Sub-barrier fusion: Probing reaction dynamics with barrier distributions

Abstract

Much theoretical work has shown that in some limit it is possible to think of heavy-ion fusion cross sections σ as coming not from a single barrier in the nucleus-nucleus potential, but from a set of eigenchannels producing a distribution D(B) of barriers. The number of eigenchannels α and their corresponding heights B α and weights w α are intimately related to the direct-reaction channels which couple to the elastic scattering. In the classical limit (ℏ → 0) tunneling is absent and each channel switches on abruptly as the incident energy E increases past the corresponding threshold B α. It has recently been shown that in this limit the quantity d 2(Eσ)/dE 2 has discontinues at these thresholds which would allow one to extract the B α and w α directly from σ. Quantum mechanical tunneling smears the threshold effect over an energy range (or tunneling width) ∼0.56 ℏ ω (where ω is the oscillator frequency related to the curvature of the barrier top). One can, however, still obtain a meaningful distribution if the spread of barrier energies is large compared with this tunneling width i.e. if the coupling is strong. We shall discuss various aspects of this approach. In particular it will be shown that rather good data are required to extract the D(B) reliably since this is related to a second derivative of the experimental data. Ironically this leads to a function which is poorly defined in the region where σ is large i.e. at energies above the conventional Coulomb barrier. The distribution corresponding to deformed nuclei (coupling to a rotational band) will be shown to be rather smooth. In contrast coupling to multi-phonon vibrational states are shown to have a more discrete structure. The double excitation of identical phonons (ignored in some models) may be important. The distribution of barriers due to multiple-neutron transfer will also obtained and we shall see that in the limit of many sequential transfer channels, the rather flat distributions proposed by Stelson may be obtained.