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Adiabatic quantum tunneling in heavy-ion sub-barrier fusion

High precision measurements of the fusion excitation functions for the reactions ^{40}Ca + ^{194}Pt, ^{192}Os clearly demonstrate that projectile excitation significantly modifies the potential barrier distribution. In sharp contrast, fusion of ^{16}O + ^{144}Sm appears to show no influence of the projectile excitation on the shape of the barrier distribution. These apparently conflicting conclusions are reconciled in this work, using realistic coupled--channels calculations, which show that high energy states produce an adiabatic potential renormalisation. This result indicates that adiabatic effects restrict, in a natural way, the states which influence the shape of a fusion barrier distribution. The analysis of barrier distributions thus offers a criterion for the relevance of the `counter term' prescription in the Caldeira-Leggett approach.

Quantum tunneling plays an important role in a range of diverse phenomena in physics and chemistry. Recent attention has been focused on tunneling in systems with many degrees of freedom [1,2]. One of the interesting aspects of the problem is in determining which of the multitude of degrees of freedom must be explicitly included in any theoretical description, and which can be omitted. In particular, it is essential to define the role of excitation energy, or the degree of adiabaticity, in limiting the effectiveness of a specific degree of freedom.
In nuclear physics, heavy-ion fusion reactions at energies near and below the Coulomb barrier provide an ideal opportunity to address this question. In order that fusion reactions occur, the Coulomb barrier created by the strong cancellation between the repulsive Coulomb force and the attractive nuclear interaction has to be overcome. Extensive experimental as well as theoretical studies have revealed that couplings of the relative motion of the colliding nuclei to nuclear intrinsic degrees of freedom cause enhancements of the fusion cross section at subbarrier energies, sometimes by several orders of magnitude, over the prediction of a single-barrier penetration model [3].
In a simple eigenchannel approach, such couplings result in the single fusion barrier being replaced by a distribution of potential barriers. A method of extracting barrier distributions directly from fusion excitation functions was proposed [4], and stimulated precise measurements of the fusion cross sections for several systems. In a rigorous theoretical interpretation, the barrier distribution representation is valid only if the excitation energy of the intrinsic motion is zero [5]. Nonetheless, this method has been successfully applied to analyse data from heavy-ion collisions, where each intrinsic degree of freedom carries a finite excitation energy. These analyses of the barrier distributions have beautifully demonstrated the effects of coupling of the relative motion to surface vibrations [6,7], their multi-phonon excitations [8,9], deformations and the associated rotational excita-tions [7,10] of the target nucleus and transfer reactions between the colliding nuclei [6,7].
Despite these successes, there are apparent conflicts regarding the role of projectile excitation. Each barrier distribution for the reactions 40 Ca + 194 Pt, 192 Os shows a characteristic structure, with a higher energy peak which has been associated with the octupole excitation of 40 Ca [11]. Calculations of fusion cross-sections for the reactions 16 O + 154 Sm, A Ge in [12,13] indicated that excitation of 16 O is important. In marked contrast, there are no specific features in the measured barrier distribution for the 16 O + 144 Sm reaction [6] which can be associated with the excitation of 16 O; indeed it was shown in Ref. [7] that a good theoretical representation of the barrier distribution is destroyed when the projectile excitation is included.
All the above conclusions were based on comparison of the experimental results with simplified coupled-channels calculations. The simplification has been achieved by using one or more of the following approximations : 1. the no-Coriolis approximation [14], where the centrifugal potential is assumed to be the same for all channels and equal to that in the elastic channel; 2. the linear coupling approximation, where the nuclear coupling potential is assumed to be linear w.r.t. the coordinate of the nuclear vibrational excitation; 3. the constant coupling approximation, where the coupling potential is assumed to be constant over the interaction range; 4. intrinsic excitation energies are assumed to be negligible or are treated approximately.
The first approximation, common to most coupled channel calculations, including those presented in this Letter, has been shown to work well for heavy ion fusion calculations [15].
Simplified coupled channel calculations [7,9,13,16] use the second approximation in conjuction with either the third or fourth. Recent studies [17,18] have shown the linear coupling approximation is not valid even in systems where the coupling is weak, and that higher order couplings strongly influence the calculated barrier distributions. It is therefore probable that in reactions with nuclei like 16  In this Letter we present the results of realistic coupled-channels calculations which demonstrate the effects of non-linear coupling and finite excitation energy of intrinsic nuclear (environmental) degrees of freedom, and resolve the apparently conflicting conclusions regarding the influence of the projectile excitation. The relevance of the 'counter term' prescription of Caldeira and Leggett [1] in heavy-ion fusion reactions is also discussed and the double counting problem of coupling effects is clarified.
The coupled channels equations are solved by imposing the incoming wave boundary condition to simulate the strong absorption inside the fusion barrier. The real nuclear potential is assumed to have a Woods-Saxon shape and the depth was chosen to reproduce the experimental fusion cross sections at high energies using the single-barrier penetration model. The values of deformation parameters are extracted from the reduced transition . The parameters of the calculations are listed in Table 1.
In order to show the inadequacies of the often used linear coupling approximation, calculations were performed for the 16  the general conclusion that the main effect of the coupling to inelastic channels whose excitation energies are larger than the curvature of the bare fusion barrier, i.e. an adiabatic coupling, is to introduce a static potential shift [19,20], and hence, the shape of the barrier distribution does not change unless the coupling form factor itself has a strong radial dependence.
In macroscopic quantum tunneling in condensed matter physics, the so-called counter term is often introduced in order to compensate for the static potential renormalization due to the coupling to the environment [1]. In contrast, in heavy-ion reactions, one usually estimates the bare potential, for example by fitting the fusion cross section at high energies, and discusses the effects of channel coupling without introducing the counter term. The mutual excitation channels up to 4 + 3 − , the former and the latter refering to the targets and the projectile respectively, are also included in the calculations. It is apparent that the projectile excitation significantly affects the shape of the barrier distribution in this case, as suggested in the simplified coupled channel calculations in Ref. [11].
As has been shown in the discussions for 16 O + 144 Sm reactions, the correct treatment of the coupling, without making the linear coupling approximation, significantly reduces the effect of projectile excitation on the shape of the barrier distribution. Calculations of the CCFUS-type, which fail in these regards, would therefore be expected to predict larger coupling effects than observed experimentally. The apparent success of the CC-FUS calculations reported in Ref. [11] was probably due to the compensation for this overestimate by the use of a smaller deformation parameter than that obtained from the octupole transition strength.
The theoretical calculations for the reactions with the 40 Ca projectile still significantly underestimate the fusion cross section at low energies, even after the excitation of the projectile is taken into account. As suggested in Ref. [11], coupling to transfer channels, which have been ignored in the present calculations, might enhance the fusion cross section at low energies.
In summary, we have performed coupled-channels calculations for the fusion reactions