Journal of Modern Physics, 2013, 4, 1-4
doi:10.4236/jmp.2013.45B001 Published Online May 2013 (
The M3Y Double Folding Dissipative Model in Agreement
with Precise Fusion Cross Sections
I. I. Gontchar, M. V. Chushnyakova
Physics and Chemistry Department, Omsk State Transport University, Omsk, Russia
Received 2013
Large numbers of precision fusion excitation functions were fitted in the literature u sing the nucleus-nucleus interaction
potential having the Woods-Saxon shape. The diffuseness of this potential fusion ranges from 0.75 to 1.5 fm. This is
much larger than the value of 0.65 fm required by the elastic scattering data. Trying to resolve this contradiction we
develop the dissipative trajectory model based on the density-dependent M3Y NN-forces folded with the nuclear matter
distribution. Resulting potential possesses the normal diffuseness about 0.65 fm. With this potential we reach the
agreement with the data for 16O+208Pb, 28Si+208Pb, 32S+208Pb reactions within 5%.
Keywords: Heavy Ion Fusion; M3Y NN-Forces; Double Folding Potential; Fusion Excitation Function
1. Introduction
Nucleus-nucleus collision is the main process from
which we obtain our knowledge about the properties of
the nuclei [1]. The fusion process, during which two
complex nuclei are converted into one excited com-
pound-nucleus, is the most probable result of such colli-
sion as the energy of the relative motion of the nuclei
exceeds the Coulomb barrier [2]. During last two decades
many interesting features of the fusion process between
two complex nuclei were discovered [3,4]. Measured
capture (fusion) cross sections are conventionally ana-
lyzed in the framework of the coupled-channels model [3,
5]. The nucleus-nucleus interaction potential is the cru-
cial element of this model. Usually, the Woods-Saxon
(WS) ansatz is used (see, e.g., Equation (2) in [5] or
Equation (1) in [6]). Its most important ingredient is the
diffuseness which finally defines the properties of the
fusion barrier: the larger is the diffuseness the lower is
the barrier and the larger is its radius (i.e. the distance
between the centers of two spherical nuclei at which the
barrier is located).
Systematic analysis of the experimental capture exci-
tation functions in [4] showed that the values of the dif-
fuseness ranging between 0.75 and 1.5 fm are needed in
order to reproduce the data. This is much larger than the
value of 0.65 fm which is required by the elastic scatter-
ing data [7]. It was discussed in [4] that these abnormally
large diffusenesses might be due to some unaccounted
dynamical effects.
Developing this suggestion, we try to analyze the pre-
cision experimental capture excitation functions using
the dissipative trajectory model with the surface friction
[8,9]. Most features of the model are specified in detail in
[6]. Therefore, in the present contribution we give only
qualitative description and concentrate on the compari-
son to the data and the conclusions which can be drown
from this comparison.
2. The Model
Within the framework of our mode l, the f ictitious Brownian
particle with the reduced mass move experiencing the
action of the conservative, dissipative, and stochastic
forces. We consider the collision process at the energies
well above the Coulomb barrier. Therefore the quantum
effects like tunneling and channels couplings are of no
importance and are not accounted for.
The collision of two spherical nuclei is considered.
That is why we account only for two degrees of freedom
corresponding to the radial and orbital motion. Our pre-
vious study [6] showed that the radial motion is rather
fast. Therefore we apply the corresponding equations in
the non-Markovian shape considering the retarding fric-
tion and colored noise for the radial momentum. These
equations finally are reduced to the system of 5 stochas-
tic differential equations with the white noise for an au x-
iliary variable. The diffusion coefficient is related to the
friction coefficient and to the temperature by the Einstein
relation (see, e.g., Equation (49) in [10]). The nuclear
potential energy is calculated using our code [11] based
on the double-folding of the M3Y density dependent
Copyright © 2013 SciRes. JMP
NN-forces with the nuclear densities [12]. Initial condi-
tions for the numerical modeling have been carefully
chosen to guarantee they do not influence the calculated
cross sections.
In our model the nucleus-nucleus potential defines
both the conservative and dissipative forces between col-
liding nu clei. Therefore, we p resent it in Figure 1 for the
case of 16O+208Pb reaction. The M3Y double folding po-
tential (solid line) is compared here with the Woods-
Saxon profile possessing abnormally large diffuseness [4]
and the standard Gross-Kalinowski potential [8]. We see
that our potential differs significantly from the predecessors.
In the M3Y double folding potential the nuclear matter
density is a crucial ingredient. This density was taken to
be proportional to the nuclear charge density. For the
latter distribution the Woods-Saxon profile (see Equation
(34) in [11]) was used. Two parameters of the profile are
related via the root mean square radius. The latter was
taken from [13]. The diffuseness for all the projectile
nuclei was taken as 0.5 fm. The diffuseness of the target
nucleus 208Pb was taken as 0.6 fm.
3. Results
We calculated the fusion (capture) cross sections for the
following three reactions for which the high precision
data are available: 16O+208Pb [14], 28Si+208Pb [15], 32S+
208Pb [16]. In Figure 2 we present the experimental fu-
sion excitation functions (upper panel) and the resulting
correlation between the data and the calculated cross
sections (lower panel) versus the relative collisio n energ y.
The latter is equal to the ratio of the collision energy to
the typical Coulomb barrier height BZ = ZPZT/(AP1/3 +
AT1/3). The cross sections for this figure are computed
using the single barrier penetration model (see Equations
(22, 24) in [6]) which does not include any dynamics.
The M3Y double folding potential was used. Here we
clearly see the essence of the abnormally large diffuse-
ness problem: the calculation with the potential whose
typical diffuseness is 0.65 fm strongly overestimates the
data. On the other hand, the Woods-Saxon profile, which
fits the fusion d ata, possesses the large diffuseness which
is in disagreement with the one required by the elastic
scattering d at a [7].
In Figure 3 we present the calculated fusion excitation
functions (upper panel) and the resulting correlation be-
tween the data and the calculated cross sections. Here
modeling was performed including the retarding friction
and corresponding colored noise.
The friction coefficient strength factors were 2.10-2
zs/MeV for the radial motion and 1.0×10-4 zs/MeV for
the orbital motion. The co rrelation time for the noise was
chosen to be an ajustable parameter. Results presented in
Figure 3 are obtained with the following values of this
time: 0.1 zs for 16O+208Pb, 0.15 zs for 28Si+208Pb and
32S+208Pb. Overall agreement of the calculations to the
data in Figure 3(b) is very good: among almost 50 points
only in 4 cases the ratio deviate from un ity by more than
5%. Note that at low values of the relative collision en-
ergy our approach is expected to underestimate the cross
sections because the quantum and coupling effects are
ignored. This is the reason why the ratio of the cro ss sec-
tions systematically decreases for 16O+208Pb reaction as
the relative collision energy goes to unity.
Figure 1. Different nuclear potentials versus dimensionless
center-of-mass distance for 16O+208Pb reaction; the vertical
lines indicate the barrier radii.
Figure 2. (a) The experimental excitation functions (the
error bars are inside the symbols); (b) The ratio of the
capture cross section calculated using the single barrier
penetration model to the experimental cross section versus
the relative collision energy.
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Figure 3. (a) The excitation functions calculated using the
M3Y double folding dissipative model; (b) The ratio of the
capture cross section calculated using the M3Y double
folding dissipative model to the experimental cross section
versus the relative collision energy.
4. Conclusions
The Woods-Saxon shape for the nucleus-nucleus interac-
tion potential was used extensively in the literature in
order to analyze the precision fusion excitation func-
tions [4,14-16]. Excellent fit was achieved with the dif-
fuseness of this potential ranging from 0.75 up to 1.5 fm.
This is significantly larger than the value of 0.65 fm with
which the elastic scattering cross sections are usually
successfully fitted [7].
This contradiction stimulated us to compose the dissi-
pative trajectory model based on the density-dependent
M3Y NN-forces folded with the nuclear matter distribu-
tion. Resulting poten tial possesses th e normal diffusen ess
about 0.65 fm.
Within the framework of our approach the nucleus-
nucleus collisi on process is modeled as the motion of the
fictitious Brownian particle experiencing the action of
the conservative, dissipative, and stochastic forces. Since
only the collision energies well above the Coulomb barrier
are considered, the quantum effects like tunneling and
channels couplings are not accounted for. Possible re-
tarding character of the dissipation and non-Markovian
nature of the noise are included.
Using this model we reached the agreement with the
data for 16O+208Pb, 28Si+208Pb, 32S+208Pb reactions within
5%. This is done with the universal strength coefficient
of radial friction 2.0x10-2 zs/MeV and with the correla-
tion time of the noise equal to 0.1 zs for 16O+208Pb reac-
tion and 0.15 zs for 28Si+208Pb and 32S+208Pb reactions.
This advance together with successful description of
the capture cross sections for 16O+92Zr and 16O+144Sm
reactions obtained in [6] seems giving a hope to resolve
the long standing problem of apparently large diffuseness
of the nucleus-nucleus potential.
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