This letter introduces a new coupling potential to explain the experimental
data over wide energy ranges for a number of systems. Within the
coupled-channels formalism, this letter first shows the limitations of the
standard coupled-channels theory in the case where one of the nuclei in the
reaction is strongly deformed and then, demonstrates that a global solution to
the problems of light heavy-ion reactions such as $^{12}$C+$^{12}$C,
$^{16}$O+$^{28}$Si and $^{12}$C+$^{24}$Mg can be found using a new
second-derivative coupling potential in the coupled-channels formalism. This
new approach consistently improves the agreement with the experimental data for
the elastic and inelastic scattering data as well as for their excitation
functions using constant or slightly energy-dependent parameters.Comment: 4 Pages, 5 Figures, submitted to the Phys. Lett. 

Boztosun

Brandan

Braun-Munzinger

Carter

Cooper

I Boztosun

Kobos

Sakuragi

Sciani

Stokstad

W.D.M Rae

Wolf

Physics Letters B

English

arXiv

AbstractA number of serious problems has continued to plague the study of the 16O+28Si, 12C+24Mg and 12C+12C systems for the last 40 years. The explanation of anomalous large angle scattering data; the reproduction of the oscillatory structure near the Coulomb barrier; the out-of-phase problem between theoretical predictions and experimental data; the consistent description of angular distributions together with resonance and excitation functions data are just some of these problems. These are long standing problems that have persisted over the years and do represent a challenge calling for a consistent framework to resolve these difficulties within a unified approach. Traditional frameworks have failed to describe these phenomena within a single model and have so far only offered different approaches where these difficulties are investigated separately from one another. The present work offers a plausible framework where all these difficulties are answered. Not only it improves the simultaneous fits to the data of these diverse observables, achieving this within a unified approach for a number of nuclear systems over wide energy ranges, but it departs for its coupling potential from the standard formulation. This new feature could provide new insight in the dynamics of these reactions. It also leads to new predictions especially concerning resonance phenomena in these systems which could form the basis for some new experimental investigation

Boztosun, I

Rae, W.D.M

Elsevier - Publisher Connector 

18 October 2001
Physics Letters B 518 (2001) 229–234
www.elsevier.com/locate/npe
A new coupling potential for the analysis of deformed light
heavy-ion reactions
I. Boztosun a,1, W.D.M. Rae b
a Department of Physics, Erciyes University, Kayseri 38039, Turkey
b Department of Nuclear Physics, University of Oxford, Keble Road, OX1 3RH Oxford, UK
Received 6 June 2001; received in revised form 18 July 2001; accepted 25 August 2001
Editor: J.-P. Blaizot
Abstract
A number of serious problems has continued to plague the study of the 16O+ 28Si, 12C+ 24Mg and 12C+ 12C systems for
the last 40 years. The explanation of anomalous large angle scattering data; the reproduction of the oscillatory structure near the
Coulomb barrier; the out-of-phase problem between theoretical predictions and experimental data; the consistent description
of angular distributions together with resonance and excitation functions data are just some of these problems. These are long
standing problems that have persisted over the years and do represent a challenge calling for a consistent framework to resolve
these difficulties within a unified approach. Traditional frameworks have failed to describe these phenomena within a single
model and have so far only offered different approaches where these difficulties are investigated separately from one another.
The present work offers a plausible framework where all these difficulties are answered. Not only it improves the simultaneous
fits to the data of these diverse observables, achieving this within a unified approach for a number of nuclear systems over
wide energy ranges, but it departs for its coupling potential from the standard formulation. This new feature could provide new
insight in the dynamics of these reactions. It also leads to new predictions especially concerning resonance phenomena in these
systems which could form the basis for some new experimental investigation.
1. Introduction
In this Letter, building on a previous paper [1], we
introduce a new coupling potential that explains the
scattering observables of the 16O+ 28Si, 12C+ 24Mg
and 12C + 12C systems over wide energy ranges in a
consistent way.
These reactions have been extensively studied both
theoretically and experimentally [2–6] and these in-
E-mail address: i.boztosun1@physics.ox.ac.uk (I. Boztosun).
1 Present address: Computational Mathematics Group, Univer-
sity of Portsmouth, Mercantile House, Portsmouth PO1 2EG, UK.
vestigations have resulted in a large body of experi-
mental data and a great number of theoretical models.
Although the theoretical models, based on dynamical
or purely phenomenological treatments, provide rea-
sonably good fits, no unique model has been proposed
that addresses the problems of such reactions in a uni-
fied way.
Consequently, the following problems continue to
exist for light heavy-ion reactions: (1) explanation of
anomalous large angle scattering data; (2) reproduc-
tion of the oscillatory structure near the Coulomb bar-
rier; (3) the out-of-phase problem between theoretical
predictions and experimental data; (4) simultaneous
PII: S0370-2693(01) 01 06 3- 2
 2001 Elsevier Science B.V. Open access under CC BY license.
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230 I. Boztosun, W.D.M. Rae / Physics Letters B 518 (2001) 229–234
fits of the individual angular distributions, resonances
and excitation functions (for the 12C+ 12C system in
particular); (5) the magnitude of the mutual-2+ ex-
cited state data in the 12C+ 12C system is unaccounted
for; (6) the deformation parameters (β values): previ-
ous calculations require β values that are at variance
with the empirical values and are physically unjustifi-
able.
In the next section, we introduce the standard and
new coupled-channels models. Sections 3–5 are de-
voted to the application of the new coupling poten-
tial to the analyzes of the 12C+ 12C, 16O+ 28Si and
12C + 24Mg reactions. Finally, we conclude in Sec-
tion 6.
2. The model
The three systems we study (16O+28Si, 12C+24Mg
and 12C + 12C) are quite different in many ways
but they share two common features: (1) the elastic
scattering data suggest that there is weak absorption
in the entrance channel in each case and (2) each
reaction involves at least one nucleus which is highly
deformed.
2.1. The standard coupled-channels calculations
In the standard coupled-channels calculations, we
describe the interaction between two nuclei with
a deformed optical potential. For all the reactions
considered, the real potential is assumed to have
the square of a Woods–Saxon shape. The parameters
of the real potential for the 16O + 28Si and 12C +
24Mg systems are fixed as a function of energy and
are not changed in the present calculations. For the
12C+ 12C system, it is slightly energy-dependent. The
parameters are shown in Table 1.
The imaginary part of the potential is the standard
Woods–Saxon shape. Only the depth or radius in-
creased linearly with energy.
Since the target nuclei 28Si, 24Mg and 12C are
strongly deformed, it is essential to treat their col-
lective excitations explicitly in the framework of the
coupled-channels formalism. We assume that the tar-
get nuclei have static quadrupole deformations, and
that their excitations can be described in the frame-
work of the collective rotational model. The empiri-
cal values for the deformation parameters (β), derived
from the known B(E2) values, are used in the present
calculations.
Using this standard coupled-channels method, we
found, as other authors had found, that it was not
possible to find a consistent solution over wide energy
ranges to the above-mentioned problems (see the
discussions of results and figures below).
2.2. New coupling potential
The limitations of the standard coupled-channels
method, on the one hand, and the resulting shape of the
compound nucleus created by the projectile and tar-
get nuclei, on the other hand, have motivated us to use
a second-derivative coupling potential. If we consider
two 12C nuclei approaching each other, the double-
folding model will generate an oblate potential which
is correct at large distances. When these two nuclei
come close enough, they create the compound nucleus
24Mg which is a prolate nucleus in its ground state.
In the standard method, the folding model yields an
oblate (attractive) potential for such a configuration.
It is not clear how well the double-folding model de-
Table 1
The parameters of the central and coupling potentials for the reactions studied
Central Coupling
V r0 a V r0 a
16O+ 28Si 750.5 0.749 1.4 155.0 0.748 0.81
12C+ 24Mg 427.0 0.865 1.187 185.0 0.710 0.62
12C+ 12C ∼ 290.0 ∼ 0.8 ∼ 1.28 210.0 ∼ 0.67 ∼ 0.68
I. Boztosun, W.D.M. Rae / Physics Letters B 518 (2001) 229–234 231
Fig. 1. For 16O + 28Si, the comparison of the standard and
new coupling potentials, parameterized as the 2nd derivative of
Woods–Saxon shape.
scribes a prolate nucleus with this oblate potential and
this may be the reason why the earlier calculations us-
ing a double-folding model in the coupled-channels
method were unable to provide a consistent solution
to the problems of such reactions. In order to describe
the above-mentioned configuration, the coupling po-
tential must be oblate (attractive) when two 12C nuclei
are at large distances and must be prolate (repulsive)
when they are at short distances [1].
The standard and new coupling potentials are com-
pared in Fig. 1. In the new calculations, we employed
the same method with the same potentials. The para-
meters are shown in Table 1. The empirical deforma-
tion parameter (β2) is used in these calculations.
3. 12C+ 12C
The first system we consider is the 12C + 12C,
which has been studied extensively over the last
40 years. Our analysis consists of a simultaneous
investigation of the elastic scattering, single-2+ and
mutual-2+ excitation inelastic scattering data from
ELab = 20.0 MeV to 126.7 MeV. In such a large
energy range, we also consider the 90◦ elastic and
inelastic excitation functions.
The conventional folding model potentials fail to re-
produce certain aspects of the data such as the repro-
duction of the gross structure in the 90◦ elastic scat-
tering excitation function and a simultaneous consis-
Fig. 2. The 12C + 12C system: the results of the single and
mutual-2+ states. The solid lines are the results of the new
coupling potential, while the dashed lines are the results of standard
coupled-channels model.
tent description of the elastic, single-2+ and mutual-
2+ states data. So far, no model has been able to pre-
dict the magnitude of the mutual-2+ state data cor-
rectly over a wide energy range. The standard coupled-
channels model using double-folding potentials under-
estimates its magnitude by a factor of 3 to 10 and the
single-2+ state results are too oscillatory with respect
to experimental data [4,7–10]. These problems have
remained unsolved so far. Clearly, the 12C+ 12C sys-
tem has numerous problems to which no consistent
global solution has been provided yet.
We have also observed such results in our standard
coupled-channels calculations, as shown only for the
mutual-2+ case in Fig. 2 with dashed lines. Varying
the parameters and changing the shape of the real and
imaginary potentials and some other attempts do not
provide a complete solution to the problems of this
reaction [1].
Using the new coupling potential, we have been able
to obtain excellent agreement with all the available ex-
232 I. Boztosun, W.D.M. Rae / Physics Letters B 518 (2001) 229–234
perimental data for the ground, single-2+ and mutual-
2+ states as well as the 90◦ elastic scattering excita-
tion function. This new approach also solves the mag-
nitude problem of the mutual-2+ excitation inelastic
scattering data, which has been outstanding problem
of this reaction. Some of the results for the single-2+
and mutual-2+ states are shown in Fig. 2 in compari-
son with the standard ones. All results can be found in
Ref. [1].
4. 16O+ 28Si
The second system we consider is 16O+28Si, which
shows anomalous large angle scattering (ALAS). In
the present work, we consider an extensive simultane-
ous investigation of the elastic and inelastic scattering
of this system at numerous energies from ELab = 29.0
to 142.5 MeV. In this wide energy range, the excita-
tion functions for the ground and single-2+ states are
also analyzed [11].
Several ad hoc models have been proposed to
explain the experimental data, but no satisfactory
microscopic models have been put forward yet. The
most satisfactory explanation proposed so far is that
of Kobos and Satchler [6] who attempted to fit only
the elastic scattering data with a microscopic double-
folding potential. However, these authors had to use
some small additional ad hoc potentials, which create
a deepening in the surface region of the potential, to
obtain a good agreement with the experimental data.
Without the additional small potentials, they could
not reproduce the experimental data. We have shown
that this deepening of the real potential in the surface
region takes into account the coupling effect in an ad
hoc way [11].
Using the standard coupled-channels method, we
obtained a very good agreement with the elastic scat-
tering data. However, for the single-2+ state data, the
theoretical predictions were too oscillatory and in par-
ticular, towards large angles at higher energies, the
out-of-phase problem between the theoretical predic-
tions and the experimental data is observed. This prob-
lem is clearly seen in the 180◦ excitation function re-
sults, shown in Fig. 3b. A number of models have been
proposed, ranging from isolated resonances to cluster
Fig. 3. The 16O + 28Si system: 180◦ excitation function for
(a) ground and (b) 2+ states, respectively. The dashed lines are the
results of standard coupled-channels calculations and solid lines are
the results obtained using new coupling potential.
exchange between the projectile and target nucleus, to
solve these problems (see Ref. [2] for a detailed dis-
cussion).
We have also attempted to overcome these prob-
lems by considering: (i) changes in the real and imag-
inary potentials, (ii) inclusion of 6+ excited state,
(iii) changes in the β2 value and some other attempts
failed to solve the problems [11]. We were unable to
get an agreement with the elastic and the 2+ inelastic
data as well as the 180◦ excitation functions simulta-
neously within the standard coupled-channels formal-
ism.
However, as shown in Fig. 3, the new coupling
potential has solved the out-of-phase problem for the
180◦ excitation functions and fits the ground state and
2+ state data simultaneously. To our knowledge, this
has not been achieved over such a wide energy range.
I. Boztosun, W.D.M. Rae / Physics Letters B 518 (2001) 229–234 233
Fig. 4. The 12C + 24Mg system: the ground and 2+ states results,
obtained using the new coupling potential.
5. 12C+ 24Mg
The final example we consider is the 12C + 24Mg
system. The angular distributions oscillate strongly
near the Coulomb barrier and the data manifest ALAS.
Our analysis consists of a simultaneous investigation
of the elastic and inelastic scattering data at numerous
energies from ELab = 16.0 to 24.0 MeV [12].
The most extensive study for this system was car-
ried out by Sciani et al. [3]. The authors could only fit
these data with Q-value dependent potential parame-
ters in a rather ad hoc fashion. Without Q-dependent
potentials, they observed that the theoretical predic-
tions and the experimental data for the elastic and in-
elastic scattering data were completely out-of-phase.
In the present calculations, we have studied the
data of Ref. [3] as well as some inelastic scat-
tering data of Ref. [13]. In the standard coupled-
channels calculations, we also found the out-of-phase
problem which was in conformity with the find-
ings of Sciani et al. However, using our new cou-
Fig. 5. The comparison of the bare potentials (solid) and the total
inverted potentials (dashed lines), found by inverting S-matrix of
the new coupled-channels calculations at ELab = 126.7 MeV for
the 12C+ 12C system.
pling potential, we obtained excellent agreement with
the experimental data. Some of the individual an-
gular distribution fits for the ground and 2+ states
are shown in Fig. 4 (see Ref. [12] for all the re-
sults).
6. Discussion and summary
The effect of this new coupling potential on the
scattering has been probed through the total inverted
potential, i.e., the dynamical polarization potential
(DPP) plus the bare potential, obtained by inversion
of the S-matrix. It is observed that for the standard
coupling potential case, the added attraction is almost
the same as the added absorption. On the other
hand, for the new coupling potential case, the added
attraction is much greater than the added absorption.
Therefore, as shown in Fig. 5, the new coupling
potential creates a deepening in the surface region
234 I. Boztosun, W.D.M. Rae / Physics Letters B 518 (2001) 229–234
of the real potential, which can be related to the
superdeformed structures of the compound nuclei [1].
To summarize, while these three systems show quite
different properties and problems, a unique solution
has come from a new coupling potential. This is very
encouraging. However, it is not possible at present to
provide a solid theoretical foundation where this term
can be derived from first principles. Further work is
needed and any insights that would lead to progress in
this direction will be greatly welcome in the future.
Despite this limitation of which we are fully aware,
the present work is still useful in that it may offer use-
ful insight in any future theoretical study. The argu-
ments in this Letter form part of a phenomenological
study of these nuclear reactions. For such reactions,
phenomenological potentials have been used for over
40 years now and continue to be used as they offer
a useful contribution towards the understanding of the
dynamics governing these systems. In such studies,
a variety of form factors have been used ad hoc to ex-
plore these reactions. It is only later on that the formal
justification of these has been developed. It is only re-
quired at first that these form factors do not lead to un-
realistic physical effects and follow broadly some ba-
sic symmetries and relations (e.g., rotational symme-
try). Furthermore, inverse scattering methods such as
those developed by Mackintosh et al. (see, for exam-
ple, the references in [14]) indicate that in most cases
the form factors have features that depart markedly
from the simple functional forms have been tradition-
ally assumed. It is in this context and spirit that our
postulating the use of such a coupling potential is pre-
sented. Within this phenomenological framework, the
relative success of our approach in improving on the
results and more importantly in tackling a variety of
problems within the same framework, argues strongly
in favor of this phenomenological adoption.
Acknowledgements
Authors wish to thank Doctors Y. Nedjadi, S. Ait-
Tahar, R. Mackintosh, B. Buck, A.M. Merchant, Pro-
fessor B.R. Fulton and Dr. Ays¸e Odman for valu-
able discussions and encouragements, I. Boztosun also
would like to thank the Turkish Council of Higher
Education (YÖK) and Erciyes University, Turkey, for
their financial support.
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A new coupling potential for the analysis of deformed light heavy-ion reactions 

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A new coupling potential for the analysis of deformed light heavy-ion reactions

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