The M3Y Double Folding Dissipative Model in Agreement with Precise Fusion Cross Sections

doi:10.4236/jmp.2013.45B001

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Journal of Modern Physics, 2013, 4, 1-4

doi:10.4236/jmp.2013.45B001 Published Online May 2013 (http://www.scirp.org/journal/jmp)

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

Email: [email protected]

Received 2013

ABSTRACT

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

I. I. GONTCHAR, M. V. CHUSHNYAKOVA

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 2.0×10-2

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.

I. I. GONTCHAR, M. V. CHUSHNYAKOV A 3

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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