Physics of Atomic Nuclei, Vol. 65, No. 4, 2002, pp. 607–611. From Yadernaya Fizika, Vol. 65, No. 4, 2002, pp. 639–643.Original English Text Copyright c© 2002 by Boztosun.

Systematic Investigation of Light Heavy-Ion Reactions*

I. Boztosun1)**

Department of Nuclear Physics, University of Oxford, UKReceived July 20, 2001

Abstract—We introduce a novel coupling potential for the scattering of deformed light heavy-ion reactions.This new approach is based on replacing the usual first derivative coupling potential by a new, secondderivative coupling potential in the coupled-channels formalism. This new approach has been successfullyapplied to the study of the 12C + 12C, 12C + 24Mg, 16O + 28Si, and 16O + 24Mg systems and made majorimprovements over all the previous coupled-channels calculations for these systems. This paper also showsthe limitations of the standard coupled-channels theory and presents a global solution to the problems facedin the previous theoretical accounts of these reactions. c© 2002 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION

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 effort 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[1–4] 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 fits, no unique model has been proposed thatexplains consistently the elastic and inelastic scatter-ing data over a wide energy range without applyingany ad hoc approaches. Consequently, the followingproblems continue to exist for light heavy-ion reac-tions: (1) explanation of anomalous large angle scat-tering data (ALAS); (2) reproduction of the oscillatorystructure near the Coulomb barrier; (3) the lack ofthe correct oscillatory structure agreement betweentheoretical predictions and experimental data for theground and excited states; (4) simultaneous fits ofthe individual angular distributions, resonances, andexcitation functions (for the 12C+ 12C system in par-ticular); (5) the magnitude of the mutual-2+ excitedstate data in the 12C+ 12C system is unaccounted for;

∗This article was submitted by the author in English.1)Permanent address: Department of Physics, Erciyes Univer-sity, Kayseri, Turkey.

**e-mail: i.boztosun1@physics.ox.ac.uk

1063-7788/02/6504-0607$22.00 c©

(6) the deformation parameters (β values): previouscalculations require β values that are at variance withthe empirical values and are physically unjustifiable.

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 find aglobal model, which simultaneously fits 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 fourdifferent reactions in detail, we will show some of theresults for the 16O+ 28Si and 12C+ 12C reactions.The details of the models and a complete set of theresults for all four reactions can be found in [6–9].

We describe the interaction between 16O and 28Sinuclei with a deformed optical potential. The realpotential is assumed to have the square of a Woods–Saxon shape:

VN (r) =−V0

[1 + exp((r − R)/a)]2, (1)

where V0 = 706.5 MeV, R = r0(A1/3p +A

1/3t ) with

r0 = 0.7490 fm and a = 1.40 fm. The parameters ofthe real potential were fixed as a function of energyand were not changed in the present calculations,although it was observed that small changes couldimprove the quality of the fits.

2002 MAIK “Nauka/Interperiodica”

608 BOZTOSUN

100

0

–100

–200

–3000 2 4 6 8 10

Radius, fm

Potentials depths, MeV

Fig. 1. For 16O+ 28Si: The comparison of the stan-dard coupling potential (solid curve) which is the firstderivative of the central potential and our new couplingpotential (dashed curve), which is parametrized as thesecond derivative of Woods–Saxon 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[4] as the sum of aWoods–Saxon 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 fixed 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(θ, φ) = r0A1/3p + r0A

1/3t [1 + β2Y20(θ, φ)], (6)

where β2 = −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 β2 was needed to be able to fit the magnitudefor the 2+ data.

P

In the present calculations, the first 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 reorientationeffects for 2+ and 4+ excited states were also in-cluded. The calculations were performed with an ex-tensively modified version of the code CHUCK [10].Using the standard coupled-channels theory, we

found, 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 first-derivativecoupling potential. The interpretation of this newcoupling potential is given in [11]. Here we employedthe same method with small changes in the potentialparameters. The empirical deformation parameter(β2) is used in these calculations.

4. RESULTS

4.1. 16O + 28Si

The first 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 to

explain these data, but no satisfactory microscopicmodels have been put forward yet. The most satisfac-tory explanation proposed so far is that of Kobos andSatchler [4], who attempted to fit 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 the

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SYSTEMATIC INVESTIGATION OF LIGHT HEAVY-ION REACTIONS 609

10

1

10

0

10

2

10

1

10

0

10

–1

10

–1

d

σ

/

d

Ω

, mb/sr

33.89 MeV

41.17 MeV

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.

200

150

100

50

Cross section, mb

0

0

+

–2

+

2

+

–2

+

80

60

40

20

020 40 60 80 100

Energy

(lab)

, MeV

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.

180-excitation functions, which are shown in the

figure. A number of models have been proposed,

ranging from isolated resonances to cluster exchange

PHYSICS OF ATOMIC NUCLEI Vol. 65 No. 4 200

between the projectile and target nucleus to solvethese problems (see [1] for a detailed discussion).We have also attempted to overcome these prob-

lems by considering (i) changes in the real and

2

610 BOZTOSUN

10

2

10

0

10

–2

10

–4

20 60 100 140

E

lab

, MeV

d

σ

/

d

Ω

, mb/sr

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 β2 value, and (iv)the inclusion of the hexadecapole deformation (β4).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 fits 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 [2]. 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 [8].

4.3. 12C + 12C

The final system we have considered is that of12C+ 12C, which has been studied extensively overthe last 40 years. There has been so far no model thatfits consistently the elastic and inelastic scatteringdata, mutual excited state data, or the resonances

P

and 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 [15–17]. 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 fit-ted by a potential, which also fits 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 find a globalpotential.Using this new coupling potential, we have been

able to fit 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 show

quite different 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.

ACKNOWLEDGMENTS

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 financial support.

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