Semiclassical approximation of cross sections with many rainbow peaks

A semiclassical formula that gives scattering cross sections down to an angular resolution of order ${\mathrm{ħ}}^{4/5}$ is derived. The formula describes rainbow peaks, which have widths of order ${\mathrm{ħ}}^{2/3}$, but not other interference effects, which give structure on a scale of order ħ. For cross sections with many rainbow peaks, it can be easier to evaluate than conventional uniform approximations. As an example, the formula is applied to a simple two-dimensional model.