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Semiclassical treatment of fusion processes in collisions of weakly bound nuclei

We describe a semiclassical treatment of nuclear fusion reactions involving weakly bound nuclei. In this treatment, the complete fusion probabilities are approximated by products of two factors: a tunneling probability and the probability that the system is in its ground state at the strong absorption radius. We investigate the validity of the method in a schematic two-channel application, where the channels in the continuum are represented by a single resonant state. Comparisons with full coupled-channels calculations are performed. The agreement between semiclassical and quantal calculations isquite good, suggesting that the procedure may be extended to more sophisticated discretizations of the continuum.


I. INTRODUCTION
The effects of channel coupling in fusion reactions induced by weakly bound projectiles have attracted great interest over the last decade [1]. Some theoretical studies predict strong influence of the breakup channel over the complete fusion (CF) cross section [2,3,4,5,6,7].
One of the reasons for this interest is the growing amount of experimental data on these processes [8,9,10,11,12,13,14], which are important both for the study of astrophysical processes as well as for the production of superheavy nuclei.
The appropriate theoretical tool to handle this problem is the coupled-channels method.
However, its implementation becomes very complicated for the breakup channel, which is specially important in the case of weakly bound nuclei, since it involves an infinite number of states in the continuum. For practical purposes, it becomes necessary to approximate the continuum by a finite set of states, as in the Continuum Discretized Coupled-Channels method (CDCC) [15]. This procedure has been extended to the case of fusion reactions in refs. [5,6,7]. Recently, a semiclassical alternative based on the classical trajectory approximation of Alder and Winther (AW) [16] has been proposed [17]. This approximation was used to calculate breakup cross sections and the results were compared with those of the CDCC method. The agreement between these calculations was very good. Since this semiclassical version of the CDCC method is much simpler, it may be a very useful tool to calculate cross sections for other channels in reactions with weakly bound nuclei.
Although the AW method has been extensively used for several nuclear reaction processes, only very recently it was applied to the estimate of the fusion cross section [18]. For this application it was considered a simplified two bound channels problem for which the fusion cross section obtained with the AW method was compared with results of a quantal coupled-channels calculation. In spite of the large simplification in the problem, the agreement between these two calculations at above-barrier energies was very good. However, the semiclassical method severely underestimated the fusion cross section at sub-barrier energies. In the present work, we also consider two channels, but instead of being both bound, one of them represents a resonance in the continuum. We have assessed the validity of the semiclassical model in this case through comparisons with results of the corresponding full quantum mechanics coupled channels calculations. We have shown in particular that the good agreement with quantum mechanical calculations can be extended to sub-barrier 2 energies through the analytical continuation of the time.
The paper is organized as follows: in section II we review the semiclassical treatment of nuclear reactions and give its extension to fusion reactions. In section III, we discuss the application of this treatment to the schematic two-channel model of ref. [18], and show that it severely underestimates the fusion cross section at sub-barrier energies. Then, in section IV, we develop a generalization of the semiclassical model which is accurate at sub-barrier energies and takes into account the width of the effective channel. The conclusions and discussion of further work are given in the last section.

II. SEMICLASSICAL DESCRIPTION OF FUSION
As this work is devoted to reactions induced by weakly bound projectiles, the variables employed to describe the collision are the projectile-target separation vector, r, and the relevant intrinsic degrees of freedom of the projectile, ξ. For simplicity, we neglect the internal structure of the target. The Hamiltonian is then given by where h 0 (ξ) is the intrinsic Hamiltonian and V (r, ξ) represents the projectile-target interaction. The eigenvectors of h 0 (ξ) are given by the equation The AW method [16] is implemented in two-steps. First, one employs classical mechanics for the time evolution of the variable r. The ensuing trajectory depends on the collision energy, E, and the angular momentum, l. In its original version, an energy symmetrized Rutherford trajectory r l (t) was used. In our case, the trajectory is the solution of the classical equations of motion with the potential V (r) = φ 0 | V (r, ξ) |φ 0 , where |φ 0 is the ground state of the projectile. In this way, the coupling interaction becomes a time-dependent interaction in the ξ-space, V l (ξ, t) ≡ V (r l (t), ξ). The second step consists of treating the dynamics in the intrinsic space as a time-dependent quantum mechanics problem. Expanding the wave function in the basis of intrinsic eigenstates, and inserting this expansion into the Schrödinger equation for ψ(ξ, t), one obtains the AW These equations are solved with the initial conditions a α (l, t → −∞) = δ α0 , which means that before the collision (t → −∞) the projectile was in its ground state. The final population of channel α in a collision with angular momentum l is P the angle-integrated cross section is To extend this method to fusion reactions, we start with the quantum mechanical calculation of the fusion cross section in a coupled channel problem. For simplicity, we assume that all channels are bound and have spin zero. The fusion cross section is a sum of contributions from each channel. Carrying out partial-wave expansions we get with Above, u αl (k α , r) represents the radial wave function for the l th -partial-wave in channel α and W F α is the absolute value of the imaginary part of the optical potential associated to fusion in that channel.
To use the AW method to evaluate the fusion cross section, we make the approximation In eq.(8),P (α) l is the probability that the system is in channel α at the point of closest approach on the classical trajectory, and T l (E α ) is the probability that a particle with energy E α = E − ε α and reduced mass µ = M P M T / (M P + M T ), where M P , M T are respectively the masses of the projectile and target, tunnels through the potential barrier in channel α.
We now proceed to study the CF cross sections in reactions induced by weakly bound projectiles. For simplicity, we assume that the GS is the only bound state of the projectile and that the breakup process produces only two fragments, F 1 and F 2 . In this way, the labels α = 0 and α = 0 correspond respectively to the GS and the breakup states represented by two unbound fragments. Neglecting any sequential contribution, the CF can only arise from the elastic channel. In this way, the cross section σ CF can be obtained from eq.(9), dropping contributions from α = 0. That is, where P Surv is usually called survival (to breakup) probability.

III. COMPARISON WITH COUPLED-CHANNELS CALCULATIONS
The accuracy of the semiclassical fusion cross section has recently been checked in a schematic two-channel calculation for the scattering of 6 He projectiles on a 238 U target at above-barrier energies [18]. The weakly bound 6 He nucleus dissociates into 4 He and two neutrons, with threshold energy B = 0.975 MeV. The elastic channel is strongly coupled to the breakup channel and the influence of this coupling on the fusion cross section is very important. In this calculation, the breakup channel was represented by a single effective state [19]. For simplicity, the effective channel was treated as a bound state but it was assumed to contribute only to incomplete fusion. Furthermore, the ingoing wave boundary condition is used in all these calculations. The CF cross section is therefore given by eq. sub-barrier energies is enhanced in the CC calculation. The contribution from CF alone (solid line) is slightly larger than the fusion cross section in the no-coupling limit (dashed line). It is clear that the semiclassical calculation does not reproduce this effect, since its CF cross section is much lower in this energy range. This deficiency is due to the fact that in the semiclassical procedure described in section II the coupling is not considered inside the barrier region, because there is no classical trajectory connecting points inside this region.
Therefore, the effective barrier lowering ocurring in the coupled-channels calculation cannot be reproduced. The semiclassical calculation shows also some instability for energies close to the barrier top (V B ≃ 20 MeV). This last behavior results from orbiting effects in the classical trajectories.

IV. THE IMPROVED SEMICLASSICAL TWO-CHANNEL MODEL
To improve the semiclassical model at sub-barrier energies, we resort to the analytical continuation method, which consists of introducing the imaginary part of the time variable However, the situation tends do be much better in a realistic treatment of the continuum, with many channels and continuum-continuum coupling, owing to the irreversibility of the breakup process. The same occurs in the two-level model if the effective channel is treated as a doorway with finite width. This will be shown below.
To further improve the semiclassical calculation, we take into account the reaction Qvalue. We assume that the excitation energy of the effective channel is the breakup threshold, B. The next step is to simulate the irreversible nature of the breakup process, identified in the CDCC calculations of Diaz-Torres and Thompson [6] by an effective channel with complex energy. To this end we use the fact that an exponentially decaying state with mean Thus it appears that fusion processes in weakly bound systems in which the elastic channel is coupled to the continuum are amenable to be treated by means of the extensions of the Alder and Winther's method presented here.
The extension of this semiclassical method to less schematic treatments of the continuum is presently under way [21]. In it the continuum is represented by a discrete set of channels, as in [6]; therefore the relative motion between the breakup fragments is more accurately described. In this way, detailed information on the positions and momenta of the projectile fragments at their creation point would be available. This would allow to study the collision of the fragments with the target, and therefore to estimate the incomplete fusion cross section as well as the sequential fusion contribution to the CF cross section. An estimate of this last process is not possible in the standard CDCC description of fusion [6], so semiclassical calculations may furnish a way to solve this problem. It should be remarked that a semiclassical study along these lines has already been implemented for calculations of the breakup cross section [17] and the results were in very good agreement with the ones obtained with CDCC calculations.
We acknowledge useful discussions with M.S. Hussein and P.R. Silveira Gomes. This work