Physics of Atomic Nuclei, Vol. 65, No. 4, 2002, pp. 607611. From Yadernaya Fizika, Vol. 65, No. 4, 2002, pp. 639643.Original English Text Copyright c 2002 by Boztosun.
dierent reactions in detail, we will show some of thens.the.8Sirealds
iths ofrgydata over a wide energy range without applyingad hoc approaches. Consequently, the followinglems continue to exist for light heavy-ion reac-: (1) explanation of anomalous large angle scat-g data (ALAS); (2) reproduction of the oscillatorycture near the Coulomb barrier; (3) the lack ofcorrect oscillatory structure agreement betweenretical predictions and experimental data for thend and excited states; (4) simultaneous ts ofndividual angular distributions, resonances, andation functions (for the 12C+ 12C system in par-ar); (5) the magnitude of the mutual-2+ exciteddata in the 12C+ 12C system is unaccounted for;
his article was submitted by the author in English.
results for the 16O+ 28Si and 12C+ 12C reactioThe details of the models and a complete set ofresults for all four reactions can be found in 
We describe the interaction between 16O and 2nuclei with a deformed optical potential. Thepotential is assumed to have the square of a WooSaxon shape:
VN (r) =V0
[1 + exp((r R)/a)]2 ,
where V0 = 706.5 MeV, R = r0(A1/3p +A
1/3t ) w
r0 = 0.7490 fm and a = 1.40 fm. The parameterthe real potential were xed as a function of eneSystematic Investigation of
I. BoztoDepartment of Nuclear Phys
AbstractWe introduce a novel coupling potential foThis new approach is based on replacing the usualderivative coupling potential in the coupled-channelsapplied to the study of the 12C + 12C, 12C + 24Mg, 16
improvements over all the previous coupled-channelsthe limitations of the standard coupled-channels theorin the previous theoretical accounts of these reactions
We investigate the elastic and inelastic scatteringof light heavy-ions, which have stimulated a greatdeal of interest over the last 40 years. There has beenextensive experimental eort to measure the elasticand inelastic scattering data as well as their 90-and 180-excitation functions. A large body of ex-perimental data for these systems is available (see and references therein). A variety of theoret-ical accounts based on dynamical models or purelyphenomenological treatments have been proposed toexplain these data [1, 5]. The elastic scattering datahave already been studied in detail using an opticalmodel and coupled-channels method.
Althoughmost of these models provide reasonablygood ts, no unique model has been proposed thatexplains consistently the elastic and inelastic scatter-ermanent address: Department of Physics, Erciyes Univer-ity, Kayseri, Turkey.-mail: email@example.com
1063-7788/02/6504-0607$22.00 cLight Heavy-Ion Reactions*
ics, University of Oxford, UKly 20, 2001
r the scattering of deformed light heavy-ion reactions.rst derivative coupling potential by a new, secondformalism. This new approach has been successfullyO + 28Si, and 16O + 24Mg systems and made majorcalculations for these systems. This paper also showsy and presents a global solution to the problems faced. c 2002 MAIK Nauka/Interperiodica.
(6) the deformation parameters ( values): previouscalculations require values that are at variance withthe empirical values and are physically unjustiable.
Therefore, in this paper, we are concerned with themeasured experimental data for 12C+ 12C, 16O+ 28Si,12C+ 24Mg, and 16O+ 24Mg in an attempt to nd aglobal model, which simultaneously ts the elasticand inelastic scattering data for the ground andexcited states in a consistent way over a wide energyrange and which throws light on the underlyingmechanism of the reactions and on the nature of theinteractions involved.
2. STANDARD COUPLED-CHANNELSMODEL
Although we have considered and studied fourand were not changed in the present calculations,although it was observed that small changes couldimprove the quality of the ts.
2002 MAIK Nauka/Interperiodica
608 BOZTOSUN 100
3000 2 4 6 8 10
Potentials depths, MeV
Fig. 1. For 16O+ 28Si: The comparison of the stan-dard coupling potential (solid curve) which is the rstderivative of the central potential and our new couplingpotential (dashed curve), which is parametrized as thesecond derivative of WoodsSaxon shape and which hasV = 155.0MeV,R = 4.160 fm, and a = 0.81 fm.
The imaginary part of the potential was taken as in as the sum of aWoodsSaxon volume and surfacepotential, i.e.,
W (r) = WV f(r,RV , aV ) (2)+ 4WSaSdf(r,RS , aS)/dr,
f(r,R, a) =1
1 + exp((r R)/a) (3)
with WV = 59.9 MeV, aV = 0.127 fm, and WS =25.0 MeV, aS = 0.257 fm. These parameters werealso xed in the calculations and only their radii in-creased linearly with energy according to the follow-ing formulas:
RV = 0.06084Ec.m. 0.442, (4)RS = 0.2406Ec.m. 2.191. (5)
Since the target nucleus 28Si is strongly deformed,it is essential to treat its collective excitation explicitlyin the framework of the coupled-channels formalism.It has been assumed that the target nucleus has astatic quadrupole deformation and that its rotationcan be described in the framework of the collectiverotational model. It is therefore taken into account bydeforming the real optical potential in the followingway:
R(, ) = r0A1/3p + r0A1/3t [1 + 2Y20(, )], (6)
where 2 = 0.64 is the deformation parameter of28Si. This value is actually larger than the value cal-culated from the known BE(2) value. However, thislarger 2 was needed to be able to t the magnitudefor the 2+ data.PIn the present calculations, the rst two excitedstates of the target nucleus 28Si: 2+ (1.78 MeV)and 4+ (4.62 MeV) were included and the 0+2+4+ coupling scheme was employed. The reorientationeects for 2+ and 4+ excited states were also in-cluded. The calculations were performed with an ex-tensively modied version of the code CHUCK .
Using the standard coupled-channels theory, wefound, as other authors had found, that it was impos-sible to describe consistently the elastic and inelasticscattering of this and other reactions we considered.
3. NEW COUPLING POTENTIAL
The limitations of the standard coupled-channelstheory in the analyses of these reactions compelledus to look for another solution. Therefore, a second-derivative coupling potential, as shown in Fig. 1, hasbeen used in the place of the usual rst-derivativecoupling potential. The interpretation of this newcoupling potential is given in . Here we employedthe same method with small changes in the potentialparameters. The empirical deformation parameter(2) is used in these calculations.
4.1. 16O + 28Si
The rst system we consider is 16O + 28Si, whichshows ALAS. In the present work, we consider anextensive simultaneous investigation of the elasticand the inelastic scattering of this system at numer-ous energies from Elab = 29.0 to 142.5 MeV over thewhole angular range up to 180. In this energy range,the excitation functions for the ground and 2+ statesare also analyzed [6, 12].
Several ad hoc models have been proposed toexplain these data, but no satisfactory microscopicmodels have been put forward yet. The most satisfac-tory explanation proposed so far is that of Kobos andSatchler , who attempted to t the elastic scatter-ing data with a microscopic double-folding potential.However, these authors had to use some small ad-ditional ad hoc potentials to obtain good agreementwith the experimental data.
Using the standard coupled-channels method,some of the results obtained for the 180-excitationfunctions for the ground and 2+ states of the reaction16O+ 28Si are shown in Fig. 2. The magnitude ofthe cross sections and the phase of the oscillationsfor the individual angular distributions are givencorrectly at most angles. However, there is an out-of-phase problem between the theoretical predictionsand the experimental data towards large angles athigher energies. This problem is clearly seen in theHYSICS OF ATOMIC NUCLEI Vol. 65 No. 4 2002
SYSTEMATIC INVESTIGATION OF LIGHT HEAVY-ION REACTIONS 609
0 50 100 150 200Scattering angle, deg
Fig. 2. The 16O+ 28Si system: The angular distribution. The solid curves are the results of standard coupled-channelscalculations, and the dashed curves are the results obtained using new coupling potential for the inelastic scattering data.The dots represent experimental data.
Cross section, mb
020 40 60 80 100
Fig. 3. The 12C+ 12C system: The itegrated cross section of the single- and mutual-2+ states. The solid curves are the resultsof the new coupling potential, while the dashed curves are the results of standard coupled-channels model. The dots representexperimental data.
80-excitation functions, which are shown in the
gure. A number of models have been proposed,
anging from isolated resonances to cluster exchange
between the projectile and target nucleus to solvethese problems (see  for a detailed discussion).
We have also attempted to overcome these prob-lems by considering (i) changes in the real and101
33.89 MeVPHYSICS OF ATOMIC NUCLEI Vol. 65 No. 4 2002
610 BOZTOSUN 102
10420 60 100 140
Fig. 4. The 12C+ 12C system: 90-excitation function forthe elastic scattering using new coupling potential (solidcurve). The dots represent experimental data.
imaginary potentials, (ii) the inclusion of the 6+
excited state, (iii) changes in the 2 value, and (iv)the inclusion of the hexadecapole deformation (4).These attempts failed to solve the problems at all[6, 12]. We were unable to get an agreement withthe elastic and the 2+ inelastic data as well as the180-excitation functions simultaneously within thestandard coupled-channels formalism. However, asshown in Fig. 2, the new coupling potential has solvedthe out-of-phase problem for the 180-excitationfunctions and ts the ground state and 2+ state datasimultaneously.
4.2. 12C + 24Mg and 16O + 24Mg
The second and third examples we have consid-ered are 12C + 24Mg and 16O + 24Mg. Fifteen com-plete angular distributions of the elastic scatteringof 12C + 24Mg system were measured at energiesaround the Coulomb barrier and were published re-cently . We have studied these 15 complete elas-tic scattering angular distributions as well as someinelastic scattering data measured by Carter et al.[13, 14] some 20 years ago. Excellent agreement withthe experimental data was obtained by using this newcoupling potential. Our model has also solved someproblems in 16O + 24Mg scattering .
4.3. 12C + 12C
The nal system we have considered is that of12C+ 12C, which has been studied extensively overthe last 40 years. There has been so far no model thatts consistently the elastic and inelastic scatteringdata, mutual excited state data, or the resonancesPand excitation functions. Another problem is the pre-dicted magnitude of the excited state cross sections,in particular for the mutual-2+ channel. The con-ventional coupled-channels model underestimates itsmagnitude by a factor of at least two and often muchmore . We have also observed this in our con-ventional coupled-channels calculations as shown inFig. 3 with dashed curves. There are also resonancesobserved at low energies, which have never been t-ted by a potential, which also ts either the angulardistributions or the excitation functions. Therefore,the experimental data at many energies between 20.0and 126.7 MeV in the laboratory system have beenstudied simultaneously to attempt to nd a globalpotential.
Using this new coupling potential, we have beenable to t the energy average of all the availableground, single-2+, mutual-2+ and the backgroundsin the integrated cross sections, as well as the maingross features of the 90-excitation function, asshown in Figs. 3 and 4, simultaneously. Our prelim-inary calculations of resonances using no imaginarypotential are promising but there are problems withthe widths of the resonances.
To summarize, while these four systems showquite dierent properties and problems, a uniquesolution has come from a new coupling potential.Although the origin of this new coupling potentialis still speculative and needs to be understood froma microscopic viewpoint, the approach outlined hereis universal and applicable to all the systems. Studiesusing this new coupling potential are likely to lead tonew insights into the formalism and the interpretationof these systems. Therefore, this work represents animportant step towards understanding the elasticand inelastic scattering of light deformed heavy-ionsystems.
Special thanks toW.D.M. Rae, Y. Nedjadi, S. Ait-tahar, G.R. Satchler, and D.M. Brink for valuablediscussions and providing some data. The author alsowould like to thank the Turkish Council of HigherEducation (YOK) and Oxford and Erciyes (Turkey)Universities for their nancial support.
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