Rainbows, supernumerary rainbows and interference effects in the angular scattering of chemical reactions: effect of varying the modulus of the S matrix in the context of Heisenberg's S matrix programme

We investigate the existence of primary rainbows, supernumerary rainbows and diffraction interference effects in the product differential cross sections (DCSs) of state-to-state chemical reactions. The rainbows can be 'pronounced' or 'hidden'. Our theoretical approach uses a 'weak' version of Heisenberg's scattering matrix programme (wHSMP) introduced by Shan and Connor 2011 Phys. Chem. Chem. Phys. 13 8392. This wHSMP uses four general physical principles for chemical reactions to suggest simple parametrised forms for the S matrix; it does not employ a potential energy surface. We use a realistic parametrization in which the modulus of the S matrix is the sum of a smooth-step function and a gaussian function; both are functions of the total angular momentum quantum number, J. We then vary the parameters in the modulus. The phase of the S matrix is a cubic polynomial in J, which is held fixed. We demonstrate for a Legendre partial wave series (PWS) the existence of primary rainbows and supernumerary rainbows (both pronounced and hidden) as well as diffraction interference effects, in reactive DCSs. We find that reactive rainbows can be complicated in their structure. We also analyse for five examples, the angular scattering using nearside–farside (NF) PWS theory, including resummations of the PWS. In addition, we apply full and NF asymptotic (semiclassical) rainbow theories to the PWS—in particular, the uniform Airy and transitional Airy approximations for the farside scattering. This lets us prove that structures in the DCSs are indeed primary rainbows, supernumerary rainbows as well as diffraction interference effects. Our calculations complement and extend those in an earlier paper by Shan et al 2018 Phys. Chem. Chem. Phys. 20 819, in which the modulus of the S matrix is held fixed, whilst the phase is varied. (paper)