measured experimentally thedifferential cross section at diferent beam energies and.analyzed the data in the framework of the models men-tioned above. The scattering matrix obtained from thesecalculations is used to extract the total reaction crosssections for this system.

separated by 10 . The resolution of the detectors was32—40 keV for 5 MeV alpha particles from the Amsource.

The pulse height spectrum measured at 80 MeV forthe detector at 15 is shown in Fig. 1. The elasticallyscattered peaks from "Al, 0, and C are labeled inthe figure. The energy resolution obtained in this spec-trum is about 800 keV. The small peak riding on the tailof elastic peak due to Al probably corresponds to theinelastic excitations in aluminum at an excitation energyof around 1 MeV.

Due to the poor experimental resolution it is not pos-sible to distinguish between the elastically scattered sil-

~8S; ~ ~At

El b80MeV 2t

lab Al

4 0

10 Blab ="5

310—

II. EXPERIMENTAL PROCEDURE

12C 16

0

The measurement was carried out using Si beam(charge state 8+) at 70, 80, 90, and 100 MeV obtainedfrom the 14 UD pelletron at Bombay. The target was

located at the center of a 100 cm diameter scatteringchamber. Targets in the form of self-supporting foils ofaluminum were prepared by vacuum evaporation. Thethickness of the target, determined by the energy shift ofz4~Am alpha peaks was 45 pg/cm . The scattered parti-cles were detected with three 200 pm thick silicon surfacebarrier detectors each subtending a solid angle of 0.129msr. The detectors were placed at a distance of 28.4 cmfrom the target at three adjacent angles in the chamber

210—

10' I I

660 ?00 laO

CHAVNEL VP

FIG. 1. Typical pulse height spectrum of the heavy-ionsystem Si + Al at 80 MeV at a detector angle 0 = 15

0556-2813/93/48(3)/1152(6)/$06. 00 48 1152 1993 The American Physical Society

48 ELASTIC SCATTERING OF Si FROM Al AT 70, 80, . . . 1153

icon particles and the recoiling aluminum nuclei at thesame scattering angle. However, the cross section, lead-ing to the scattering of Al into the detector at forwardangles, is estimated to be orders of magnitude smallerthan the cross section for the scattering of Si into the de-tector at forward angles. Specifically the optical modelcalculation shows that at 80 MeV beam energy the crosssection for the scattering of Al at an angle of 30 (c.m. )is six orders lower than the scattering of Al at an angleof 150 (c.m. ). Thus in the extraction of the data thecontribution of Al scattered into the detector is ignored.The angular distribution is measured in steps of 1 . Itis also important to know the detector angles accurately.The angular offset is determined to be less than 0.1, byrepeating a few data points on either side of the beam.The measured data are shown in Fig. 2 as the ratio cr/OR,where o R is the Rutherford cross section. The ratio variesfrom 1 to 3 x 10 with typical errors of 3—5% upto anangle of 30' and 5—10% for angles beyond 30'.

III. MODELS FOB. THE DESCRIPTION QFELASTIC SCATTERING

The nucleus-nucleus differential cross section is de-scribed as

der/dO =( f(0)(

where the scattering amplitude f (0) for the nonidenticalspinless nuclei is given by

f(0) = fc(0) + ) (2l + 1) exp(2icri)

x (Si —l)Pi(cos 0). (2)

Here fc(0) is the Coulomb scattering amplitude, o'i isthe Coulomb phase shift, and k is the wave number. Thescattering matrix Si is given by Si = exp(2ibi), where hiis the nuclear phase shift. The total reaction cross sectionis given by

cr„= —„)(2t + 1)(1 —~Si

~). (3)

010

Ai( Si, Si)

AL

The calculation of the nuclear phase shift is describedbelow in various models such as the optical model, theparametrized phase shift model, its variant the general-ized Fresnel model, and the Coulomb-modified Glaubermodel.

A. Optical model

010 eV

The potential employed in the optical model analysisof this system is parametrized as

V(r) = Vco i(r) —VRfIt(r) —iVIfI(r). (4)

010

where Vc „~ is the Coulomb potential of uniform chargedsphere with radius B, = 1.3(A„+A~ ), Az and A& are1/3 1/3

the projectile and the target mass numbers, and VR andVI are the strength parameters of the real and imaginaryparts, respectively. The Woods-Saxon form factor fx(X = I, R) in the standard form is given by

1+ exp(5)

-110— in terms of the radius parameter Bx = rx (A„+At )1/3 1/3

and the diffuseness parameter a~. By numerically solv-ing the Schrodinger equation with the optical potential,the scattering matrix S~ is obtained.

B. Parametrized phase shift model

-310 i I i I i I I I I I I I I

50 100

~c m(d'g )

150

In the parametrized phase shift model (PPSM) thescattering matrix is described in terms of a few param-eters. We use the Frahn and Venter parametrization [2]for the S matrix

FIG. 2. Elastic scattering angular distribution at difFerentenergies for Si + Al system fitted by optical model (solidline) and Coulomb-modified Glauber model (dashed line).

Re S) ——

1+ exp

1154 CHARAGI, GUPTA, BETIGERI, FERNANDES, AND KULDEEP

and C. Generalized Fresnel model

1Im S) ——p—

1+exp(7)

where l~ is the grazing angular momentum and 4 theangular momentum distribution width.

When the imaginary S~ is zero, the generalized Fresnelmodel (GFM) expressions can be derived by making afew reasonable assumptions from the PPSM model de-scribed above. The advantage of GFM lies in its simplic-ity. The differential cross section in the GFM in termsof the Fresnel integrals C = C(~W~) and S = S(~W~) isgiven by [6]

a(e) 1+0.5[(0.5 —C) + (0.5 —S)']I"'+ (C+ S —1)E for e & eR,OR(e) 0.5[(0.5 —C) + (0.5 —S) ]I" for e ) eR,

where the grazing angle 0~ ——2arctan&

"0 5 and the19+0.5Z Z 2

Sommerfeld parameter g = ~&' . The function I'"' is

given by

where yo is given by

7r pl (0)p2 (0)al a 2

G + Q1 2(i7)

lrA(eR —e)[ ( R )j»nh[~~(eR —e)]

' (io)

lr(lg + 0.5)2 / 2E lr 4k~ ( lg + 0.5 3(lg + 0 5) )

+

(i2)

while the argument of the Fresnel integrals is given by

w = (".")"(e- e.).(lr sin eR)o s

The GFM expressions have their validity when theCoulomb interaction plays a dominant role. High-precision approximation formulas for Fresnel sine and co-sine integrals are given in [7]. A closed form expressionfor the reaction cross section in GFM is

Recently we have described [11] a method for the cal-culation of fNN(0). The N Kscatte-ring amplitude inthe case of nucleus-nucleus scattering will be differentin general from free scattering. However, as the inter-action determining elastic scattering takes place in thelow-density region of the nuclear overlap, the effectiveN-N scattering amplitude can be approximated by thefree scattering amplitude.

The forward N-N scattering amplitude is written interms of the effective N-N cross section o~~ and o.~~the ratio of real to the imaginary part of forward N-Nscattering amplitude as

krrNN (& + &NN)NN 0 4'

The reaction cross section in CMGM formalism is givenby [5]

D. Coulomb-modified Glauber model (CMGM) cr = lr 1 — ' '"'[a, + a2][lngo+ 0.5772].

In terms of the optical limit to the Glauber model [8—10] the nuclear phase shift can be obtained with the mod-ification for Coulomb interaction as given by

7r fNN(0)y(b', )k

where fNN(0) is the forward nucleon-nucleon scatteringamplitude and y(bI) is the overlap integral of nuclear den-sities along a straight line characterized by the distanceof closest approach 6& given as

The potential in the CMGM model is given as [9]

V(r) = —si2 0NN(ANN + 2)al+a2

( r'x pl (0)p2 (0)a 1a2 exp

I al+a. )(2o)

This potential should be compared with that of the op-tical model.

kb,' = ll + [rI2 + (l + 0.5)2]0 s.

For a Gaussian density distribution of the form

(14)

E. Comparison of the models

p, (r) = p, (0) exp( —r /a, ), (15)

x(bI) = ~«xpt —bI'/( l+ '. )1 (16)

where p, (0), a, are adjusted to reproduce the surface tex-ture of the realistic nuclear density. The overlap integralof the nuclear densities can be written as [8]

The models described here are having different num-ber of parameters. The optical model used by us hasfour parameters and it involves numerical computationof scattering matrix by solving the Schrodinger equation.The parametrized phase shift model has three parame-ters, while the generalized Fresnel model has only twoparameters. In principle the Coulomb-modified Glauber

48 ELASTIC SCATTERING OF 2 Si FROM Al AT 70, 80, . . . 1155

Beam energy(MeV)

708090100

VR

(MeV)

54.0359.4461.8570.02

~R +R Pl Ql

(fm) (fm) (MeV) (fm) (fm)

1.15 0.63 23.50 1.15 0.631.15 0.63 31.15 1.15 0.631.15 0.63 36.75 1.15 0.631.15 0.63 31.24 1.15 0.63

TABLE I. Parameters of the optical model potential forSi beam on Al target.

d = [l(0.9) —l(0.1)]/4.4. (23)

A. The optical model analysis

where l(0.9) is the l value for which ~S(l)~2 = 0.9 andl(0.1) is the l value for which ~S(l)

~

= 0.1. The measureddata on o/oR shown in Fig. 2 are characterized by theFresnel diffraction oscillatory pattern at forward anglesfollowed by an exponential fall.

model has no free parameter; however, we use one freeparameter. This model has maximal input which is takenfrom other independent set of measurements and in thisspirit it is a microscopic model. It does not involve nu-merically solving the Schrodinger equation. It has been ahigh-energy physics model and works with sub-Coulombenergies as well with suitable modifications. The datataken in the present experiment has been particularlyanalyzed in terms of the four models and an explicit com-parison is made among them.

Table I gives the fitted values of the various parametersof the optical model potential for different energies. Wehave fixed rR ——rl ——1.15 fm and aR ——ar = 0.63 fm.The optical model fits to the elastic angular distributionsare shown in Fig. 2 as the solid lines.

B. The PPSM analysis

The fits to the angular distributions at four differentenergies are given in Fig. 3 as the solid lines and labeled

IV. ANALYSIS OF THE DATA

The measured experimental data are analyzed in termsof the four models discussed above. In the analysis theparameters of a model are varied to get the best fit tothe data by minimizing the y, which is defined as

010

P PSMQ F' M

1 ) . (&,'„i —&th)

(Ao.,'„)2 (21) 010

where i is the summation index over N data points.The physically relevent quantities which can be extractedby fitting the data are the scattering matrix elementsas a function of L. These quantities should agree witheach other for all the models. The value of l for which

~S&~ = 0.5 is defined as ii~2. The quantity d describesthe diffuseness of the l distribution and is defined below.B, and d are given by

010

to

10

kR, = q+ [ri + lii2(ii(2+- 1)] (22)

and -']

10

TABLE II. Parameters of the parametrized phase shiftmodel (PPSM) and the generalized Fresnel model (GFM) for

Si beam on Al target.

@lab(MeV)

70

80

90

100

Methodused

PPSMGFMPPSMGFMPPSMGFMPPSMGFM

20.7820.0526.6026.5531.4731.3135.7036.02

0.5641.2861.1681.5731.761.9731.6232.593

0.0010.0

1.2030.0

1.4370.02.170.0

-310

1OO9 (deg )

t t I

150

FIG. 3. Fit to the elastic scattering angular distributionof Si + Al. The solid line is the parametrized phase shiftmodel (PPSM) fit to the data and the dashed line is the gen-eralized Fresnel model (GFM) fit to the data.

1156 CHARAGI, GUPTA, BETIGERI, FERNANDES, AND KULDEEP 48

Si beamenergy (MeV)

708090100

&NNcalculated

0.1530.15080.15140.1542

&NNfitted

0.80.60.60.8

&NNcalculated

(mb)2599235221501980

TABLE III. Table of o.~~ and o.~~ calculated by amethod given in Ref. [11]along with the fitted values of aiviv.

10—

0.5—

OPTICAL

ppS gGFMCMGIvI

28SI ~ 27AI

100 MeV

as PPSM. The fitted values of the parameters L~, 4, andp are given in Table II.

C. The GFM analysis 26 30 34 38 42

l

46 50

The fits to the data using GFM are given as the dashedlines in Fig. 3. The fitted values of the parameters tg and4 in this model are given in Table II.

D. The CMGM analysis

In principle the CMGM is a parameter-free model [11].However, in the analysis of present data we have usedo.~~ as the free parameter. The o.~~ calculated in theefFective range approximation [11] along with the A~~fitted in the present analysis are listed in Table III. Alsogiven in the table are o~~ values used in the presentwork which are calculated according to Ref. [11]. Thefits to the elastic data are shown as dashed lines in Fig.2.

V. DISCUSSION

The fits obtained by all the four models at different en-ergies yield comparable y values as listed in Table IV.

FIG. 4. Plot of the function ~S(l)~ with l for Si + Alsystem at 100 MeV.

The characteristic derived quantities such as ii~2, B„d,and o„are also listed for all energies in the same ta-ble and they are model independent to a large extent.The scattering matrix elements from different models arecompared and found generally to be in agreement witheach other. In Fig. 4, ~S~I is plotted as a function of lfor the laboratory energy of 100 MeV. The y is similarfor the optical model, the PPSM, and CMGM; however,the y for GFM is about three times the value of othermodels. The y for GFM is large as it sets the parameterp of the PPSM as zero and therefore the model has lessflexibility. This explains the fact that ~S~~ obtained fromGFM is not agreeing well with that from other models.It is of interest to compare the potential obtained in theGlauber model and the one used in the optical model.Both the models have a complex potential; however, themethods of solving the Schrodinger equation are differ-

TABLE IV. Table of the quantities such as Izyz, R„and d defined in the text, and o, thereaction cross section deduced from the scattering matrix. Also given in the Table is the minimum

per point and o„as defined in the text.

Si beamenergy (MeV)

70

80

90

100

Methodused

OPTICALPPSMGFM

CMGMOPTICAL

PPSMGFM

CMGMOPTICAL

PPSMGFM

CMGMOPTICAL

PPSMGFM

CMGM

20.7021.3021.1821.7526.927.5

27.9327.6431.9032.8033.0532.4035.6

36.5038.336.5

B,(fm)9.659.749.729.819.649.749.819.769.639.789.829.719.529.679.969.67

d2.020.51.082.192.041.071.362.142.161.541.652.172.111.542.172.24

o7

(mb)674664669704979962100498512021224124511981345136415051365

x'1.230.751.111.581.291.441.482.550.781.251.081.410.7

0.612.470.82

usingEqs. (24,25)

674+14

982+15

1217+19

1371+42

o~using

Eq. (26)

639

930

1157

1338

48 ELASTIC SCATTERING OF ~ Si FROM Al AT 70, 80, . . . 1157

210—28 27Al

(24)

where the summation i is over different models. Also

110— (25)

The error on o„values are around 2—3%. The averageR, is around 9.72 fm with variation of +0.2 fm at allthe energies and for all the models. The reaction crosssection using this R, can be calculated from

, ( Z„Zte' io„=vrR,i

I—E. B,)(26)

10

The values of cr obtained using Eq. (26) are smaller by3—6% from the values obtained using Eq. (24).

VI. CONCLUSIONS

-210 I I I I I I I I I

3 5 7 9 11

I (frn)FIG. 5. The variation of the real part of the potential with

distance (fm). The solid line is the optical model calculationand the dashed line is the Glauber model calculation.

ent. As there exist several ambiguities in the potentialdescription of the scattering, they can explain furtherdifFerences in the deduced potentials in the two models.For 100 MeV laboratory energy, a comparison is madein Fig. 5 for the real potential obtained in the Glaubermodel (dotted line) and the optical model (solid line). Inthe tail region sensitive to the data, the two agree well.The ratio n~~ ——0.8 is equal to the ratio of the real tothe imaginary potentials in the Glauber model, whereasthe shapes of the two potentials remain the same. Inour optical model analysis also the shapes of the realand the imaginary potentials are the same; however, theratio of the real to the imaginary potentials is around2, which is widely different from the ratio 0.8, obtainedin the Glauber model. This demonstrates the ambigu-ity in the potentials. The average value of the reactioncross section at each energy is calculated by taking yweighted average of the reaction cross sections obtainedby different models. Thus

The elastic scattering angular distribution data aremeasured over a wide angular range for four differentenergies, i.e., 70, 80, 90, and 100 MeV. The data areanalyzed in the framework of the optical model, thePPSM, the GFM, and the Coulomb-modified Glaubermodel (CMGM). The comparable fits to the data wereobtained in all these models and in general scattering ma-trix elements were in agreement from model to model forall these energies. The characteristic features of the Siare described in terms of the strong absorption radius R,and d. A value of R, equal to 9.72 fm and the diffusenessd in / space of about 2 can describe the data reason-ably well at all the energies. The reaction cross sectionobtained at each energy is also model independent andcan be considered as another characteristic of the scatter-ing matrix. Out of all the models the CMGM providesan economical way of describing the scattering data asit has only one free parameter. The GFM however hasthe simplicity of a closed form for which calculations canbe made by using the characteristic quantities R, and

The advantages of the optical model and CMGM liein their capability to generate the distorted waves whichcan be used in the calculations of direct reactions.

ACKNOWLEDGMENT

The authors gratefully acknowledge valuable com-ments made by Dr. B. K. Dain on the manuscript.

[1] G.R. Satchler, Direct Nuclear Reactions (Oxford Univer-sity Press, Oxford, 1983); P. Schwandt, Indiana Univer-sity Cyclotron Facility Report No. 81-3, 1981 (unpub-lished).

[2] W.E. Frahn and R.H. Venter, Ann. Phys. (N.Y'.) 24, 243(1963).

[3] D.A. Bromley, Treatise on Heavy ion Science (Ple-num,New York, 1984), Vol. I, p. 159.

[4] R.J. Glauber, Lectures on Theoretical Physics (Inter-science, New Y'ork, 1959), Vol. I.

[5] S.K. Charagi and S.K. Gupta, Phys. Rev. C 41, 1610

(1990).[6] W.E. Frahn, Nucl. Phys. A 302, 267 (1978).[7] Handbook of Mathematical Functions, edited by M.

Abramowitz and I.A. Stegun (Dover, New York, 1965).[8] P.J. Karol, Phys. Rev. C ll, 1203 (1975).[9] A. Vitturi and F. Zardi, Phys. Rev. C 36, 1404 (1987).

[10] J. Chauvin, D. Lebrun, A. Lounis, and M. Buenerd,Phys. Rev. C 28, 1970 (1983).

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