Rainbow angles

Several theories have been developed to explain the rainbow phenomena, including the Lorenz-Mie theory, Airy s theory, the complex angular momentum theory that provides an approximation to the Lorenz-Mie theory, and the theory based on Huy gen s principle. Among these theories, only the Lorenz-Mie theory provides an exact solution for the scattering of electromagnetic waves by a spherical particle. The implementation of the rainbow thermometry for droplet temperature measurement necessitates two functional relationships. One relates the rainbow angle to the droplet refractive index and size, and the other describes the dependence of the refractive index on temperature of the liquid of interest. The former can be calculated on the basis of the Lorenz-Mie theory, whereas the latter may be either found in reference handbooks/literature or calibrated in laboratory. 

ANGULAR DISTRIBUTION OF THE SCATTERED LIGHT RAINBOW ANGLES... 

Rainbow angles are not evident in matrix elements for nonspherical particles. 

Figure 27. Calculations for He (2 S) + He from ungerade potential only compared to experiment. Classical rainbow angles are indicated by vertical arrows.

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. 

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 ... 

Rainbow angle 9r dependent on the energy E and the potential well depth e. 

The partial wave sum is now reduced to a sum over few pole contributions in the complex plane of /.. The contribution of a single pole to the phase shift function and the deflection function can be obtained from the parameterization (55). Fig. 10 illustrates the result. (/) is essentially a pulse centred at / = Re (Xp — ) with the depth 2/Im Xp and the width 2 Im Xp. Now one proceeds as follows. Starting with N poles, which are placed on a small circle centred at 7.p in the complex /-plane, the number of these poles (N) and the real and imaginary part of the central pole (/p) are derived from semiclassical quantities. The rainbow angle is given by 9r = 2N/lm Xp,... 

In the region of backscatter the intensity of secondary refraction is only dominant in a narrow range above the Rainbow angle for perpendicular polarization. The optimum location of the receiving optics however strongly... 

For atypical interatomic potential such as a 6-12 potential, 0(ft) looks like figure A3.11.6 rather than A3.11.5. This shows that for some 0 there are three b (one for positive 0 and two for negative 0) that contribute to the DCS. The 0 where the number of contributing trajectories changes value are sometimes called rainbow angles. At these angles, the classical differential cross sections have singularities. 

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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