Theory of forward glory scattering for chemical reactions

The theory of forward glory scattering is developed for a state-to-state chemical reaction whose scattering amplitude can be expanded in a Legendre partial wave series. Two transitional approximations are derived that are valid for angles on, and close to, the axial caustic associated with the glory. These are the integral transitional approximation (ITA) and the semiclassical transitional approximation (STA), which is obtained when the stationary phase method is applied to the ITA. Both the ITA and STA predict that the scattering amplitude for glory scattering is proportional to a Legendre function of real degree or, to a very good approximation, a Bessel function of order zero. A primitive semiclassical approximation (PSA) is also derived that is valid at larger angles, away from the caustic direction, but which is singular on the caustic. The PSA demonstrates that glory structure arises from nearside-farside (NF) interference, in an analogous way to the two-slit experiment. The main result of the paper is a uniform semiclassical approximation (USA) that correctly interpolates between small angles, where the ITA and STA are valid, and larger angles where the PSA is valid. The USA expresses the scattering amplitude in terms of Bessel functions of order zero and unity, together with N and F cross sections and phases. In addition, various subsidiary approximations are derived. The input to the theory consists of accurate quantum scattering matrix elements. The theory also has the important attribute that it provides physical insight by bringing out semiclassical and NF aspects of the scattering. The theory is used to show that the enhanced small angle scattering in the F + H 2(v i = 0, j i = 0, m i = 0) → FH(v f = 3, j f = 3, m f = 0) + H reaction is a forward glory, where v i, j i, m i and v f ,j f, m f are initial and final vibrational, rotational and helicity quantum numbers respectively. The forward angle scattering for the H + D 2(v i = 0, j i = 0, m i = 0) → HD(v f = 3, j f = 0, m f = 0) + D reaction is also analysed and shown to be a forward glory, in agreement with a simpler treatment by D. Sokolovski (Chem. Phys. Lett., 2003, 370, 805), which is a special case of the STA.