On the Wong cross section and fusion oscillations

TL;DR

This paper revisits the Wong formula for heavy-ion fusion cross sections, improves it by including energy dependence, and analyzes the resulting oscillations due to angular momentum quantization effects.

Contribution

It generalizes the Wong formula to include energy-dependent barrier parameters and provides analytic expressions for fusion oscillations caused by angular momentum discreteness.

Findings

Energy-dependent Wong formula matches full quantum results.

Oscillations in fusion cross sections are explained analytically.

Discrepancies in light systems are addressed.

Abstract

We re-examine the well-known Wong formula for heavy-ion fusion cross sections. Although this celebrated formula yields almost exact results for single-channel calculations for relatively heavy systems such as $^{16}$O+$^{144}$Sm, it tends to overestimate the cross section for light systems such as $^{12}$C+$^{12}$C. We generalise the formula to take account of the energy dependence of the barrier parameters and show that the energy-dependent version gives results practically indistinguishable from a full quantal calculation. We then examine the deviations arising from the discrete nature of the intervening angular momenta, whose effect can lead to an oscillatory contribution to the excitation function. We recall some compact, analytic expressions for these oscillations, and highlight the important physical parameters that give rise to them. Oscillations in symmetric systems are discussed, as are systems where the target and projectile identities can be exchanged via a strong transfer channel.