Rainbows supernumerary

The former referred to as supernumerary rainbows are sensitive to the attractive part of the potential. The latter usually called rapid oscillations, determine bg and therefore the absolute scale of the potential. For angles 9 greater than the rainbow angle there is only one contribution to the classical path so that the cross section is monotonic. The repulsive part of the deflection function and therefore of the potential is probed. These features are summarized for energies E greater than the orbiting energy orb ... 

Supernumerary rainbows, interference between b2 and b3, very sensitive to the form of the attractive part of the potential near the inflection point. 

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 measurable quantities are then calculated with the help of these functions. For instance, the rainbow scattering is given by Equation (16). Because of the behaviour of Af and the smaller amplitude of the second term, the positions of the supernumerary rainbows zN are given by the zeros of A lzN) (maxima) and the zeros of A,(zN) (minima). Now zN is calculated using (19) and the function (48) with the result... 

Fig. 7. Measured differential cross sections for NaHg of five different energies in the centre of mass system (Buck and Pauly, 1971). Supernumerary rainbows and rapid oscillation (at E = 0-25 eV and 0-19 eV) are well resolved.

Upper and lower bounds for Dq AtH ) are reasonably well established from photo-ionization and flowing-afterglow measurements " 4.2 eV > Dq(AtV ) > 3.4 eV. Several groups have scattered protons oif argon. Wherever the analysis has depended exclusively on the location of the supernumerary rainbows, the derived values for Do(ArH ) are all < 3.4 eV. In one study, " superior resolution has revealed the rapid oscillations in the differential cross section and an analysis based on their location gives Do(ArH ) = 3.85 eV, agreeing with a theoretical calculated value of 3.9 Zero-point-energy... 

Descartes analysis was a remarkable feat, but was deficient in two main respects. It said nothing about the colors of the rainbow nor about the supernumerary rainbows which sometimes appear faintly in the interior of the primary bow and on the exterior of the secondary bow. Newton resolved the problem of the colors,but the supernumerary rainbows are a wave interference effect and their explanation came more than 100 years later. 

Airy s equation gives an intensity which oscillates about the geometrical result, having its strongest peak just before the rainbow angle and decaying exponentially beyond it. This is illustrated in Fig. 2. The weaker peaks describe quite accurately the positions and intensities of the supernumerary rainbows and the dependence of the intensity on the radius and wavelength resolves several other problems associated with the description of the rainbow. 

Ford and Wheelerin their semiclassical treatment of quantum mechanical rainbow scattering remark that the necessary mathematics is not essentially different from Airy s treatment of the reflection and refraction of light. Indeed, their semiclassical formula for the cross section contains the Airy function or rainbow integral. The semiclassical cross section is illustrated schematically in Fig. 3c. The supernumerary rainbows of the wave mechanical theory are clearly resolved in the Na-Hg cross section measurements of Buck and Pauly, reproduced here in Fig. 5. 

Fig. 5. Measured differeatial cross sections for NaHg at five different energies in the center-of-mass system. Supernumerary rainbows are well resolved. (Reprinted with permission from reference 35, figure 7.)...

Recently, HAS experiments have been reported [82], which allow a rather accurate determination of the He-IiF potential and, furthermore, the corrugation of the liF(OOl) surface. In these experiments, a helium atom beam with Idloelectronvolt energies was scattered under grazing incidence at a LiF(OOl) surface, giving rise to supernumerary rainbows in the detected angular distributions. 

Second, the calculated (as well as the measured) distributions are remarkably smooth although often more than fifty or so rotational states are populated. If so many quantum states take part in a collision, one intuitively expects pronounced interference oscillations. The reason for the absence of interferences is the uniqueness between 70 and j one and only one trajectory contributes to the cross section for a specific final rotational state. If two trajectories that lead to the same j had comparable weights, the constructive and destructive interference, within a semiclassical picture, would lead to pronounced oscillations (Miller 1974, 1975 Korsch and Schinke 1980 Schinke and Bowman 1983). These so-called supernumerary rotational rainbows are well established in full collisions (Gottwald, Bergmann, and Schinke 1987). If the weighting function W (70) is sufficiently wide that both trajectories contribute to the dissociation cross section, similar oscillations may also exist in photodissociation (see, for example, Philippoz, Monot, and van den Bergh 1990 and Miller, Kable, Houston, and Burak 1992). 

Fig. 2. Supernumerary rotational rainbow oscillations for the j = 0 2 transition in He + Na2 collisions. The dots are the...

---