Folding model analysis of 32S + 32S elastic scattering at 70, 90, 97.09, 120 and 160 MeV

Nuclear Physics A473 (1987) 353-364 North-Holland, Amsterdam

FOLDING

MODEL

ANALYSIS

OF ?3+%3

AT 70, 90, 97.09, B. BILWES, Groupe

de.7 Basses Energies, F. BALLESTER. lnstituto

120 AND

R. BILWES

and

ELASTIC

SCARERING

160 MeV

L. STUTTGE

Centre de Recherches Nucle’aires and UniversitC Louiv Pasteur, Strashourg, France J. DiAZ, J.L. FERRERO, C. ROLDAN and F. SANCHEZ

de Firica Corpuscular,

Uniuersidad Received

de Valencia-CSIC,

20 March

Burjasot,

Valencia, Spain

1987

Abstract:

have been measured at 70, 90, Angular distributions for the 32S(‘7S, “S) elastic scattering 97.09, 120 and 160 MeV incident lab-energies. The data have been analyzed with the folding model, using the M3Y and DDD interactions. A good reproduction of the data is obtained if a renormalization coefficient for the real part of the optical potential is introduced. Moreover the application of the dispersion relation proposed by Mahaux et al. and which uses the imaginary part does not seem to give renormalization coefficients as important as those found in the analysis of the data.

E

NUCLEAR

REACTIONS

‘%(“S, “S), &,=70, 90, 97.09, a( E, 0). Folding model analysis.

120, 160 MeV;

measured

1. Introduction

The folding

model for the real part of the optical

potential

1-J) has proved to give

a good description of the elastic scattering for many systems in a large colliding energies. Moreover this kind of semi-microscopical model gives interpretation of the data than phenomenological optical potentials, since it as basic physical ingredients the nuclear densities folded with a realistic

range of a better contains nuclear

interaction in the correct way. Consequently, when discrepancies between the folding model and the data appear, it can be suspected that a new phenomenon not contained in the model is involved. The major discrepancies that have been found up to now consist in the need of a renormalization coefficient for the real part of the potential consistently different from unity. It can be less than one and weakly energy dependent ‘,4-h) or greater than unity and strongly energy dependent ‘-I”). Different interpretations have been proposed to explain these results. The most plausible one in the case of a renormalization coefficient less than unity, which appears in systems involving as projectile d, t, ‘.‘Li and “Be, is the existence of strong coupling to break-up channels effects. In fact, it has been found by S-matrix inversion methods ‘I) that coupling to break-up channels produces repulsive potentials. On the other hand, the existence of a strongly energy-dependent renormalization coefficient in 0375.9474/87/$03.50 ,Q Elsevier Science Publishers (North-Holland Physics Publishing Division)

B.V.

354

B. Bilwes et al. / Folding model analysis

some rather bound systems such as a + 160, ““Ca, ‘“O+4”Ca [ref. I”)], 32S+40Ca [ref. ‘)I, I60 +“‘Ni [ref. “)I and IhO + 2oxPb [ref. “)I has been interpreted as due to the coupling to inelastic and transfer channels. It emerges as a consequence of a dispersion relation which links the real and imaginary parts of the optical potential ‘“.‘2,13). The validity of the dispersion relation is subject to a certain number of hypotheses. It has also been confirmed by S-matrix inversion methods that coupling to inelastic and transfer channels produces attractive polarization potentials 14). The study of both Coulomb and nuclear polarization potentials and its numerical computation have received considerable attention recently ‘5m25),hence experimental data in which these potentials can be tested are of relevant importance. The “anomalous” systems are perhaps the most adequate for this purpose because of the large intensity of the polarization effects. However, the finding of anomalous systems has been up to now a purely experimental matter and to our knowledge, no present theory can predict reliably which system will be “anomalous” and which will not. In particular, it is not known how bound a system has to be for anomalies to be present. We know that “S + 4”Ca is an “anomalous” system. Hence it would be interesting to investigate if there is some kind of anomaly in a less bound system; a good candidate is the “‘S+“S system. In this paper we present measurements of “Sf3’S at 70, 90, 97.09, 120 and 160 MeV and analysis of the data with the folding model.

2. Experimental The experiments

method

were made at the MP Tandem

of the CRN-Strasbourg.

The data

have been taken by using the kinematical coincidence method as described in [refs. 7,26)]. However it is worth giving some additional remarks. The Q-energy resolution was about 300 keV; the events were stored in steps of 1” c.m. For each angular distribution three or four arrangements of the detectors were sufficient. The error bars include both statistical and other estimated errors. The error on angles and the angular resolution were of 0.4” c.m. The ‘?S targets were made of 2-4 kg/cm’ of “S onto a 20 pg/cm’ carbon foil by using the implantation technique at the Bernas laboratory in Orsay. In fig. 1 an energy spectrum at 120 MeV is shown and in fig. 2 a mass spectrum at the same energy is exhibited in which the good quality of the data can be seen. The angular distributions were normalized to the Rutherford cross section at small angles. The normalization error is estimated to be less than 3% except at 70 MeV and 160 MeV. At 70 MeV due to the strong oscillating pattern, the angular resolution deteriorates the accuracy of the normalization value. It was estimated to be of 5%. At 160 MeV the smallest angle data measured are already affected by the nuclear potential. However as we used the same target as that at 120 MeV for which we could normalize the measured data to the Rutherford cross section, we could in a

B. Bihes

32S( 32s ,32S 132s

et al. / Folding

35.5

model analysis

120 flEV-LAB

32” -
cx +“Si

flEV) Fig. 1. Energy

spectrum

for 7rS+‘zS

at 120 MeV of laboratory

energy

first step compute the normalization at 160 MeV within better than 10% accuracy. Then, in a second step we renormalized the data by minimizing the ,$ of the best fit; the renormalization of the data varied by less than 5%. We think that this last renormalization

value is accurate

to better than 5%.

3. Analysis In a first step we analyzed our data by using Woods-Saxon potentials. The reasons to do so were twofold: first of all, the purpose was to determine at each energy the radius of sensitivity, defined as the radius at which the real parts of different optical potentials fitted to the data cross approximately each other; in the second place, it was to estimate how good the best fits obtained with the folding model should be expected. One of the potentials obtained for each energy is shown in table 1. The other potentials were derived by changing one parameter of the real part, say the diffuseness, and fitting the remaining parameters in order to recover the best fit as done in ref. “). In this way a family of potentials linked by an ambiguity and giving a similarly good description of the data was obtained for each energy. In fig. 3 the family obtained at 90 MeV is shown and it is seen how well the sensitivity radius

356

B. Bilwes et al. / Folding

model analysis

100

8

/

16 a+;

0

21

28

IGg. 2. Mass spectrum

32

36

for “S+

“S

LO

TABLE Best fit optical

potentials

for “S+“S

&I

68

at 120 MeV of laboratory

52 I( energy

1 at 90, 97.09,

120 and 160 MeV

W (MeV) 90

49.9

1.231

0.6

39.55

1.3

0.37

97.09

19.9

1.33

0.6

23.1

1.35

0.32

1.4

120

19.9

1.42

0.395

21.8

1.385

0.266

2.0

160

15.0

1.265

0.601

4.87

1.313

0.786

1.8

3.4

is determined. We have found for the sensitivity radius a value of Rs = 9.7510.10 fm and no significant variation with energy has been observed. The importance of determining the sensitivity radius comes from the fact that the optical potential is sensitive to the data only within a small region of its range. Thus, although WoodsSaxon, folding potentials and any other model give very different results in the inner region, the value of the potential at the sensitivity radius should be nearly the same for any reasonably good description of the data. In other words the position of the sensitivity region, the strength of the potential within the region and any energy

B. Bilwes et al. / Folding

model ana/_vsis

357

-U(REU lO(

Fig. 3. Family

dependence

of real Woods-Saxon

of these strengths

potentials

for “S+

“S at 90 MeV laboratory

are almost model-independent

features

energy.

for any model

which fits the data. All the optical model calculations both with phenomenological and folding model potentials were done by use of the ECIS code *‘). We have found that at 70 MeV there is no sensitivity to the real part of the optical potential. In fact, if it is set to zero, a very good fit to the data can be obtained by varying the parameters of the imaginary part. The reason is that the scattering at this energy is dominated by the Coulomb interaction and the imaginary potential is needed only to reproduce the small departure of the data from the Mott scattering. Unfortunately, due to the errors on angles and on the absolute normalization of the data we were unable to extract with enough reliability the parameters of the imaginary potential. For this reason we do not include these data in our analysis. In a second step, we analysed the data by using folding model potentials. These potentials were computed with the help of the DFPOT code *“). The interaction used was the standard M3Y ‘.2”) given by

B. Bihes

358

et al. / Folding

model unalysis

where the 6(r) term accounts for knockon exchange and Jo0 = -262 MeV . fm’. In fact, this term has a weak energy dependence which has been ignored because of its negligible effects in the range of energies studied here. The charge density for “S was taken from an electron parametrized in the Fermi parabolic form,

scattering

experiment

“‘)

where w = -0.213, c = 3.441 fm and a = 0.624 fm. The nucleon density was computed by scaling the charge density to the nuclear mass and correcting for the finite size of the proton in the standard way ‘). This procedure is justified because proton and neutron densities are very similar in the sd-shell ‘I). Thus, in the case of identical particles, the folding potential reads, V,(r)

=

dr, dr2 p(rI)p(rA%,3drId ,

where r,? = lr+ rz - r,(. The Coulomb potential was computed with the same folding procedure using the Coulomb force. The results differ from those obtained with the standard method of considering a point charge onto a charged sphere of radius R, + R2 only in the inner region and the potentials obtained by both procedures give practically the same cross section. The imaginary potential was taken of Woods-Saxon shape,

The total potential

is, V(r) = NV,(r)+iW(r)+

where N is a renormalization the data. In a first set of calculations

coefficient

V,.(r),

that can be allowed

we fixed the renormalization

to vary in order to fit coefficient

to one and

we fitted the parameters of the imaginary potential; an attempt to fix the reduced radius in order to lower the number of parameters was unsuccessful. For best fit, the potentials found are given in table 2. In a second step, aiming at determining the best value of the renormalization coefficient, searches were done on the imaginary parameters over a grid of N values in steps of 0.1. In this way we obtained at each energy a near parabolic curve giving ,$/n versus N where n is the number of experimental points (fig. 4). By this method the best renormalization coefficient at each energy is determined. For the best fit potentials with four parameters are given in table 3. In fig. 5 are shown the experimental data and the calculated curves for three ( W, rw, aw) and four (N, W, ru., au,) parameters fitted. The curve at 70 MeV is calculated with the Coulomb potential only.

B. Bilwes et al. / Folding model analysix

359

TABLE 2 Best fit imaginary potentials for’%+%? at 90,97.09, 120 and 160 MeV by taking the real part calculated with the unrenormalized M3Y interaction E

W

r&t

au

(MeV)

(MeV)

(fm)

(fm)

90 97.09 120 160

4.40 7.25 9.79 60.94

1.427 1.281 1.329 1.112

0.508 0.757 0.644 0.555

X2/n 5.4 3.5 7.8 2.9

6-

2-

Fig. 4. Curves

of X2/n versus

N for ‘zS+32S

at 90, 97.09 and 120 MeV laboratory

TABLET Best fit parameters for the renormalized M3Y interaction ‘*S + % at 90, 97.09, 120 and 160 MeV

N

90 97.09 120 160

1.8 1.9 1.7 0.9

W

a,

(MeV)

c&

(fm)

109.25 62.5 16.7 60.7

1.286 1.325 1.243 1.044

0.325 0.303 0.424 0.657

for the

X2/n 2.3 1.7 2.9 2.8

energies

E. Bilwvs et al. / Folding model analysis

360

Wmb/sr) SCATTERING

f

10

IlEU-LAB

90

IlEU-LAB

91,09

120

i

REV-LAB

IlEU-LAB

I

160

tlEV-LAB

t

Fig. 5. Experimental data for 32S+3*S at 70, 90, 97.09, 120, and 160 MeV laboratory energies compared with predictions using the folding model: renormalized (full line) and unrenormalized (broken line) M3Y interactions.

We performed also calculations with the usual hypothesis of equal geometry i.e. by taking the imaginary part proportional to the real one, and we fitted the real and imaginary renormalization coefficients. The results are given in table 4. It is seen that at all energies the renormalization coefficient of the real part is at the same time larger than unity and larger than the coefficient of the imaginary part.

TAHLE~

Best fit parameters obtained by taking the imaginary potential proportional to the real potential E ( MeV)

N,

N,

x’l n

90 97.09 120 160

1.35 1.455 1.544 1.03

0.72 0.737 1.022 0.42

3.4 2.6 2.7 3.4

B. Bilwes et al. / Folding model analysis

361

Finally we studied also whether or not a density dependence of the interaction would produce any significant effect. To this end we used the DDD interaction given by ‘),

-?.Sr

u,(r) = 6839 $-

1887 !-2.5r

’

-2 5, 0~(r)=6893$1938~

2.5r

’

and JO,,= -213 MeV . fm3. The results obtained are shown in table 5. It appears that the density dependence of the interaction produces slightly smaller renormalization coefficients although the general trend is the same. TABLE 5 Best fit parameters E

W

N

(MeV)

using the DDD interaction

(MeV)

ru (fm)

(&

X2/n

90

1.5

35.6

1.321

0.359

97.09

1.6

168.9

1.250

0.335

1.6

2.3

120

1.4

62.63

1.239

0.460

3.5

160

0.8

10.9

1.236

0.653

2.6

In order to check consistency between the real and imaginary parts we have applied the dispersion relation to the values of the imaginary parts at the sensitivity radius and we have compared the predicted values of the real part to the experimental ones. The dispersion relation in the so-called subtracted form links the real and imaginary potentials through the equation,

where

E, is a convenient

reference

energy

AVEb(r; E)=

and

V(r; E)-

V(r; Es)

and P means the principal value of the integral. We assume for W(E) a linear schematic model in which W(E) is represented by n linear segments, the vertices being the “experimental” values. The integral can be computed easily and the result is AV,(r;

n;:;

E)=lln T

n::i

j(E,+, - E)/(E, I(Ei+, - E,)/(E,

- E)IW,‘“‘(l/(En

- E)jyz

- E,)~~(~~)~l/(E,

-E,)Iya

’

B. Bilwes et al.1 Folding model

362

analysis

where Wi(E) = :+‘I,“~E 1+1 I

-Ed)+

W,

and ( Ei, Wi) are the “experimental points” for the imaginary potential. The energy E,, is the energy at which it is assumed that the imaginary potential vanishes, i.e. W(E,) = 0. In the calculations we have taken 160 MeV for the reference energy, assuming that at this energy the real potential is the bare folding potential, i.e. N = 1. The calculations have been done both for the M3Y and DDD interactions. At the top of fig. 6 are shown the results of calculations of the dispersion relation in the case of the M3Y interaction using as input the experimental imaginary potentials given at the bottom of the figure and two different values for the energy at which the imaginary potential vanishes (60 MeV: dashed curve, 70 MeV: full curve). The error on the dispersion relation due to a 5% variation of the data normalization at 160 MeV is represented by a dashed area. In fig. 6 we also represented the relation dispersion calculated taking the “experimental” normalization factor of the real potential at 160 MeV, i.e. N = 0.9 (thin curve). A 10% variation of the minimum value of the x2/n curves was adopted to determine the error bars of the “experimental points”. N

, .

LOI-

,

,

, ,

,

I , ,

, , ,

, , , , , , ,

T T

1 MeV

0.5MeV

60

80

100

120

140

160

180

200

ElMeV-LAB1

Fig. 6. “Experimental points” for the real and imaginary potentials for 32S+32S at 90, 97.09, 120, and 160 MeV at 9.75 fm and predicted real part by the dispersion relation. The curves for the real part have been calculated with a 3-segment imaginary part for the M3Y interaction. The energy at which the imaginary potential vanishes was 70MeV (full curve) or 60MeV (dashed curve). The dashed area corresponds to an error of 5% on the data normalization at 160 MeV. The thin curve is the analogous of the full curve but normalized to the “experimental” value of N at 160 MeV (0.9 instead of 1.) The error bars correspond to a 10% variation of the minimum value of the x2/n curves of fig. 5.

B. Bilwes et al. / Foldiq

model analy.~is

363

4. Conclusions

The elastic scattering angular distributions for the 32S+32S at 70, 90, 97.09, 120 and 160 MeV have been measured. The data have been analyzed with the folding model, using the standard M3Y interaction and the density-dependent DDD interaction. It has been found that both interactions need to be renormalized by an energy dependent coefficient in order to reproduce satisfactorily the data. This result is similar to that found for “S + 4”Ca in earlier works ‘). The values of the real potential found at the sensitivity radius have been compared to the values predicted by a recently derived ‘“,‘2,11) dispersion relation which uses as input the imaginary potential at the sensitivity radius. It has been found that the dispersion relation reproduces the general trend of the fitted renormalization coefficients but the absolute values are underestimated. A slightly better agreement is observed in the case of the DDD interaction. However, to check the evolution at lower enenergy very high precision data are needed. This remains a challenging experimental problem in the particular case of the symmetric mass system. This paper has been supported by the CAICYT (Spain) under project 3/5151985/87 and by the IN2P3-CNRS (France). The authors wish to express their gratitude to Professor C. Mahaux for stimulating discussions, and to Dr P. Grang.4 for the careful reading of the manuscript. Two of us (JD and FS) are grateful to Ministerio de Education y Ciencia (Spain) for fellowships and to CRN (Strasbourg) for hospitality.

References

1) G.R. Satchler and W.G. Love, Phys. Reports 55 (1979) 183 M. El-Azab Farid and G.R. Satchler, Nucl. Phys. A438 (1985) 525

2)

3) 4) S) 6) 7) 8) 9) IO) 11) 12) 13) 14) IS) 16) 17) 18) 19) 20)

M. El-Azab Farid and G.R. Satchler, Nucl. Phys. A441 (1985) 157 J. Cook, Nucl. Phys. A375 (1982) 23X J. Cook and R.J. Griffiths, Nucl. Phys. A366 (1981) 27 C.L. Woods, B.A. Brown and N.A. Jelley, J. of Phys. G8 (1982) 1699 A. Baeza, B. Bilwes, R. Bilwes, J. Diar and J.L. Ferrero, Nucl. Phys. A419 (1984) 160; B. Bilwes, R. Bilwes, J. Diar and J.L. Ferrero, Nucl. Phys. A449 (1986) 519 J.S. Lilley, B.R. Fulton, M.A. Nagarajan, I.J. Thompsonand D.W. Banes, Phys. Lett. 151B (1985) 1x1 B.R. Fulton, D.W. Banes, J.S. Lilley, M.A. Nagarajan and I.J. Thompson, Phys. Lett. 162B (1985 ) 55 C. Mahaux, H. Ngi, and G.R. Satchler. Nucl. Phys. A449 (19X6) 354 A.A. loannides and R.S. Mackintosh, Phys. Lett. 169B (1986) II3 C. Mahaux, H. Ng6 and G.R. Satchler, Nucl. Phys. A456 (1986) 134 M.A. Nagarajan, C.C. Mahaux and G.R. Satchler, Phys. Rev. Lett. 54 (1985) 1136 A.A. Ioannides and R.S. Mackintosh, Phys. Lett. 161B (1985) 43 M.S. Hussein, A.J. Baltz and B.V. Carlson, Phys. Reports 113 (1984) 133 C.L. Rao, M. Reeves III and G.R. Satchler, Nucl. Phys. A207 (1973) 182 P.W. Coulter and G.R. Satchler, Nucl. Phys. A293 (1977) 269 Z. El-Itaoui, P.J. Ellis and B.A. Mughrabi, Nucl. Phys. A441 (1985) 511 0. Tanimura, R. Wolf, R. Kaps and LJ. Mosel, Phys. Lett. 132B (1983) 249 R. Wolf, 0. Tanimura and U. Mosel, Nucl. Phys. A414 (1984) 162

364 21) 22) 23) 24) 25) 26) 27) 28) 29) 30) 31)

B. Bit&s

et al. 1 Folding model analysis

R. Wolf, 0. Tanimura, R. Kaps and U. Mosel, Nucl. Phys. A441 (1985) 381 S.B. Khadkikar, L. Rikus, A. Faessler and R. Sartor, Nucl. Phys. A369 (1981) 495 A.M. Kobos, G.R. Satchler and R.S. Mackintosh, Nucl. Phys. A395 (1983) 248 A.M. Kobos and R.S. Mackintosh, Phys. Rev. C26 (1982) 1766 G.R. Satchler, Nucl. Phys. A409 (1983) 3c B. Bilwes, R. Bilwes, V. D’Amico, J.L. Ferrero, R. Potenza and G. Giardina, Nucl. Phys. A408 (1983) 173 J. Raynal, Phys. Rev. C23 (1981) 2571 J. Cook, Comp. Phys. Comm. 25 (1982) 12.5 G. Bertsch, J. Borysowicz, H. McManus and W.G. Love, Nucl. Phys. A284 (1977) 399 G.C. Li, M.R. Yearian and I. Sick, Phys. Rev. C9 (1974) 1861 W.J. Thompson and J.S. Eck, Phys. Lett. 67B (1977) 151