Glory angle

In order to remedy the recognized deficiencies of equation (4) for the scattering cross section, such as unphysical discontinuities at 6 = 0, the so-called glory angle [19], and at angles where d3/db = 0, called rainbow angles [19], as well as the lack of the interference between the various trajectories in the sum of equation (4), semiclassical corrections such as the uniform Airy or Schiff [20] approximations can be included. 

Fig. 2. Schematic diagram of classical trajectories and the corresponding deflection function for a realistic interatomic potential. Special trajectories which lead to forward rainbow (br) and glory (bt) scattering are marked. In addition the paths contributing to scattering at an angle of observation 9 are drawn.

Fig. 5. Differential cross section (weighted with sin 9) as a function of the deflection angle 9 and the reduced parameter A = krm which is proportional to the velocity. The calculation was performed for a LJ 12-6 potential with B = 5000. In the upper part the total cross section multiplied by v° is plotted versus A. The close connection between the number of supernumerary rainbows and the number of the glory undulations is clearly demonstrated. Note that another rainbow oscillation is buried under the forward diffraction peak and not shown in the figure.

The advantage of semiclassical corrections is the inclusion of quantum effects to the differential cross section in the small scattering angle, the so-called forward peak character of the differential cross section. Furthermore, in the particular case of the Schiff approximation, the glory and rainbow angle effects in the interference are accurately represented. This behavior can be observed in Fig. 3, where the absolute direct differential cross section obtained with ENDyne goes through the experimentally determined absolute cross sections. 

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