Dynamical effects of the repulsive core in 54.7 MeV 16O + 28Si large-angle elastic scattering

Nuclear Physics A433 (1985) 495-510 0 North-Holland Publishing Company

DYNAMICAL EFFECTS OF THE REPULSIVE CORE IN 54.7 MeV I60 + **Si LARGE-ANGLE ELASTIC SCATTERING V. N. BRAGIN * and R. DONANGELO ** The Nlels Bohr Institute, University of Copenhagen, Copenhagen, DK-2100 Copenhagen Q, Denmark

Received 6 June 1984 Abstract: The inclusion of a short-range repulsive term in the real part of the optical potential has previously been shown to be needed to explain the 54.7 MeV I60 + “Si large-angle scattering data. The dynamical effects arising from this repulsive core are studied here. The changes in the angular distribution for this system brought about by the presence of the core are shown to be due to an increased reflectivity in the low-angular-momentum partial waves affecting mostly the nearside component of the scattering amplitude. From this study we draw inferences about the expected behaviour of the elastic scattering cross section for heavier systems where anomalous large-angle scattering is also observed.

1. Introduction

The cdllision between two heavy’ions can proceed in many different ways. Two colliding nuclei may maintain their internal structure while being elastically scattered. If the incident energy is high enough they are much more likely to undergo excitation, transfer or even break-up processes. Eventually they might enter into close contact creating either a compound nucleus or a dinucle’ar system with suppressed memory of their initial structure. In most heavy-ion collisions all these possibilities are realized. This is the reason why a typical elastic scattering angular distribution has a diffractional pattern with a rapid falloff of the cross section at large angles. One immediate explanation of this experimental fact is that for most colliding nuclear systems very strong absorption of partial waves occurs just at the edge of the target nucleus. Thus any analysis of heavy-ion elastic scattering in the presence of strong absorption would provide us with only limited knowledge about the optical potential since it cannot probe the potential deeper than it is allowed by the “strong absorption radius”, The surprising observation of anomalous large-angle scattering (ALAS) in the 160 + “Si system ’ ) prompted many experimental investigations into this * Permanent address: I. V. Kurchatov Institute of Atomic Energy, Moscow, USSR. ** Permanent address: Instituto de Fisica, Universidade Federal do Rio de Janeiro, Rio de Janeiro, RJ, Brazil. 495

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phenomenon in heavy-ion collisions with several theoretical models king put forward in an attempt to find an authentic explanation for ALAS (see BraunMunzinger and Barrette 2, and references therein). The subject still continues to attract attention 3-11). Although there is as yet no unique and widely accepted understanding of ALAS it seems well established that it can hardly happen in the presence of strong absorption. So one has to admit that strong enhancement of the scattering cross sections at large angles has to be related. to some process(es) taking place at comparatively short distances between the centers of mass of the colliding nuclei. Thus ALAS carries valuable information about either the optical potential at small separations or about the intermediate system created when the two nuclei start to overlap, A proper choice between the two above-mentioned alternatives (it might be their combination as well) seems to be one of the most important problems in ALAS at the time. When considering heavy-ion scattering in terms of the optical model we have only one dynamical variable: I, the distance between the centers of mass of the nuclei. Only the elastic channel is treated explicitly, with all others being taken into account through the optical potential. If absorption is strong enough we are free to make various assumptions about the optical potential. Its behaviour and even validity at short distances are of minor importance in this case provided that a good guess has been made in the grazing distance region. Application of the optical model to ALAS brings about very important changes. The potent.ial is surface transparent now, and a very large mean free path must be associated with this weak absorption. To be consistent with the basic ideas of the optical model one must admit that in this case there is quite a notable probability for the projectile to plunge deep into the target nucleus and turn up again after a while in one piece. No matter how fascinating this picture might be it is very difficult to imagine the nuclei remaining in their ground states after such an encounter. As there is no way to overcome the above-mentioned contradiction with the help of absorption one has to look for a way out in the refractive or reflective features of the real part of the potential. If we persist in applying the optical model to ALAS with surface transparency we ought to reconsider the behaviour. of the potential at short distances. When the two nuclei approach each other and start to overlap, a very strong resistance has to develop to prevent their interpenetration and to maintain the nuclear shapes unchanged. The most natural way to simulate this in the framework of the optical model is to introduce a short-range repulsive core 3*12-16) into the real part of the optical potential. In the following section we discuss how such a repulsive core can be estimated and parametrized and how it can help in better understanding the experimental angular distribution for I60 + 28Si elastic scattering. The third section is devoted to the analysis of dynamical effects caused by the repulsion and their qualitative

V. N. Bragin, R. Donangelo / Dynamical effects

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influence on the angular distribution. A summary of the results and the main conclusions are given in the last section.

2. The optical potential The calculation of the potential acting between two nuclei is among the most important and challenging problems in nuclear physics at the present time. The main difficulties arise from the fact that the heavy-ion potential appears as a result of numerous nucleon-nucleon interactions and these same interactions determine the internal structure of the colliding nuclei. Thus the interaction between heavy ions originates from the solution to a many-body problem. Substitution of this by a one-particle optical potential is a strong assumption indeed. In particular, at short distances one must take special care about such many-body effects as the Pauli principle and the consequences of the antisymmetrization. We might learn how to do this in a proper way from the existing microscopic approaches to the theory of nuclear reactions. Here we refer to the genuine microscopical models based on an approximate solution to the many-particle Schrddinger equation with consistent treatment of the Pauli principle (the method of hyperspherical functions, RGM, GCM, etc). Although we can hardly expect to find the exact solution to this problem at the time it seems well established that all of these models predict similar qualitative behaviour of the heavy-ion potential 17-34). Namely, it has a shallow attractive well near the surface of the target nucleus and a strong repulsive core in the region where the nuclei start to overlap. It is only fair to mention that this idea is not a new one ; for example, the importance of the repulsion between two composite fermionic systems was discussed by Zeldovich as far back as 1959 [ref. ““)I. The above-mentioned microscopic models would give us important knowledge about the overall behaviour of the potential while the macroscopic models based on the sudden approximation are most suitable for obtaining some quantitative estimates. Here we refer to those models in which the Brueckner energy-density formalism or the density-dependent Skyrme interaction are applied 36-48). These approaches take the Pauli principle into account when constructing the energydensity functional or the density-dependent interaction. They predict qualitatively similar results although they lead to different estimates of the optical potential parameters. That is why in this paper we adopt the following method for extracting information about the optical potential for 160+ 28Si elastic scattering. First, we define a parametrization of the potential which is based on some of the above-mentioned macroscopic models and make first rough estimates for the free parameters. Then we adjust the parameters to get the best description of the experimental data. As in our earlier studies 3*15) we assume the following structure of the optical

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potential for 160 + ‘*Si elastic scattering:

WI = L.&) + V,(r) + I/co,,(r)+ i W(r), where core(r) is the short-range repulsive core, V,(r) is the nuclear attraction, V,,,,(r) is the Coulomb interaction, and W(r) is the imaginary part of the potential taking into account the effect of the inelastic channels. Using the results of the studies by Ng6 et al. 36,37), where the Brueckner energydensity formalism and the sudden approximation were applied to calculate the real part of the heavy-ion potential, we assume that the nuclear attractive potential has the following form:

-

‘Vexp[ - (r - R,)2/a$],

r > R, r -C R,,

(2)

where R, = r&4$+ A$), and A,,, are the projectile and target mass numbers, respectively. The depth V, diffuseness a,, and reduced radius rv are free parameters of the model. For the Coulomb interaction we use the potential of a uniformly charged sphere of radius Rcou, = R,. There are several reasons for our choice of the parametrization for the nuclear attractive potential V,(r). First, when calculating the heavy-ion potential the authors of refs. 36*37)use d an expression for the energy density E(C) which leads to the correct values of the binding energies for nuclei from I60 to 238U within the framework of the Brueckner theory and also nucleon density distributions which are in good agreement with the experimental data on electron scattering. Second, the potential from 36*37) provides very reasonable estimates for the interaction barriers for many pairs of nuclei, and these estimates were found to be in good agreement with the experimental ones. Finally, the potential from 36,37) is consistent with the predictions of the “proximity force theorem” 43). For the numerous pairs of nuclei investigated in refs. 36,37) very stable and universal estimates of the above-mentioned parameters were found. Namely, the parameters determining geometrical features of the potential are almost constant: rv 1: 1 fm and a, N 1.9 fm, while the depth of the potential changes regularly as a function of the mass numbers:

where V, N 30 MeV is also a universal value. All these arguments persuade us that estimates (2) and (3) might be considered as both reliable and physical. As in our earlier studies 3, “) we assume that the repulsive core as a function of the separation between the centres of mass of the nuclei is proportional to the volume of their overlap: (4)

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V. N. Bragin, R. Donangelo 1 Dynamical effects

where I”,,, is the height of the core and the form factor &_,&) is given by the following expression derived under the assumption that the nuclei are spherical and homogeneous :

1,

f,,,,(r) =

x < xg

I

0.75a(a+ l)(l -x)2x-1[1 0,

-d(l -x)(1 +x/3)],

x, 1,

(5)

where x = r/Rcore is the ratio of r to the core radius R,,,, = R,, a = RJR, > 1 is the ratio of the target and the projectile radii, R,,, = r,At,,, d = a(a+2+ l/a), and x0 = (a- l)/(a+ 1). From the above expression one can immediately see that j, > 1 correspond to iralues of r > R, = R,+ R,. There is no repulsion between the nuclei in this case as they do not overlap, b&Jr) = 0. The values x < x0 or r < R,- R, correspond to complete overlap in which case the repulsion achieves its maximal value, V,,,,. The height of the core V,,,, is assumed to be a free parameter of the model. Anyway it can be roughly estimated as Vcore = *A,K,

(6)

where K N lOCL300 MeV is the bulk modulus of nuclear matter [see ref. 3, for details]. For the imaginary part of the optical potential we apply the conventional Woods-Saxon expression W(r) = -W[l+exp((r-R,)/a,]-‘,

(7)

where R, = r,(A$ + A#), and w rw, and a, are free parameters of the model. Thus the optical potential for the elastic scattering 54.7 MeV I60 on 28Si, as defined by the above formulas, has seven free parameters: V,,,,, r! rv, and a, for the real part, and w rw, and a, for the imaginary part. The starting values for the parameters V, ry, and a,, listed in table 1, were calculated according to the results of Ng6 et al. 36*37). A simultaneous search on all of the seven parameters was undertaken to get the best possible fit to the experimental data ‘). This was done with an appropriately modified version of the optical model code GENOA [ref. “)I. TABLE1 The parameters of the optical potential for 54.7 MeV I60 + **Si elastic scattering K,, phenomenological calculated

CMeVl 78.19

V [MeV]

F” [fm]

av [fm]

W [MeV]

rw [fm]

a, [fm]

45.3175 41.3126

0.9733 1.0000

1.7515 1.9245

12.8154

1.0565

0.5020

V. N. Bragin, R. Donangelo / Dynamical effects

500

The final values of the parameters are listed in table 1. Comparison of the calculated and experimental angular distributions is discussed in the following section. We would like to finish this part with the remark that the parameters obtained from the search are in good agreement with the starting calculated. ones. The reduced radii ry almost coincide in both cases. The depth of the phenomenological potential is a little larger than that of the calculated one, but this is well compensated at the potential surface by the somewhat smaller diffuseness a,. 3. Dynamical effects of the repulsive core Once we have decided that the presence of a repulsive core in the heavy-ion potential is an expected feature of that potential, and some of the physical reasons of its existence have been discussed above, the question of how does this core affect the dynamical behaviour of the scattering system comes naturally to mind. To untangle the effects of the repulsive core from those of the other terms in the optical potential several means are at our disposal. We can thus study, as a function of the core height, the classical deflection function, the transmission coefficient I&I, and, of course, the actual elastic cross section. In relation to this last the picture may be considerably clarified if we decompose it into its near and far contributions, since much of the structure in the elastic cross section comes from the interference between these two terms. As the near/far decomposition has not yet become a standard tool in data analysis we proceed to give a short summary of its basic formulas and ideas. Hussein and McVoy have recently presented a report 50) dealing extensively on the application of this decomposition and we refer to it for further details. The near/far decomposition of the elastic scattering amplitude,

can be viewed as a separation of the contributions arising from the refraction and diffraction of partial waves coming nearest to and furthest from the detector (see fig. 1). The expressions for these components are 50) fn(e) = &I

f,(e) = &5X (21+ l)S,E+)(cos@, I

I

(21+ i)s,Qj-‘(case),

(9)

where S1 is the usual S-matrix element and or*)

= +[P,(COS

0) f (2/7d)Q,(COS

e)]

with P, and QI being the Legendre polynomial and the Legendre function of the

V. N. Bragin, R. Donangelo / Dynamical effects

501

DETECTOR

A

with

core +

nearside

core

I I L

without

core

t

I Fig. 1. Schematic representation of the classical trajectories contributing to 90° elastic scattering. The full circle indicates the range of the repulsive core, the dashed one that of the nuclear attraction. The trajectory labehed “farside” corresponds to a deflection angle 0 = -90” and is seen not to touch the core. The “nearside” trajectory with smaller impact parameter is reflected by the core out of the most absorptive region. The dashed line shows how the same trajectory would look but for the presence of the core.

second kind, respectively. The difficulties with the Coulomb part in S, have been solved by Fuller 51) who has calculated the near and far components of the scattering amplitude for the point Coulomb potential case in closed form. An immediate consequence of eqs. (9) and (10) is that fr(fI) =fN( -0) which, consistently with the picture suggested by fig. 1, leads to the semiclassical statement that fFSN(0) are the contributions to the scattering amplitude due to trajectories which have negative and positive deflection angles, respectively. These two components are thus expected to probe the heavy-ion potential in different ways. A

502

V. N. Bragin, R. Donangelo / Dynamical

effects

careful look at the near and far cross sections rrN,r = IfN,r(@lz and the deflection function 8 = 8(b)’ might then shed considerable light on the way the optical potential is working. In the particular analysis of the dynamical effects of the core we are envisaging, we should especially consider cases where these effects are not hidden from view by the absorption. The most favourable systems for studying the repulsive core are those for which the imaginary part of the optical potential has retracted inside the real part, leading to a surface-transparent potential. The last one is a wellestablished feature of the scattering system ‘60+28Si at E = 54.7 MeV. Thus we can expect that in this case the deeply penetrating partial waves can probe the core. Starting from the lit to the experimental angular distribution obtained in the previous section we investigate the effect of the core by varying its height and looking at the resulting changes in the normal (o(e)) and near/far (o,,,(f3)) cross sections, in the classical deflection function and in the reflection coefficient. In figs. 2 and 3 the elastic cross sections calculated for a variety of core heights (V,,,, = O-400 MeV) have been plotted together with their corresponding near/farside decompositions. All of the other parameters of the optical potential have been kept constant in these calculations (as they are listed in table 1). If there is no core (V,,,, = 0) the calculated angular distribution has a regular diffractional pattern which comes from the Fraunhoffer crossover of the near and farside components near 0 N 90”. In this angular region both components are very close to each other. Thus we can observe here the expected large oscillations which become less pronounced as we move to larger angles (0 N 135”). At the most backward angles again we can see very large oscillations since the near and farside components make comparable contributions to the scattering amplitude. In the entire angular region 90° < 0 < 180” the cross section is scaled by the farside component while the nearside one brings about the interference which is responsible for the oscillations. The structure of the oscillations is regular and it is strictly related to the value of the grazing angular momentum (I, N 24). While reasonably good either at low (0 < 90°) or large (0 > 150°) angles, this calculation fails to reproduce the amplitude of the oscillatory structure at intermediate angles. Consideration of the near and farside components indicates the small nearside contribution in this angular region as the main reason for not having a strong near-far interference leading to larger oscillations. If we now include in the optical potential a repulsive core of height V,,,, N 60-80 MeV, this will bring about strong changes in the calculated angular distributions. We can see from fig. 2 that in this case the farside contribution does not change much. It becomes a little larger, and a weak oscillatory structure appears in it at the most backward angles. Meanwhile the core raises the nearside contribution significantly. A classical illustration of this behaviour is given in fig. 1. As the near and farside contributions approach each other their interference causes notable

503

V. N. Bragin, R. Donangelo / Dynamical effects

changes in the calculated

angular

appears in the angular

range

calculated

describes

cross section

distributions.

Very irregular

oscillatory

90” < 8 < 180”. For core heights the oscillation

amplitudes

structure

in this range

the

much better than when

the core is absent. We consider it important to underline here that although the elastic scattering angular distribution is rather sensitive to the repulsive core height, it changes very continuously with the value of this parameter. For example, when we change the core height from I&,, N 60 MeV to 80 MeV the calculated angular distribution varies smoothly: no sudden fluctuation-like wiggles arise. The Iit to the data remains quite stable. From the estimates (3) and (6) given above one can see that the ratio I&,/V,, which controls the pattern of the calculated angular distribution, is likely to become larger as the masses of the colliding nuclei increase. It is then possible to investigate what we expect for heavier colliding systems by further increasing the core height in the ‘60+28Si case. First, we outline the region of the core heights I&,, 2: 110-140 MeV. As can be seen in fig. 3 the near and farside contributions closely approach each other throughout the angular region 0 v 100°-180°. The calculated angular distribution becomes rather regular with very pronounced oscillations arising in this backward hemisphere (compare this with the case Vcore = 0). As we proceed to increase the core height and set V,,,, > 150 MeV we obtain quite a new pattern for the angular distribution. The ratio of near and farside contributions to the large-angle scattering amplitude is reversed. The average behaviour of the cross section in this area is controlled by the nearside component now, while the farside one causes most of the oscillatory structure. Besides the dependence of the angular distribution pattern on the ratio V,_,,,/VN it is very sensitive to the strength of the absorption. This can be easily seen from fig. 4 in which several angular distributions calculated without absorption (W = 0) are plotted together. In this case the near and farside components have reversed their ratio already at the core height VcOre= 70 MeV. Very large oscillatory cross sections arise in the backward hemisphere. At the same time we can see pronounced oscillations in the forward-angle region which differ very much from the results of calculations with absorption. In fig. 5 we have plotted the reflection coefficients IS,1 calculated for several core heights. The main change due to the repulsive core is a large increase in the reflection coefficients at the lower angular momenta. The core also gives rise to the more pronounced fluctuations in the angular momentum dependence of IS,1 in the region 1 < 1, which were shown to be necessary for the existence of ALAS 52). Fig. 6 illustrates the changes in the classical deflection function due to the presence of the repulsive core. We immediately see that the most important changes take place at lower impact parameters. These changes are reflected especially in the large-angle cross section since in the presence of the core we have classical trajectories with low impact parameters contributing to large-angle scattering. Without the core there are none of this kind. This implies that the

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V. N. Bragin, R. Donangelo / Dynamical effects

r”“““““““1 I60 + **Si

for different Fig. 2. Angular distributions for the elastic scattering of 54.7 MeV I60 + ‘*Si calculated cross sections (full curves) as values of the core height V’,,, = O-100 MeV. Shown are the calculated well as then nearside (dashed) and farside (dot-dashed) contributions. The data points are taken from ref. I).

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V. N. Bragin, R. Donangelo / Dynamical effects

MeV

54.7

*

0

I I 30

8 I. 60

. 1. 90 Srn

8 18. 120

I”1 150

160

(ded

Fig. 3. The same as in fig. 2, but for the core height

V,,, = 125400

MeV.

angular region 80” -C 8 < 150°, which is the most sensitive to the value of the core height, is the one where it should be most easy to read the information carried from the interior of the optical potential by these not totally absorbed lower angular momentum partial waves.

V. N. Bragin, R. Donangelo / Dynamical effects

506

0

l.~l.,l,.i..l..l] 30 60

90 &.

120

150

160

(de@

Fig. 4. The same as in figs. 2 and 3, but with no imaginary part in the optical potential.

It is necessary to mention that although most of the oscillatory structure seen in figs. 2 and 3 is due to the near/far interference, it is also obvious that both the near and far contributions have an oscillatory structure of their own. The nearside oscillations at small angles (0 < 40”) take the well-known Fresnel diffractional

V. N. Bragin, R. Donangelo / Dynamical effects

507

Fig. 5. The reflection coefficients (S,I calculated for different core heights leaving the other parameters of the potential unchanged.

pattern which is produced by the mutual interference of waves scattered through positive angles. The nearside oscillations at large angles appear because of interference between different terms of the Poisson representation of the nearside scattering amplitude, more precisely those corres~nding to the m = 0 and - 1 components 53). In the absence of a core the farside cross section is seen to be relatively structureless. The observation of increasing oscillations in the farside component as the core height increases is consistent with the larger reflection coefficients for the low partial waves shown in fig. 5. This makes it possible for the most internal of the two trajectories contributing to the farside cross section indicated by figs. 2 and 3 to be partially reflected and to interfere with the outermost one. 4. Conclusions A short-range repulsive component in the real part of the nucleus-nucleus potential should be a general feature of the interaction between two elastically

508

V. N. Bragin, R. Donangelo / Dynamical effects

-60

-

b (fm) Fig. 6. Effect of the repulsive core on the classical deflection function. The impact parameters labelled by b, _3 correspond to the classical trajectories with deflection angles 0 = k90”.

scattering heavy ions to prevent strong mutual overlap of their internal wave functions as they enter into close contact. It is, though, only for systems with anomalously weak absorption in the surface region that the effects arising from the repulsive core can be detected. Thus we have chosen the 160+ 28Si system to study the dynamical effects associated with the presence of this repulsive core in the optical potential. The height, radius and difuseness of the core were estimated by fitting the calculated angular distribution for 54.7 MeV ‘60+2*Si elastic scattering to the experimental one. The nuclear attractive part of the potential was found to be in good agreement with the results of calculations performed within the framework of the Brueckner energy-density formalism and the sudden approximation. The inclusion of the core in the optical potential was seen to drastically change the pattern of the calculated cross section and appears to improve the overall agreement with the experimental data. The decomposition of the elastic scattering amplitude into its nearside and farside components, together with consideration of the classical deflection function, provided information on the way these changes are brought about. The presence of the core was sho.wn to affect mostly the low-l partial waves by

V. N. Bragin, R. Donangelo 1 Dynamical effects

509

reflecting them into positive angles and out of the most absorptive region. Therefore, the nearside contribution to the scattering amplitude appears to be much more sensitive to the core parameters than the farside one. The transparency of the potential for the low-l partial waves is remarkably increased by the core. This indicates that the nearside component contains most of the information on the physical origin of ALAS probed by deeply penetrating waves. It may be pointed out that while the core height is likely to rise linearly with the projectile mass, the depth of the nuclear attractive potential is expected to change much more slowly. This implies that the elastic scattering angular distribution for a system such as 12C+ 28Si must have some resemblance to that of the I60 + 2sSi system, but with a smaller core height, while for heavier systems we might expect the pattern of angular distribution illustrated in fig. 3 for the larger core height provided that these systems remain surface transparent. It is worth mentioning that the recently obtained data on the 28Si + 28Si scattering s4) present strong oscillations in the angular distributions similar to those shown in fig. 3 for the case V’,,, N 150 MeV, and that the recent data on 28Si + 32S and 34S + 32S scattering 55) bear a striking resemblance to the angular distributions calculated with the core height I&,, > 200 MeV. We would like to thank Profs. D. P. Grechuhin, A. A. Ogloblin, V. I. Man’ko and M. V. Zhukov for their interest in this work, Profs. S. Bj@-nholm, J. Bondorf, R. A. Broglia, and A. Winther for discussions, and Dr. B. S. Nilsson for valuable advice. We thank the NBI staff for hospitality in granting us use of all necessary facilities to accomplish this work. V.N.B. acknowledges financial support from the University of Copenhagen. R.D. acknowledges financial support from the Brazilian Research Council (CNPq) and the Universidade Federal do Rio de Janeiro. References 1) P. Braun-Munzinger, G. M. Berkowitz, T. M. Cormier, J. W. Harris, C. M. Jacheinski, J. Barrette and M. J. Levine, Phys. Rev. Lett. 38 (1977) 944 2) P. Braun-Munzinger and J. Barrette, Phys. Reports 87 (1982) 209 3) V. N. Bragin, Sov. J. Nucl. Phys. 37 (1983) 722 4) V. Shkolnik, D. Dehnhard and M. A. Franey, Phys. Rev. C28 (1983) 717 5) S. Kahana, J. Barrette, B. Berthier, E. Chavez, A. Greiner and M. C. Mermaz, Phys. Rev. CZS (1983) 1393 6) S. Kahana, G. Pollarolo, J. Barrette, A. Winther and R. A. Broglia, Phys. Lett. 133B (1983) 283 7) G. Pollarolo and R. A. Broglia, Copenhagen preprint NBI-83-26 8) A. M. Kobos, G. R. Satchler and R. S. Mackintosh, Nucl. Phys. A395 (1983) 248 9) A. M. Kobos and G. R. Satchler, Nucl. Phys. A427 (1984) 589 10) M. C. Mermaz, E. R. Chavez-Lomeli, J. Barrette, B. Berthier and A. Greiner, Phys. Rev. C29 (1984) 147 11) L. F. Canto, R. Donangelo, M. S. Hussein and A. Lepine-Szily, Phys. Rev. Lett. 51 (1983) 95 12) K. A. Gridnev and A. A. Ogloblin, Sov. J. Part. Nucl. 6 (1975) 393 13) A. I. Baz, V. Z. Goldberg, K. A. Gridnev, V. M. Semjonov and E. F. Hefter, Z. Phys. A280 (1977) 171

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