Sub-Barrier Fusion and Elastic Scattering in S + Ni Systems
CRN/PN 90-23 • STRASBOURG! "^^^^^_H ^^_H I
Sub-Barrier fusion and elastic scattering in S + Ni systems
RJ. TIGHE, JJ. VEGA, Genbao UU, A. MORSAD*. and JJ. KOLATA Physics Department, University of Notre Dame, Notre Dame, In. 46556
S.H. FRICKE, H. ESBENSEN, and S. LTVNDOWNE Physics Division, Argonne National Laboratory. Argonne, IL 60439
P/jys. Rev. C (to be published)
Centre de Recherches Nucléaires, IN2P3-CNRS/Université Louis Pasteur, B.P. 20. F67037 STHASBOURG CEDEX, France
CENTRE DE RECHERCHES NUCLEAIRES
STRASBOURG
IN2P3 UNIVERSITE
CNRS LOUIS PASTEUR SUB-BARRIER FUSION AND ELASTIC SCATTERING in S+Ni SYSTEMS
RJ. Tighe, JJ. Vega, Genbao Liu, A.Morsad*. and J J.Kolata Physics DepL, Univ. of Notre Dame, Notre Dame, IN 46556
S.H. Fricke, R Esbensen, and S. Landowne Physics Division, Argonne National Laboratoiy, Argonne, IL 60439
ABSTRACT
32 The elastic scattering of S on 58.64^1 a^j fusiOn excitation functions for 32s+58,64i»ji and 34S^64Ni have been measured at energies near the Coulomb barrier. Our results differ in several important respects from previous measurements on these systems. Coupled- channels calculations which explicitly allow for collective inelastic excitations and single- nucléon transfer reactions simultaneously reproduce the main features of the new elastic scattering and fusion data. I. INTRODUCTION
In recent years there has been considerable interest in heavy-ion reactions at energies near to and well below the Coulomb barrier. Observations of enhanced low energy heavy ion fusion rates, as compared with the predictions of conventional barrier penetration models, have accelerated the exploration of how nuclear structure influences the fusion process. At the opposite end of the reaction spectrum, optical model analyses of heavy ion elastic scattering data are found to require strongly energy-dependent complex potentials in the region of the barrier. In particular, the magnitude of the imaginary part of the optical- model potential has been found to decrease, and the magnitude of the attractive real potential to increase, as the bombarding energy is lowered. It is natural to expect that increased low energy fusion rates should be correlated with increased attraction in the optical model potential1, since both phenomena reflect the dynamic polarizability of the colliding nuclei. The interpretation of these recent developments relies upon having a consistent body of data for both fusion and elastic scattering cross sections. The measurements of the S + Ni systems by the Legnaro group2»3 have played a prominent role in discussions of the physical basis behind the observed phenomena. An analysis of the 32St58^64Ni elastic- scattering data presented in Réf. 2 tried to directly correlate the energy dependence of the optical model potentials with the corresponding low-energy fusion data3 using an energy- dependent barrier penetration model as suggested in Réf. 1. However, the theoretical basis of this procedure has been questioned in Réf. 4. A later analysis of the 32St5^64Ni elastic scattering and fusion data was carried out by Udagawa, et al.5 in the context of an absorption model, and a recent discussion of this particular analysis is given in Réf. 6. Quite recently7, the Legnaro group has published tables of their 32St58^64Ni elastic scattering data. There are, however, some unusual features of the 32St58^64Ni data presented in Refs. 2 and 3 which must be clarified before one attempts to draw conclusions from them. In the former case, elastic scattering angular distributions for 32S + 58-64Ni, near to and below fhe Coulomb barrier, are characterized by strong deviations from Rutherford scattering at angles that are well forward of the expected "grazing" angle. For example, the 32 58 experimental S + Ni elastic angular distribution at E0n = 56.7 MeV (4 MeV below the barrier) begins to deviate from Rutherford scattering at 9crn = 80°, whereas "normal" nuclear potentials predict little or no deviation even at 8cm = 160°. Coulomb excitation of low-lying excited states of the target and/or projectile, which might conceivably provide a mechanism for explaining part of the observed behavior8, is in fact far too weak to make a significant impact on the predictions. As emphasized in Réf. 5, this seems to imply that an anomalously large, unknown direct reaction process is occurring in these systems. With regard to the fusion channel, we note that the barrier parameters extracted from the analysis presented in Réf. 3 do not follow the systematics as closely as one might expect, nor do they show the same sensitivity to nuclear structure effects that we have found in detailed 9 10 studies of the ^Cl + 58.60.62.64Ni ^112?Ai + 70.72,73.74,76Ge systems - in the same general mass and charge regime. In order to gain a better insight into the behavior of the S + Ni systems, we have made new measurements of the elastic scattering and fusion channels near the barrier. Our measurements disagree with the previous results in several important respects, and in fact do not show the unusual features noted above. The new elastic scattering and fusion data have been simultaneously analyzed in the context of a coupled-channels barrier penetration model, and a good overall description of the data is achieved. The results of the experiments, presented in the next section, are followed by a discussion of the theoretical calculation.
II. EXPERIMENTALMETHODANDRESULTS
A. Elastic Scattering
As mentioned above, the low-energy elastic scattering results of Réf. 2 could possibly be interpreted5 by invoking a significant flux-loss to unmeasured reaction channels. We therefore used a kinematic coincidence technique, which should be a sensitive way to search for strong inelastic and/or transfer channels that might be responsible for the observed anomalies. Two sets of two silicon surface-barrier position sensitive detectors (PSDs) each were placed on either side of the beam at a distance of 131mm from the target. Both detectors of one set were collimated using an array of six slits, each 2.4mm wide by 12.7mm high and separated from the adjoining slit by a center-to-center distance of 9mm. Thus, twelve angles could in principle be measured simultaneously with a single setting of the detector array, but the angle of incidence on the outermost slit of each detector was such that it did not give reproducible results and only ten angles were simultaneously measured. The two-detector array was oriented such that the plane of the collimator was perpendicular to the line from the target to the center-point of the array, and the length of this line was 131mm. The solid angle subtended by each slit therefore varied according to its distance from the target, and also because of the changing orientation of the normal to each slit relative to the target direction. The two PSDs on the opposite side of the beam were uncollimated. Each was 52mm long by 15mm high, and this array was also perpendicular at its centerpoint to the line to, and at a distance of 131mm from, the target. The coincidence efficiency in this geometry can be computed from kinematics. However, die elastic-scattering angular distribution was in fact determined from the singles rather than the coincidence counting rate so that efficiency corrections were not needed in this case. The center-of-mass (cm) angular range from 40° to 160° was covered with four settings of the detector arrays. There was considerable overlap between detector angles measured at these settings, so that many normalization checks were possible. We were also able to obtain elastic scattering information by de;ecting the recoiling Ni ions in many cases. The data set typically included 45-50 individual measurements at each beam energy. The targets consisted of 187 jig/cm2 58Ni and 160 jig/cm2 64Ni self-supporting foils. The isotopic purity of the 58Ni foil was >99.8%, while that of 64Ni was 97.3%, with 1 % 58.60NJ and 0.7% 62Ni contaminants. Further discussion of the determination of the thickness and isotopic purity of the targets can be found in Réf. 9. The beam current was collected in a magnetically-suppressed Faraday cup. The absolute elastic-scattering cross section was obtained by normalizing to Rutherford at the most forward angles. In all cases, the ratio to Rutherford was found to be very close to unity over the cm range from 40° to 80°, so that at least 15 data points could be used to determine the normalization at each energy. The elastic-scattering angular distributions obtained in this experiment are compared with the results of the previous work, as tabulated in Réf. 7, in Fig. 1. Consider, for 32 58 example, elastic scattering in the St Ni system at Ecn,= 56.0 MeV (corrected for energy loss to the center of the target). In marked contrast to the Legnaro results at nearly the same energy, no deviation from Rutherford scattering beyond the experimental uncertainty is observed over the entire cm angular range from 40° to 160° covered in the present experiment. 32 Strong population of the first excited states of S or 58,64Nj v{a inelastic scattering was not evident at any angle or beam energy and for either target, except for the two most backward angles at the highest energy measured for 32S + 58js|i. On the other hand, from the latter two data points it was determined that the experimental energy resolution (1.2 MeV FWHM) was marginal for separating thes? HVO states from the clastic yield. Therefore, we compare our data (Fig. 1) with theoretical predictions for true elastic scattering (dashed curves), as well as with the sum of elastic scattering plus inelastic scattering to the first excited states of 32S and 58.64Ni (solid curves). A detailed discussion of these calculations is given in the next section. Not surprisingly, given the observation of essentially pure Rutherford scattering for 32 58 St Ni scattering at E61n= 56.0 MeV in Fig. 1, we were also unable to find any transfer channel with measurable cross section at this particular energy, which is five MeV below the Coulomb barrier. Some transfer yield could be ascertained for 32S + 64Ni at die highest energy studied, but no evidence was found for die existence of anomalously strong direct reaction processes in die systems studied. For each elastic angular distribution shown in Fig. 1, there is disagreement between our data and the results of the Lcgnaro group7, which in general fall from Rutherford much foster at backward angles. Nb realty satisfactory explanation for these discrepancies exists. In the present experiment, ten angles were measured simultaneously, extensive overlap measurements were taken from one angular setting of die detectors to die next, and recoil cross sections (corresponding to scattering to the opposite side of the beam) were also sometimes obtained.
B. Fusion Measurements.
The evaporation residues (ER) from fusion are emitted in a narrow cone within a few degrees around the beam axis, and their cross section is relatively small at energies below the barrier. Therefore, direct detection of residues becomes difficult in the presence of a large background arising from slit scattering and other similar types of events. To accomplish the separation, the evaporation residues emerging from the target were deflected out of the direct beam by means of an electrostatic deflector. The value of die potential applied to the electrode plates was selected so as to maximize the yield from each target. The separated residues were then identified in a time-of-flight (TOF) and energy spectrometer, which consisted of a microchanncl plate and a silicon surface barrier detector (SSB) which together defined a Im flight path. The whole apparatus (TOF arm, electrostatic deflector and target chamber) was rigidly attached and designed to rotate as a unit, making it possible to measure the angular distribution over ±10° with respect to the beam. For the systems studied in the present experiment, and in the energy range we are interested in, the ER cross section may be equated with the total fusion cross section since fission of the compound nucleus is negligible. Absolute differential cross sections were obtained by normalizing the ER yield to the elastic scattering cross section at forward angles, which is purely Rutherford. The elastic yield was measured with an array of four monitor counters symmetrically distributed with respect to the beam axis and at a laboratory angle of 15°. With these monitors, both die beam position on target and the product of the integrated beam current times the target thickness can be determined with a high degree of accuracy (1%). A detailed discussion of the superiority of this four-monitor normalization procedure is given in Réf. 11. The transmission probability of the ER through the recoil velocity spectrometer was determined empirically by elastic scattering of ions of similar atomic and mass numbers, and kinetic energy. To accomplish this, we measured die Rutherford scattering of 10^Rh ions on 60Ni at bombarding energies of 42,39, and 36 MeV and at a laboratory angle of 9.75°. These three energies cover the range of interest for the recoiling compound nuclei of the fused systems. In order to investigate how strongly the transmission depends on the mass of die analyzed products, we also used a 81Br beam at 45 and 42 MeV. No measurable mass dependence was noted over this range. We define the transmission probability T as the ratio of the number of particles detected at the SSB detector to that initially traveling within the 58 nsr solid angle determined by the entrance slit of die TOF arm. The empirical value was determined to be T = 0.780 ± 0.045, measured at the deflection potential corresponding to the peak yield. The magnitude of this potential was consistent between the fusion-product and elastic-scattering experiments, indicating that the charge-state distributions were very similar. The experimentally measured transmission probability was confirmed by measuring the 3SQ + 58Ni reaction, which was previously analyzed by Scobel, et a/.12 using a different experimental technique. For this purpose, angular distributions Of35Cl -f 58Ni evaporation residues were obtained at cm energies of 60.8,61.4, and 62.0 MeV (corrected for energy loss in the target). The values obtained for the integrated fusion yields were 13.8+1.4 mb, 23.2±2.5 mb, and 36.7±3.8 mb, respectively, in good agreement with the values of 14.4±2.9 mb, 28.9±4.3 mb, and 44.2±5.5 mb interpolated from the cross sections reported in Réf. 12. As a final check on our transmission determinations, a Monte Carlo model was used to simulate the performance of the spectrometer under different operating conditions. A detailed report of this study will appear elsewhere. Here it can be said that, as a result of this theoretical analysis, it is known that the transmission is not strongly dependent on any parameters other than the electrostatic rigidity E/q of the ion and the voltage on the deflector. Fortunately, even the latter dependence is weak and the transmission is stable against ±5% deviations in the applied voltage. In a preliminary experiment, it was determined that the shape of the ER angular distributions did not change appreciably over the energy range of interest in this work, so that it was sufficient to measure them at convenient energies. Typical angular distributions are shown in Fig. 2. Excitation functions for all systems were measured at an angle of 6lab - 3°, which is near to the maximum of the do/d0 angular distribution. In order to convert this single angle excitation function into a total fusion yield, the ER angular distributions were fitted with a Gaussian function and then integrated to get the total fusion cross section at one energy, which could be used to deduce the absolute cross sections at other energies. Beam energy losses in the targets were corrected for by an iterative procedure, taking into account the slopes of the excitation function. At each step, corrected beam energies were obtained by weighting the energies from the previous step by the experimental fusion cross section, and averaging over the energy loss in the target. This process was repeated until self-consistent results were obtained. The targets used in this experiment were the same as those used for the elastic scattering measurements described above. The systematic errors associated with the determination of the absolute fusion cross section are estimated to be 8%. The principle sources of error are: 6% from the uncertainty in the transmission measurements, 3.5% from error introduced in the reduction of single- angle cross sections to total cross sections, and 4.5% due to angular uncertainty in the positioning of the monitor counters. The total fusion cross section measured for the three systems are listed in Table I, and plots of these data are shown in Fig. 3 compared with the results of other work. The error bars displayed on our measurements do not include the systematic error in the total yield. The fusion excitation functions shown in Fig. 3 for 32S
+ 58,64Ni an(j for 34s + 64^i do not agree with the published results of the Legnaro group^, but are in excellent agreement with the earlier fusion data of Gutbrod, et al. ^ for
III. COUPLED-CHANNELS CALCULATIONS
In this section, we describe the coupled-channels calculations which are compared with our S+Ni elastic scattering and fusion reaction data. These calculations are similar in scope and methodology to those recently carried out for Ca+Ca systems14. The model consists of a real, energy-independent ion-ion potential and explicit couplings to inelastic excitation and single-nucléon transfer reaction channels. The fusion process is simulated by imposing ingoing-wave boundary conditions in all channels at a separation distance inside the Coulomb barrier. In this way, the main reaction processes are accounted for without introducing phenomenological imaginary potentials. The "rotating frame approximation" is used to make the coupled-channels calculations more tractable. In this approximation, one neglects die change in the centrifugal potential in a reaction channel with respect to the elastic channel due to angular-momentum transfer. This allows one to reduce the number of sub-channels to just one for each spin state. See Réf. 14 and references therein for further discussion.
A. Parameters of the Calculations
We will now specify the various parameters which enter into the present set of calculations. They define the nuclear ion-ion potential, the macroscopic coupling parameters for inelastic excitations, and the spectroscopic information for the single-particle transfer form factors. The ion-ion potential is based on a semi-empirical parametrization of folding-model potentials used in fits to heavy-ion elastic scattering data (see Refs. IS and 16). The formula given in Réf. 16 is:
r R(Al A2) Un(D --31.67 J^iVSyjl + exp( - ' f MeV
where: R(A) = (1.233A!/3 - 0.98A-1/3 ) fm (2)
and:
R(Ai A2) = R(Ai)+ R(A2)+ AR. (3)
The diffuseness parameter is a = 0.63 fm and the nominal value of AR given in Réf. 16 is AR = 0.29 fm. For the present set of calculations, the value AR = 0.20 fm has been used. This was determined by adjusting AR such that the full coupled-channels calculation gave a 2 reasonably good description of the 3 S+^Ni elastic scattering data at Ecm=62.5 MeV, as shown in Fig. 1. This was the only parameter that was adjusted in order to compare with the present set of elastic scattering and fusion reaction measurements. All remaining parameters were fixed by independent means. It should be noted, however, that the isotope dependence of the radius parameter in Eq. (3) was modified to better account for the detailed changes in the structure of the different nuclei. To be precise, the potential radii for the various combinations were generated using:
instead of Eq. (3). Here, Rnns(A) refers to the root-mean-square matter radii obtained from shell-model calculations17. The actual values that have been used are 3.12, 3.18, 3.67, and 3.87 fin for 32S,34S,58Ni,and 64Ni, respectively. This scaling procedure has been found to be reliable for other reactions involving the Ni isotopes (see Refs. 1 8 and 19). The inelastic excitation channels taken into account included the low-lying 2+ and 3~ states in both the target and the projectile. The usual collective-model prescription of deforming the ion-ion potential radius was followed to obtain the coupling interactions (see e.g. Réf. 14). The Coulomb and nuclear deformation amplitudes which were used are given in Table E. We note that the nuclear deformation parameters for the Ni 2+ states are larger than the corresponding Coulomb ones, as opposed to some earlier Ni+Ni calculations where they were taken to be equal1 WQ. This change was dictated by a new coupled-channels analysis of the O+Ni elastic and inelastic scattering data carried out in Réf. 19. It has also been found that these new parameters lead to a systematic improvement in calculations of low energy Ni+Ni fusion cross sections and at the same time give very good agreement with the elastic scattering data21. No experimental values exist for the deformation amplitudes of the 3' state in 34S, so we have taken them to be the same as in 32S. The single nucléon transfer reaction channels were chosen on the basis of available spectroscopic information from light-ion reaction studies, also taking into account considerations of matching conditions. For all three combinations of isotopes in this work, one-neutron pickup reactions to strongly populated single panicle states are more favorably matched than the similar neutron stripping reactions. Likewise, single proton stripping is favored over proton pickup. Consequently, we have restricted the single particle transfer channels in each case to be neutron pickup and proton stripping. The spectroscopic factors and excitation energies for the specific transfer processes considered in the calculations are collected in Table III. We have taken into account all of the possible projectile-target transitions implied by the individual states listed in Table III. However, this was done JM an approximate way. Transitions having similar Q- values were taken to define a single transfer channel at an effective Q-value. The corresponding coupling interactions were obtained as a geometric average of the microscopic form factors computed from overlaps of single-particle orbitals. We refer to Réf. 20 for a more detailed discussion of this procedure. Test calculations of total transfer cross sections indicate that this prescription is accurate to within 5% over the range of energies covered in the present measurements. The actual effective channels used in the transfer calculations are summarized in Table IV. For example, in the single-neutron pickup reaction for 32S-I-58Ni, all 16 transitions corresponding to the states listed in Table III were combined into one channel with an effective Q-value of -5 MeV. This technique of "bunching" the single- particle transitions, and the use of the rotating frame approximation, are the ways in which we reduce the coupled-channels problem to manageable proportions.
B. Comparison wjth the Measurements.
We now discuss the comparison of the calculations with the data, beginning with the elastic scattering cross sections for 32S+58Ni. As noted above, the only free parameter in the calculation (which fine-tunes the radius of the ion-ion potential) was adjusted once to 32 58 fit the S^ Ni elastic data at Ec1n= 62.5 MeV. This comparison is shown by the solid curve in the middle of Fig. 4. It may be noted that, while the data for 6cm>l 10° are nicely reproduced, the calculation produces too much rise above the Rutherford cross section around 6cm= 110°. In a conventional optical-model analysis, one could increase the absorption in the nuclear surface region to dampen this effect. Here we do not have this freedom, as explicit couplings to the main direct reaction channels have already been made. In fact, our calculation of the single-nucléon transfer cross section for 32S-J-58Ni (Fig. 5) reproduces the available data2^ very well, so this aspect appears to be under control. There is, however, a subtle effect in the way the transfer channels influence the elastic scattering angular distribution, which can be seen by comparing the dashed and solid 32 58 = curves in Fig. 4, for S+ Ni at Ecm 62.5 MeV. The dashed curve includes only coupling to the inelastic channels. G" each case, the curves represent the prediction for true elastic scattering plus the cross section for exciting the first 2+ state of the projectile and target). Adding the transfer couplings lowers the backward-angle cross section as expected from an absorption effect, but also increases the rise above the Rutherford value. This is the sign of a refractive effect, indicating that the transfer couplings are effectively producing an attractive polarization potential in the elastic channel. It may be that the combination of transfer transitions into single effective transfer channels, and possibly also the use of the rotating frame approximation, causes an overestimate of this polarization effect.
10 The energy dependence of the elastic scattering cross section for 32S+58Ni is followed reasonably well by the calculations as one goes from Ec1n= 62.5 MeV to Ecm= 56 MeV (Fig. 4). We note again that these calculations do not use energy-dependent parameters. The previous elastic scattering measurements were analyzed using energy-dependent optical model potentials in Refs. 2,5, and 7. Because of the differences between the present measurements and those of the Legnaro group2»7, the significance of these potentials is questionable. On the other hand, since the present data set does not cover a wide energy range in detail, it would be premature to try to extract an energy-dependent optical-model potential from them. We now look to the isotope dependence of the elastic scattering cross section as one changes the target from 58Ni to ^4Ni. As shown in Fig. 4, the calculations reproduce this change rather well, giving a good account of the data at E01n= 58.5 MeV and the correct average behavior at Ec1n= 54.5 MeV. There are basically two features that account for the differences between the 32S+58Ni and 32S-J-64Ni cases. One is clearly the size difference modeled in Eq. 4. The other turns out to be the more significant role played by the transfer reactions when 64Ni is the target, since the inelastic excitation of 58Ni and 64Ni are quite similar as indicated in Table II. To appreciate these points, one may compare the 32S-I-64Ni elastic scattering results at 32 58 Ecm= 58.5 MeV to those for S+ Ni at Ecm= 59.3 MeV in Fig. 4. The lower energy in the 64Ni case would normally imply a smaller deviation from Rutherford scattering, but this is offset by the larger size Of64Ni. The size effect can be judged by comparing the dashed curves, which only allow for inelastic couplings in the two cases. It is clear from Fig. 4 that the increased size of 64Ni alone is insufficient to explain the large deviation from Rutherford scattering that is observed experimentally. The agreement with the data is achieved by requiring couplings to the single-particle transfer channels. This is a nice result from the theoretical model. If one were to use conventional optical-model potentials to fit both data sets, as was done for the earlier measurements in Refs. 2,5, and 7, this effect would be masked by an abrupt change in the parameters in going from 32S+58Ni to 32S^-64Ni. A similar result has recently been reported in an analysis of 2851+5S-64Ni elastic scattering3^. On the other hand, it is disturbing that the predicted single-particle transfer cross sections for ^5+04Ni are larger than the available measurements2^ indicate, as shown in Fig. 5. Actually, these transfer data are for higher energies than the elastic scattering measurements shown in Fig. 4. At these higher energies, we expect that additional couplings (in particular the coupling of the single particle to the two-particle channels) will reduce the single-particle transfer yields (sec Réf. 20 ).
11 The comparisons of the calculated fusion cross sections with the data are shown in Fig. 6 for the three systems 32S-^-64Ni and 34S-^64Ni. In all cases, the dotted curves show the cross section corresponding to the flux which penetrates past the Coulomb barrier without allowing any inelastic excitation or transfer processes. This gives reasonable agreement with the fusion yields at energies above the barrier, but severely underpredicts the low-energy cross sections. Adding the ineLstic couplings enhances the low-energy cross section significantly as shown by the dashed curves. The solid curves are the results from the full c.i'culation with both inelastic excitation and single-nucléon transfer couplings. It is seen that the transfer couplings have a rather small effect for 32S+58Ni, which is already well-described by the calculation which includes only inelastic excitation. There is a more notable effect of the transfer couplings for the 328+64Ni case, which is in accord with the larger transfer cross sections as compared with 32S+58Ni (Fig. 5). It is interesting that the single-particle transfer couplings appear to be required to obtain 32 64 agreement with the S^ Ni fusion data at the relatively high energy OfE01n= 58.5 MeV where they were essential for obtaining agreement with the elastic scattering data. The overall agreement with all three fusion excitation functions is very satisfying, particularly as it is obtained simultaneously with good fits to the elastic scattering data as shown in Fig. 4. On the other hand, the full calculations tend to overpredict the fusion data for all three cases at the highest energies. For heavier systems or at higher energies, one might attribute this to fission competing with evaporation in the decay of the compound nucleus. Here, however, the fission mode is expected to be negligible1'. It would be interesting to see if the trend indicated by the present data set continues at even higher energies. It should also be noted that there is still a discrepancy with the 32S+64Ni fusion data at the lowest energies. This indicates that additional couplings are needed in the calculations. One possibility is that a direct two-particle transfer process helps to enhance the low-energy fusion rate20. It is known29 that the two-nucleon transfer cross sections are larger for 32S^64Ni than for 32S+58Ni. Finally, we note that we were unable to obtain a consistent description of the previous fusion and elastic scattering data 2^3-7 within the framework of the coupled-channels barrier penetration model used here. For example, adjusting the potential to fit the elastic scattering for 32S+58Ni resulted in an overprediction of the fusion data. A similar inconsistency is apparent in the recent coupled-channels calculations presented in Réf. 7.
12 IV. CONCLUSIONS
32 Elastic scattering measurements have been made for S + 58,64^i at several energies near the Coulomb barrier. Our results display several significant differences with respect to earlier measurements2'7. In particular, there is no evidence in our data for the strong deviation from Rutherford scattering at the lower energies. Consistent with this observation, we were unable to locate the anomalously large direct reaction cross section hypothesized in Réf. S. Finally, we also observe a different trend in the energy dependence of the elastic scattering angular distributions. It is our conclusion that optical- model potentials generated from analyses2-5-7 of the previous elastic scattering data arc incorrect. Fusion excitation functions near to and well below the barrier have been obtained for 2 3 S+ 58,64^i and 343 + 64^i, covering the cross section range from 300 jib to 300 mb. Our fusion yields are larger than those of the Legnaro group3, particularly at the lower energies where dynamical coupling effects should play an important role, but agree very well with the values at higher energies reported in Réf. 13. Intercomparison of various data sets suggests the possibility that the targets used by the Legnaro group may have been thicker than anticipated. Because of the disagreements with existing fusion and elastic scattering data, we conclude that previous attempts3-5-7 to correlate these channels are unreliable. A coupled-channels analysis of the new experimental data has achieved a very good simultaneous description of elastic scattering and sub-barrier fusion, which gives confidence that the theoretical model is fundamentally correct. The dynamical couplings are clearly required to explain the low-energy fusion rates, and the predicted increase in the single-nucléon transfer strength in going from 32S+58Ni to 32S+64Ni accounts for a significant part of the difference in the corresponding elastic-scattering angular distributions. Our attempts to correlate the previous elastic scattering and fusion measurements2-3 within this model were unsuccessful. While generally good agreement with the new experimental data is obtained, there are also some deficiencies in the details of the model results. They include a tendency to overpredict the rise above Rutherford scattering in the elastic channel, too little fusion cross section for 32St^Ni at the lowest energies, too much single-nucléon transfer for 32S+^*Ni at higher energies, and overprediction of the fusion data at the highest measured energies in all three systems. It would be interesting to sec if the latter trend persists over a wider range of energies.
13 This work was supported by the U.S. NSF under Contract No. PHY88-03035, and by the U.S. Dept. of Energy under Contract No. W-31-109-ENG-38.
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* Present address: Centre de Recherches Nucléaires and Université Louis Pasteur, 67037 STRASBOURG Cedex, France.
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15 TABLE CAPTIONS
I. Total fusion cross sections for the three systems measured. The uncertainties quoted here do not include the estimated 8% systematic error in the overall normalization.
D. Multipolarities and excitation energies of the low-lying states in the S and Ni isotopes.
Also given are the Coulomb and nuclear coupling strengths, oc and On, used in the calculations. They are related to the deformation lengths 8 by o2 = 62 /4jt.
HI. Particle and hole states which result from proton stripping and neutron pickup reactions. The spectroscopic strengths C2S are taken from Refs. 22-28. The excitation energy and spectroscopic factor for the 5/2* state in 65Cu was obtained by averaging over three close-lying states. The full spectroscopic strength was used for the 3/2+ state of 33P as no data are available.
IV. The effective transfer channels and corresponding Q-values which are used in the calculations.
FIGURE CAPTIONS
Fig. 1. Comparison of the present elastic scattering measurements for S -f Ni systems (solid points) with the data of Réf. 2 (open triangles). The solid and dashed curves are the result of a coupled-channel calculation (see text). The dashed curve represents true elastic scattering, while the solid curve includes the cross section for exciting the first 2+ state in the projectile and the target
Fig. 2. Angular distributions of fusion products observed in the present experiment. The corresponding cm energies are 68.3 MeV for 32S+58Ni, 67.5 MeV for 32St64Ni, and 63.5 MeV for 34St64Ni, respectively. The curves are the gaussian fits described in the text.
Fig. 3. Fusion cross sections measured in the present experiment, compared with the data of Refs. 3 and 13. In the latter comparison, the cross sections measured at the same energies as in Réf. 13 agree within the size of the symbols so that the present measurement is suppressed on the figure.
16 Rg. 4. Comparison of the coupled-channels calculations with the elastic-scattering angular distributions of the present experiment. The dashed curves correspond to the calculations in which only coupling to inelastic channels is allowed, while the solid curves include transfer couplings. Ih both cases, the predictions include the cross section for exciting the first 2+ state of the projectile and target
Fig. 5. Comparison of thesingle proton stripping and single neutron pickup transfer data of Réf. 29 with the predictions of the present coupled-channels calculations.
Fig. 6. Comparison of the calculated 32S + W&tii and 34S^f64Ni fusion cross sections with the results of the present experiment The dotted curve is die no-coupling result, the dashed curve includes coupling to the inelastic channels, and the solid curve is the full calculation including transfer couplings.
17 Table I
System Ecm 07*« System Ecm G fu* System •Ecm */«• (MeV) (mb) (MeV) (mb) (MeV) (mb)
"S+"Ni 53.18 .19 ±.03 32S-J-64Ni 54.27 1.40 ±.21 32S-H88Ni 55.60 .26 ± .05 53.83 .62 ± .08 54.92 4.20 ±.34 56.30 .67 ± .11 54.47 1.88 ± .16 55.56 9.54 ±.66 56.95 2.99 ± .32 55.11 5.49 ± .41 56.21 16.8 ± 1.2 57.51 7.91 ± .75 55.73 12.2 ±0.9 56.96 28.9 ±2.0 58.10 15.4 ±1.3 56.36 21.7 ±1.6 57.51 41.3 ±2.8 58.74 24.5 ± 1.8 57.00 34.9 ± 2.6 58.17 59.4 ±4.0 59.37 41.7 ± 2.9 57.64 53.5 ± 3.9 58.34 71.8 ± 5.0 59.97 53.6 ± 3.8 58.29 67.7 ±5.0 59.50 99.1 ± 6.7 60.62 67.8 ± 4.6 58.94 94.5 ± 6.8 60.16 126 ±9 61.28 91.9 ± 6.1 59.58 117 ±8 60.82 145 ± 10 61.91 118 ±8 60.24 151 ± 10 61.50 168 ± 12 62.53 137 ±9 60.88 173 ± 12 62.16 192 ± 13 63.17 152 ± 12 61.53 190 ± 13 62.82 226 ± 16 63.83 173 ± 12 62.19 214 ± 14 63.49 239 ± 16 64.47 192 ± 13 62.18 221 ± 15 64.16 249 ± 18 65.10 211 ±15 62.84 235 ± 15 64.82 274 ± 19 65.75 227 ± 15 62.84 239 ± 17 65.49 289 ± 20 66.41 255 ± 16 63.49 244 ± 16 66.16 326 ± 22 67.05 279 ± 18 66.82 334 ± 22 67.69 302 ± 20 67.48 341 ± 24 68.33 324 ± 20 Table II
E' "S 2+ 2.23 0.315 0.315 3~ 5.01 0.448 0.448 "S 2+ 2.13 0.263 0.263 3- 4.62 0.448 0.448 58Ni 2+ 1.« ,36 0.280 3~ 4.47 0.260 0.260 84Ni 2+ 1.34 0.251 0.298 3~ 3.56 0.238 0.238 Table III Neutrons Protons J* E* (MeV) C2S J* E' (MeV) C2S 31 1+ "S I+ 0.000 0.950 P 0.00 1.90 1+ 3 + 0.842 0.325 2 1.27 2.00 T- 5 + 2 2.93 0.575 2 2.23 3.70 s- 3 3.22 0.500 3 + 1 + «s 3 0.000 0.525 "P 2 0.00 1.80 1+ 3 + 1.570 0.225 2 1.43 4.00 T- 5+ 2 1.99 0.9375 2 1.80 3.40 r 2.35 0.500 3- 59 3~ "Ni 2 0.000 1.04 Cu 2 0.000 0.49 s- 1~ 2 0.769 1.05 2 0.491 0.40 5- 2 1.010 0.21 2 0.912 0.74 T- 2 2.553 3.10 1- 85 3- "Ni 2 0.0000 0.47 Cu 2 0.000 0.77 3- 1- 2 0.0869 3.43 2 0.7706 0.65 3 — 5- 2 0.1555 2.42 2 1.59 0.65 3~ 2 0.5175 0.82 I~ 2 1.0009 0.52 Table IV Channel Number of Qg.*. Qeff Transitions (MeV) (MeV) "S+MNi —» 33S (all) + 57Ni (all) 16 -3.577 -5.00 —» 31P (|+) + 58Cu (all) 3 -5.448 -5.70 —» 31P (f+,f+) + 58Cu (all) 6 -7.60 «S+"Ni —» 33S (1""",I+) + "3Ni (all) 10 -1.018 -1.60 —> "s (f ~>D + "Ni M) 10 ~4-40 —t 31P (all) + 85Cu (all) 9 -1.413 -2.70 "S (|+) + 63Ni (all) 5 -2.674 -2.92 (|+,i~,D + 83Ni (all) 15 -4.95 33P (all)+ 85Cu (all) 9 -3.431 -4.40 32S^58Ni i.o 0.5 •Ec.mf 56.0 MeV *Ec.m=56.6MeV 1.0 MeV 0.5 *Ec.m.= 59.8 MeV i.o 0.5 Ec.m* 62.5 MeV b Ec m = 62.7 MeV \ w»IH* b 0.2 « +'"Ni 1.0 0.5 • Ec.mr 54.5 MeV i * E . = 54,5 MeV i c m 1.0 « « ****fify 0.5 Hi (I -Ec.m = 58.5 MeV f 0.1 40 60 80 100 120 140 160 F,3 i. T T 32 58 S + Ni IO V) Eio4 C? 34 64 S+ Ni "0IO4 (cleg) io; 34 64 i o S+ Ni, > IO * •t »t 10 »4 IO T to . 10 32- 58K1. S+ Nl 10 .£. • Present work o « Réf. 3 10 * Réf. 13 -i IO 50 55 60 65 Ec-m.(MeV) I I I I I 32 58 S+ Ni , * A .. T • . .» • %» • » A.« » »T« Ec.mf 59-3 MeV 32 64- Ec.mf 54.5 MeV E= 58.5 MeV o.a - 40 60 80 100 120 140 160 0_c.m_ . (deg) 10' 32S + 64Ni 10' 10 32S + 58Ni b IQl 1 10O 10-1 55 60 65 70 75 Ecm (MeV) 5 IO' 64 10' S+ Ni -à IO 32 64 S+ Ni '§ IO CO =3 IOV 32 58 S+ Ni 3 IO Itf 10" L. 50 55 60 65 7O c.rn.( MeV ) Ft •). t