Energy-dependent renormalization coefficients of folding-model description of 32S+40Ca elastic scattering

Nuclear 0

Physics

A419 (1984) 412428

North-Holland

Publishing

Company

ENERGY-DEPENDENT OF FOLDING-MODEL

A BAEZA

RENORMALIZATION COEFFICIENTS DESCRIPTION OF 32S + 40Ca ELASTIC SCATTERING

‘, B. BILWES ‘, R. BILWES ‘. J. DiAZ

’ and J. L. FERRER0

’

’ Inst~tutode Fisica Corpuscular, Unrversidad de Valencia, Burjasot, Valencm, Spam ’ Groupe des Basses Energies-Cenire de Recherches Nuclkaires and Universitk Louis Pasteur, Strasbourg, France

Received 16 May 1983 (Revised 6 September 1983) Abstract: Angular distributions for the elastic scattering of 32S on 40Ca have been measured at “S bombarding energies of 100, 120 and 151.5 MeV. Optical model analyses were performed using Woods-Saxon and double-folding potentials which yield a good description of the angular distributions. The energy dependence of the phenomenological optical potentials and renormalization coefliclents of the double-folded potential were found. These results are anomalous m comparison with the elastic scattermg of heavy proJectiles (with A > 12) on the same target nucleus.

E

NUCLEAR

REACTIONS 40Ca(3ZS,3ZS), E = 100. 120, 151.5 MeV; Double-foldmg model, deduced optical model parameters.

measured

u(B).

1. Introduction During recent years several elastic scattering experiments of heavy ions on 40Ca have been performed at various incident energies ‘). All these systems present the typical Fresnel diffraction pattern. However, recently measured ‘*C +40Ca and 160 +40Ca elastic scattering data show oscillations for O/CJ~< lo-* [ref. ‘)I. The elastic scattering of 32S+40Ca has only been partially studied at energies near the Coulomb barrier. The first data concerning this system were measured by Gutbrod et al. 3, at energies between 80 and 90 MeV with poor energy resolution and the data do not resolve the elastic from the first inelastic peak at backward angles. These experimental points are few and cover an angular range up to a value of o/oR 21 0.1. The more recent data of Richter et al. “) at Elab = 100 MeV cover up to a/~~ N lo-*. The 32S+40Ca system does not exhibit any oscillatory structure in these two references. The optical potentials proposed by them have a different behaviour in the tail region at 100 MeV. 412

A. Bae:a et al. 1 4oCa132S, j2SI

413

Considerable success has been achieved in fitting heavy-ion elastic data for many systems by generating the real part of the potential from a double-folding model with a two-body interaction known as M3Y 5, and using a Woods-Saxon form for the imaginary part. However, some systems have been found to be “anomalous”, in the sense that the real part of the potential must be multiplied by a renormalization coefftcient different from 1 in order to obtain a good agreement with the experimental data. Energy-independent coeflicients less than 1 were found for 6Li, ‘Li and 9Be projectiles ‘). In addition, a variation of this coefficient with energy has been found in the analyses of “C + ‘?Si [ref. “)I and ‘(‘0 + 28Si [ref. ‘)I. This fact was associated with a back-angle enhancement and strong oscillations at large angles, probably due to unexplained processes which interfere with the shape elastic scattering. In this paper we report our measurements of the elastic scattering of “S on 40Ca at energies of 100, 120 and 151.5 MeV which extend the previous data at 100 MeV to higher energies and to values of cr/crR v 10p3. We extend the data to energies well above the Coulomb barrier in order to investigate the possible existence of oscillatory structure. We have analyzed them using two different shapes for the optical potentials : a standard phenomenological Woods-Saxon and a doublefolded form. 2. Experimental

procedure

The 32S beam of the MP tandem of the CRN Strasbourg was collimated on target to a disc of 1 mm diameter by means of a diaphragm which was aligned to the geometrical center of the Ortec chamber. The targets were obtained by evaporating 40Ca onto a 20 pg/cm2 carbon foil. The thickness of 40Ca was 20 pg/cm’. Two position sensitive detectors (48 x 9 mm’) in coincidence, set at 140 mm from the target on each side of the beam, were used to identify the reaction products by the kinematical coincidence method *. 9). Fig. 1 shows some mass spectra obtained at incident energies of 100, 120 and 151.5 MeV. The elastic and inelastic scattering and cc-transfer reactions are the most important processes at the lowest incident energy. At 120 MeV, one-, twoand three-nucleon transfers also become important and finally at 151.5 MeV the elastic peak maintains the same FWHM, but the other peaks disappear in a continuous background. An extensive study of these spectra has shown that background is due in part to some inaccuracy in the energy calibration for highly negative Q-values, but the main contribution is due to reactions with three (or more) particles in the exit channel. To extract the elastic and quasielastic data, it was necessary to select events within a window set on Q-values deduced from the energy pulses of the detectors. One gated spectrum is shown in fig. 2.

414

A. Baeza et al. / 40Ca(32S,

-cl l

32S)

A. Baeza et al. / 4oCa(32S, I

M2

“S)

1

I

I

32S +‘OCa 151.5 MeV-Lab 44.5”s 8, s 63.5” 31.5”s 8,s 50.5” -8
: m b2 -

@

LO

@

+(32,40)

t 38 36

t

(36,361

0

3b

32 30 28 26 24

22 c 22

26

30

34

38

42

b6

M,

(amu) Fig. 2. Gated

mass spectrum

obtained

by putting a window the energy pulses.

on the Q-value

determined

directly

from

The Q-value used to obtain the corresponding differential cross section was calculated from the exact masses of the events, the projectile mass, the incident energy, and the scattering angles of the particles. The resolution on the Q-value, which depends on the geometrical characteristics of the experimental set-up (the position resolution of the detectors and the angular straggling in the target), was always sufficient to resolve the ground state and the first excited level of projectile and target, as shown in fig. 3. This resolution was much better than that obtained directly from energy pulses. The accuracy of the angular calibration was tested using the angular correlations between the elastically scattered particles. The angle of the center of the detector was modified if necessary (always less than 0.2O) to take into account the misalignments of the experimental set-up. The spatial geometrical resolution (0.5 mm) of the detectors produces an angular straggling in the target also contributes to spread of w 0.2O. The angular deteriorate the angular resolution, mainly for the low-energy scattered particle and this effect can only be efficiently reduced using thinner targets. When the incident

416

A. Baeza et al. / 4oCa(32S,

energy increases,

the angular

straggling

decreases

32Si

slowly lo), but the greater

slope of

dE/dB produces a larger spread of the Q-values. Typical FWHMs of the Q-value were 0.5 and 1.2 MeV at 100 and 151.5 MeV incident energies, respectively. The total angular resolution was better than 0.5” in the lab system which allowed us to obtain the experimental cross section in lo steps in c.m. Three different geometrical arrangements of the detectors were necessary to obtain each angular distribution. The different parts were matched using the overlapping angular range of measurement. Absolute values of the elastic cross sections were obtained by normalizing to Rutherford scattering at the most forward angles. The error of normalization was less than 3 “/,. The error bars include the statistical error plus the relative normalization errors between the different measurements.

N (a.ul

LEVEL

S

LEVEL! i

500

I

I I I I

I I

I

400

32s+%a

j

I 1

I

120 MeV-Lab

!

300

200

100

-6 Fig.

-5

-4

-3

-2

-1

0

1

ahkv)

3. Q-value spectrum calculated from the position pulses for a selected area in the spectrum. The level schemes of the nuclei are shown at the top of the figure.

mass

A. Baeza et al. / “°Ca(32S,

3. Optical model analysis with Woods-Saxon

32S)

417

phenomenological

potentiak

Previously measured angular distribution for the elastic scattering of “S on 40Ca [ref. “)I and those of the present study are illustrated in lig. 4. The optical potentials obtained by Gutbrod et al. 3, and Richter et al. “) (also shown in fig. 4) do not give good fits to all the angular distributions we have measured. We do not show the experimental data of ref. 4, since our data agree with them in the common angular range, i.e. 0 < 100” c.m. The optical model code GENOA l1 ) was used to fit the data, using the standard Woods-Saxon form for both the real and imaginary part of the nuclear potential. A spherical charge distribution was used for the Coulomb potential with R, = 1.2(Af+A$). As the first step of the analysis, extensive parameter searches were performed by fitting the 100 MeV angular distribution. Some parameter sets obtained by minimizing x2/N are reported in table 1. As expected, continuous and discrete ambiguities are evident in the results illustrated in table 1. It is not our purpose here to study these ambiguities which have been discussed in detail elsewhere 12). In a second step we tried to fit simultaneously energies. We only succeeded when an energy

the angular distribution at the three dependence of the imaginary part of

1 100 MeV- Lab

81.5 MeV-Lab

10

50

70

90 110 8 c.m.(deg) 30

50

70

90

110

8 c.m(d~g)

Fig. 4. Experimental angular distributions for the elastic scattering of 32S by 40Ca. Left side, data from ref. 3); right side, our data. The curves are the predictions obtained using the optical potentials proposed in ref. 3, (solid lines) and in ref. 4, (dashed lines).

extracsted

A. Baeza et al. 1 4oCa(32S,

418

‘=S)

TABLE 1 Optical

potential

parameters

“) for “S+“‘Ca

1

Set

Pl P2 P3 P4 P5 P6

together with the corresponding and 151.5 MeV

values

x’IN (MeV)

l+

(MeV)

(fL,

(2,

(MeV)

,fk,

(f&

27.2 30.7 39.0 150 9.40 250

1.38 1.43 1.41 1.35 1.45 1.18

0.447 0.316 0.322 0.327 0.428 0.479

7.80 2 32 2.31 7.84 33.5X).183& m 187.G1.20E3, m

1.45 1.43 1.41 1.35

0.260 0.316 0.322 0.327 0 348 0.479

“) “l(r) = - V (r. R,. a,)-IW~ Cr.R,, a,).

x2/N b, at 100. 120

1.38 1.18

R,,, c = T~,~,JA:‘~ +A;/Q

f’(r,R,a)

100

120

1515

1.7 1.6 1.6 1.6 3.6 4.8

2.4 22

3.1 2.1

= (l+exp(r-R)/u)-’

the Woods-Saxon potential was included. Fig. 5 shows the fit obtained with parameter set P5 reported in table 1. All these sets of potential parameters allow a good reproduction of the data of Gutbrod et u/. 3, at lower energies. There are several definitions of distances that are relevant for the elastic scattering of heavy ions. One of these is the strong absorption radius R,, defined as the distance of closest approach for a particle moving along a Rutherford trajectory having an angular momentum of I+. The I, is determined by the transmission coefficient ‘T;+= 0.5. As we can see in table 2, the R,, and the values of the real potential at this TABLE 2 Values

of some characteristic

quantities

at the strong absorption sets of table 1

Set

E,,, (MeV)

Rs, “1 (fm)

Pl P2

100 100 100 100 100 120 151.5 100 120 151.5

10.53 10.59 10.60 10.58 10.55 10.43 10.28 10.53 10.39 10.20

P3 P4 P5

P6

“)R

SA

&+[s2+ili.2(I I,2+l)]“‘*I k

radius

-~UW

R,,

for the optIca

(MeV) 1.64 0.73 0.73 0.8 1 0.89 1.14 1.55 0.76 1.01 1.51

potential

VW’(R,,) 8.54 13.25 16.93 19.2$ 2.40 2.40 2.52 2.08 2.33 2.91

A. Bue-_a et al. / 4oCa(32S,

I

30

50

3zS)

419

\

70

90

110

8 cm. (dcg) Ftg

5. 32S+10Ca

angular distributions at labelled energies together wzth preduxlons for the global fit parameter set P.5 of table 1.

the

optical

model

point are very similar for all sets which give a good fit to the experimental data. The dispersion in R,, obtained at 100 MeV is 0.7%. Furthermore the ratio of the real to the imaginary parts of the potential is always greater than 1 and this distance, i.e. all these potentials are surface transparent. We have also found that at each energy all the real parts of the potentials which give a good fit, cross each other within a relatively small area in the Y- 1/ plane and define another relevant distance called by Satchler the sensitivity radius R, [ref. 13)]. The values obtained by us for the three energies analyzed are R, = 10.4 fm. V(R,)= - 1.45 MeV at 100 MeV, R, = 10.05 fm, V(R$)= -2.18 MeV at 120 MeV, and R, = 9.25 fm, V(R,)= -5.74MeV at 151.5 MeV.

A. Barzu et al. / 4oCa(32S, 32S)

420

TABLE 3 Total

reaction cross sections

(in mb) Energy (MeV)

Method opttcal model

Potenttal Pl P2 P3 P4 P5 P6 foldmg “)

100

120

151.5

146 759 755 760 765 791 738

1175 1157 1144

1578 1588 1603

quarter-pomt method 14)

762k5

Frahns’s closed formahsm ‘s,*‘)

741

1167$-7 1152

1609f8 1584

SOD method “)

780& 30

1230+80

1700+ 120

“) Values obtained from set I1 reported m table 4.

n)

Fig. 6. Constant contours of .$/iv in the R$,,-G, plane. G and R are the co~~ponding points using the potentials sets Gutbrod et al. ‘) and Richter et al. 4), respectively. Dots correspond to intermediate potentials obtained by fitting the elastic angular distributions at 100 MeV, asterisks to the best fits, and open circles to the simultaneous tits for the three energies (see table 1).

A. Baeza et al. / 40Ca(32S, “Sj

421

Comparing these values with the corresponding R,, values of table 2 we observe that the sensitivity radius moves faster with energy to the inner part of the nucleus than the strong absorption radius. In contrast, the imaginary parts of the potentials do not cross in a definite region. The total reaction cross section is another important quantity. The values obtained by optical model analysis for the selected potentials of table 1 are compared to those obtained by the Fresnel quarter-point method 14), by Frahn’s closed formalism ’ 5, 16) and by the sum-of-differences method “) (SOD), in table 3. All potentials provide values of total reaction cross sections which agree with the mean value within 4%. The unambiguous behaviour of the above-mentioned quantities is summarized in

lh 32S +%a 11

07

/,/ :I’-

1

100MeV-El,b

...“.... ---__--

P4 p3 p5

0.5’

0.2

Fig. 7. Reflection coefficients for 32S+40Ca at E,,, = 100 MeV calculated with three of the potential sets reported in table 1.

422

A. Bae;a et al.

f 4oCa(32S, “S)

fig. 6, which shows contours of constant $/N as a function of RSA--CJ,values for “S + 40Ca at Elab = 100 MeV. The same behaviour is true at higher energies. Since there exists a direct relation between the S-matrix provided by the optical model potential and the predicted values of the total reaction cross sections, we could expect an unambiguous determination of the S-matrix. In fig. 7 we show different reflection coefficients obtained from some potentials reported in table 1. A considerable uncertainty exists in the determination of the S-matrix elements for the low partial waves but these elements are well defined in the range where 0.5 < /S,l < 0.8, which corresponds to the surface waves. These results confirm that the cross sections are in this case sensitive only to a few surface partial waves. In order to connect the different relevant distances, we have used the semiquantitative perturbation technique developed by Moffa et al. 18) and improved by Cramer et al. 19). We have determined the radial regions of the potentials which are sensitive to the elastic scattering data. In fig. 8, we show the evolution with the incident energy of this sensitivity region (for us the sensitivity region is defined by the radii where the perturbation changes x2/N by an amount equal to half of his maximum value). As one can see, all the unambiguously determined radii previously defined lie inside this region. This region shifts toward lower r-values when the incident energy increases. We notice that an inward shift of only 0.5 fm is explored by increasing the incident energy from 100 to 151.5 MeV.

4. Optical model analysis with double-folded potentials The real part of the optical potentials were computed using the DOLFIN code “) with the standard M3Y interaction 5): -4r

z>(r)= 7999e

4r

- 2.5r

-2134e ~

2.5~.

-2626(r),

where the last term is a pseudo-potential included to take into account exchange corrections. A standard Woods-Saxon form was used for the imaginary part. Densities of Fermi shape were used:

( >

p(r) = p(0) lt

f

1

1 +exp((r-c)/a)’

The parameters, obtained by fitting electron scattering experiments are w = - 0.10117, c = 3.6758 fm and a = 0.5851 fm for 40Ca [ref. ““)J and w = -0.213, c = 3.441 fm and a = 0.624 fm for 32S [ref. “‘)I. Other densities used produced only small changes in the renorm~i~tion coefficient without altering the conclusions obtained below. Since the densities taken from the literature were charge densities, standard corrections ‘) were made to obtain the distribution of center of mass of the

A. Ram2

ef

al 1 *°Ca(32S, 32Sj

-V(r) -W(r) (MeV)

423

32s +%a

100MeV- Lab

r(rm)

-V(r) -W(r) (MeV)

120 MeV- Lab

I 0.1

I 12

I”

r (fin)

-V(r) -W(r) (MeV)

1 0.1 001

Fig. 8. Radial sensitivity of optical model calculations with potential P5 of table 1 at 100, 120 and 151.5 MeV 32S+4”Ca elastic scattering. The strong absorption radius and sensitivity radius are also reported.

TABLE

parameters using real potential

N

Set 100

100 120 151.5

1.03 0.96 1.57 1.15 0.90

double-folding real and the radius of

W (MeV)

4.67 9.22 10.74

comparison between double-folding real

X2/N

(Iii,

1.48

1.30 1.30 1.30

4

0.470 0.507 0.334 0:446 0.507

8.0 3.7 4.8

standard WoodsNQ(R,) at

-

)

(fm)

(MeV)

10.40

10.40 10.05 9.25

2.18 5.74 1.45 2.18 5.74

I 44 1.66 5.04

424

1515 MeV-Lab

I

30

Fig. 9. 3’s +“OCa angular

50

70

90

110 8 cm

distributions at labelled energies together predictions for parameter set I of table 4.

(deg)

with the folding-model

nucfeons. Neutron and proton densities were assumed to be equal because there exists evidence that this holds for nuctei of the sd shell with M = Z [ref. ““>]. An attempt was made to fit the experimental angular distributions at each energy by varying the three imaginary potential parameters, keeping the renormalization factor N fixed to 1. However, we only obtained good fits for the data at 120 MeV and 151.5 MeV. We subsequently tried to fit the data by varying also the renormalization factor N. The results are shown in table 4 (set I) and in fig. 8. A renormalization coefficient larger than 1 is necessary at 100 MeV. At the same energy a value of N = 1.26 was obtained by Satchler ‘) by fitting the data of Richter et aE.4), which only extend up to 100” c.m., and using different n&ear densities, but his calculations were a

A. Baeza es ai. / 40Ca(32S, “Sj

425

in to accurately both oscillations about the Rutherford cross section and the fall-off in the shadow region. A further attempt was made to fit our three angular distributions by fixing the imaginary reduced radius to the value yi = 1.3 fm used in ref. 5). The results and quality of fits were essentially the same as those obtained by varying the four parameters, as seen in table 4 (set II), and the corresponding curves are visually indistinguishable from that presented in fig. 9. To confirm the anomaly observed in the renormalization coefficient, we have fitted the data of Gutbrod et al. 3, at 80 MeV, 85 MeV and 90 MeV and we have obtained renormalization factor values of N = 1.5, N = 1.6 and N = 1.8 respectively, which are consistent with the results obtained by us at 100 MeV. A possible cause of the energy dependence found could be the neglect of density dependence in the interaction. In order to check this, we have used the DDD interaction ’ ) : v(r, p) = r~l(r)+t~2(r)e-oP,

with e-2

L:,(r) = 6839:

-18872.5

+(r) = 6893q

- 1938 ___ e

5r

P

MeV,

-2.5

2.5

MeV,

and b = 41.4 fm3. The values obtained for the renormalization factor in this case are 1.72, 1.28 and 1.35. Thus the N energy dependence remains. In order to obtain further insight on the role played by the finite size nucleon charge distribution and the pseudopotential 6(r), we made two additional fits, with and without the 6(r) term, taking the nucleons as pointlike. In both cases good fits were obtained with an energy-dependent renormalization coefficient N = 1 and 100 MeV, N = 0.684 at 120 MeV and N = 0.662 at 151.5 MeV for the first case and N = 1.4 at 100 MeV, N = 1.0 at 120 MeV and N = 0.94 at 151.5 MeV for the second case. The anomaly remains and the different corrections only produce a uniform shift of the renormalization coefficient. In fig. 10 some best-fit Woods-Saxon potentials and the renormalized folding potential that we obtained at 100 MeV (set II in table 4) are compared. Although they are different in the inner region, all these potentials are very similar in the sensitivity region defined in sect. 3. The values at the radius of sensitivity of different Woods-Saxon potentials R, are given in table 4 for the three energies we studied.

A. Baeza et al. / 4oCa(32S, 32S)

426

-V(r) (MeV: I....,

..**

10:

..

FOLDING **.a.* *.*. ‘*.* ‘*** “., ‘t.

PL ---__---_______+__

Pl

--~---_

5 a* 5 :

to2

10’

*** ‘.

\ i

t

‘.

\

t

\ \

\

P5

too

L

12

r(fm)

Fig. 10. Comparison between the tad regions of the real parts of the best fit Woods-Saxon potentials (reported in table I) and the renormalized double-foldmg potential (set II m table 4) fitted to the elastic scattering of 100 MeV 32S by “Ca.

5. Conclusions Precise measurements of the elastic scattering of 32S on 40Ca were made at incident energies near to and above the Coulomb barrier. In the angular range explored, which corresponded to g/crR 2 10v3 at 100 and 120 MeV. and g/gR 1 lob4 at 151.5 MeV, no oscillatory structure was observed. in spite of the a-like character of both projectile and target 23). A different region of the optical potential was determined at each energy by the available angular range of data. These regions of sensitivity present three remarkable aspects: (a) The unambiguously obtained radii from the Woods-Saxon potential 0%~ and R,) lie inside these regions at each energy. (b) They move in by only 0.5 fm when incident energy increase by 50%. (c) R, moves faster than R,, inside these regions when the energy increases. Equivalent fits were obtained using both the Woods-Saxon and double-folding forms for the real part of the optical potential, and a Woods-Saxon form for the

A. Baeza et al. / 40Ca(32S, 32S)

427

imaginary part. Both kinds of potentials agree very well in the sensitivity region and are surface transparent, although they are very different in the interior. While at 120 MeV and 151.5 MeV, the M3Y interaction seems to be adequate (a renormalization factor of N - 1 was obtained). at 100 MeV a value of N = 1.57 was necessary to fit our data. We found also values greater than one when fitting the data of Gutbrod et al. 3). As our data do not present any oscillatory structure in the measured angular range, it is difficult, at first sight, to attribute this energy dependence of the renormalization coefficient to other processes interfering with the elastic scattering in the shadow region. Another possible reason could be the energy independence of the nn interaction at low energies. The M3Y interaction, howeve’r, has proved to be adequate at energies near the Coulomb barrier 5). Our interpretation is that there exists a significant contribution of the real part of the polarization potential. increasing the strength at the nucleus surface of .the effective potential near the Coulomb barrier. From the results without the 6(v) term, we see that the contribution of the polarization potential could be simulated using a strong energydependent 6(r) term. The authors of Strasbourg (Valencia) and CAICYT and support.

are indebted to I. Linck, L. Kraus and the technical staff of CRN for their expert assistance during the experiments, to the UPV JEN (Madrid) for the access to their computing centers, and to the the Cooperaci6n Cientifica Hispano-Francesa for their financial

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