arXiv:1001.2658v2 [nucl-th] 18 Mar 2010 Stochastic Semi-Classical Description of Fusion at Near-Barrier Energies Sakir Ayik $^{1}$, Bulent Yilmaz2 , $^{3}$, and Denis Lacroix3 $^{1}$Physics Department, Tennessee Technological University, Cookville, TN 38505, USA $^{2}$Physics Department, Ankara University, 06100 Ankara, Turkey and $^{3}$Grand Acc´el´erateur National d’Ions Lourds (GANIL), CEA/DSM-CNRS/IN2P3, BP 55027, F-14076 Caen Cedex 5, France (Dated: November 20, 2018) Fusion reactions of heavy ions are investigated by employing a simple stochastic semi-classical model which includes coupling between the relative motion and low frequency collective surface modes of colliding ions similarly to the quantal coupled-channels description. The quantal effect enters into the calculation through the initial zero-point fluctuations of the surface vibrations. A good agreement with results of coupled-channels calculations as well as experimental data is obtained for fusion cross sections of nickel isotopes. The internal excitations in non-fusing events as well as the fusion time are investigated. PACS numbers: 24.10.Lx, 25.60.Pj, 25.70.Jj Over the past years various models with different lev- els of approximation have been developed to explain the enhancement of sub-barrier fusion cross sections close to the Coulomb barrier energy. The description of fusion reactions is generally realized by phenomenological mod- els such as barrier penetration models (see ref. [1] for a review) or more sophisticated approaches like coupled- channels approach [2–4]. These models provide some qualitative as well as quantitative explanations to the enhancement of the cross sections as a result of coupling to selective collective modes. In such macroscopic ap- proaches, one of the main ingredients is the nuclear inter- action potential which can be taken as the Bass potential [5], the proximity potential [6, 7], or the double-folding potential [8]. Some microscopic approaches equipped with full quantum treatment such as Time Dependent Hartree-Fock model are also used to give quantitative ex- planations of nuclear fusion reactions by providing a con- nection between the macroscopic and microscopic phe- nomena [9–13]. In this paper, we investigate fusion reactions of heavy- ions by employing a simple stochastic semi-classical model. In this model, the coupling between the rela- tive motion and low frequency collective surface modes is incorporated into the description in a manner similar to quantal coupled-channels calculations. Quantal effects enter into the calculation through the initial zero-point fluctuations of the surface modes. In the calculations, we include the coupling of relative motion with the low- lying quadrupole and octupole surface modes. The zero- point fluctuations of surface vibrations are simulated in a stochastic approximation by generating an ensemble of trajectories in accordance with the Gaussian distribu- tion of zero-point fluctuations. The zero-point quantum fluctuations of surface modes lead to barrier fluctuations, which enhance the fusion cross-section at near and sub- barrier energies. This stochastic semi-classical model has been proposed in ref. [14] (see also [15] for a detailed re- where R represents the relative distance between two cen- ters of mass of the colliding nuclei and P is the corre- sponding relative momentum. In this expression, the first and second terms are the radial kinetic energy and the rotational kinetic energy with orbital angular momen- tum l . The quantities V$_{C}$ ( R ) and V$_{N}$ ( R, Ω , α$_{iλ}$ ) repre- sent the Coulomb potential energy and the nuclear inter- action potential. The parameters set Ω = { Ω 1 , Ω 2 , Ω 3 } In order to describe heavy-ion collisions and fusion at near-barrier and sub-barrier energies, we follow the idea originally proposed by Esbensen et al. [14]. In this semi- classical model, heavy-ion collisions are described by a Hamiltonian in which the relative motion is coupled to a number of low-lying collective modes (surface vibrations) of the colliding ions. Treating the surface vibrations in harmonic approximation, the semi-classical Hamiltonian for the colliding ions is given as, view). However, only a few applications of the model have been carried out so far [16–18]. This is probably due to insufficient computation power at the time it was proposed. In this work, employing this stochastic model, we investigate fusion reactions of nickel isotopes at near- barrier and sub-barrier energies. We compare our re- sults with the experimental data as well as the quantal coupled-channels calculations with the same input pa- rameters. In section two, the model is briefly described. In section three, calculations of the fusion cross-sections of nickel isotopes are presented. In section four, further applications of the approach are illustrated and conclu- sions are given in section five. I. INTRODUCTION II. STOCHASTIC SEMI-CLASSICAL MODEL H = P 2 2 µ + l ( l + 1) ℏ 2 2 µR 2 + V$_{C}$ ( R ) + V$_{N}$ ( R, Ω , α$_{iλ}$ ) + 2 ∑ i =1 N − 1 ∑ λ =0 [ Π 2 iλ 2 D$_{iλ}$ + 1 2 C$_{iλ}$α 2 iλ ] , (1)