INDEX spheroidal

Figure 13.3.10 shows the typical spheroidal deformation of stainless steel particles with treatment time. As compared with the result of copper particles, the rate of spheroidal deformation was lower. Thus, the stainless steel particles were easier for stepwise adjustment of the shape index with treatment time. 

The classical method of solving scattering problems, separation of variables, has been applied previously in this book to a homogeneous sphere, a coated sphere (a simple example of an inhomogeneous particle), and an infinite right circular cylinder. It is applicable to particles with boundaries coinciding with coordinate surfaces of coordinate systems in which the wave equation is separable. By this method Asano and Yamamoto (1975) obtained an exact solution to the problem of scattering by an arbitrary spheroid (prolate or oblate) and numerical results have been obtained for spheroids of various shape, orientation, and refractive index (Asano, 1979 Asano and Sato, 1980). 

Noctilucent cloud particles are now generally believed to be ice, although more by default—no serious competitor is still in the running—than because of direct evidence. The degree of linear polarization of visible light scattered by Rayleigh ellipsoids of ice is nearly independent of shape. This follows from (5.52) and (5.54) if the refractive index is 1.305, then P(90°) is 1.0 for spheres, 0.97 for prolate spheroids, and 0.94 for oblate spheroids. 

Some insight into how departures from Rayleigh theory affect linear polarization can be obtained from calculations of Asano and Sato (1980) for randomly oriented oblate spheroids with refractive index 1.33, which is near enough to that of ice, axial ratio a/c = 5, and size parameter 2ma/ = 5 for a... 

The refractive index n of ZnS, which determines its scattering properties, is 2.37 and is much greater than that of plastics and binders (n = 1.5-1.6). Spheroidal ZnS particles have their maximum scattering power at a diameter of 294 nm. Barium sulfate does not directly contribute to the light scattering due to its relatively low refractive index (n = 1.64), but acts as an extender, and increases the scattering efficiency of the ZnS. 

Cadmium yellow consists of pure cadmium sulfide (golden yellow color) or mixed crystals of zinc and cadmium sulfide [8048-07-5], (Cd, Zn)S, in which up to one-third of the cadmium can be replaced by zinc. The density of this pigment is 4.5-4.8 g/cm3 and its refractive index is 2.4-2.5. The prevalent parcticle size is approx. 0.2 pm with cubic to spheroidal habits. Cadmium yellow is practically insoluble in water and alkali, and of low solubility in dilate mineral acid. It dissolves in concentrated mineral acid with generation of hydrogen sulfide. 

Cadmium red consists of cadmium sulfoselenide [12656-57-4], [58339-34-7], Cd(S,Se), and is formed when sulfur is replaced by selenium in the cadmium sulfide lattice. With increasing selenium content, the color changes to orange, red, and finally dark red. The density of these pigments increases correspondingly from 4.6 to 5.6 g/cm3 and the refractive index from 2.5 to 2.8. The crystals have cubic or spheroidal habits, the prevalent particle size is 0.3-0.4 pm. 

Figure 2. Mixture of unreacted kaolinite and of spheroids (high index faces exposed) of zeolite omega obtained in the first stages of the crystallization. Scale bar 5 urn.

The direction of the principal axes of the index of refraction tensor n can be described by the indicatrix. For isotropic crystals the indicatrix is a sphere. For positive uniaxial crystals it is a prolate spheroid (ns > n0j) for negative uniaxial crystals it is an oblate spheroid (nol > n,). For orientations away from the principal axis orientations, the extraordinary ray will have a refractive index h - intermediate between nm and ne. 

This section presents results performed on specially characterized hematite spheroidal particles of major/minor axes 0.13 0.02 pm and 0.07+ 0.01 pm respectively. The particles are suspended in water. The illumination source is a lOmW stabilized 632.8mn HeNe laser. The relative refractive index of the particles at this wavelength is n = 1.698 0.0149. T he detectors employed are photon-counting photomultipliers with red-enhanced photocathodes. 

Figure 2 The scattering problem illustrated for a spheroidal particle with orientation e and effective refractive index = n — i k. The surrounding medium is nonabsorbing, with the real refractive index ne, The speed of light within the medium is c = Co/r e, where Cq is the speed of light in vacuum. The incident plane wave has frequency v (ie, wavelength X = dv) and a wave vector collinear to o. A propagation direction of the radiation scattered by the particle is denoted as a. 0 is the angle between a and co. ...

The radiation characteristics of axisymmetric spheroidal microorganisms, such a C. reinhardtii (Fig. lA), with major and minor chameten a and b can be predicted numerically using (i) the T-matrix method (Waterman, 1965 Mackowski, 1994 Mishchenko et al., 2002, 1995), (ii) the discrete-dipole approximation (Draine, 1988), and (iii) the finite-difference time-domain method (Liou, 2002). Most often, however, they have been approximated as homogeneous spheres with some equivalent radius r and some effective complex index of refraction nix = n +ikx (Pettier et al., 2005 Berbero u et al., 2007 Dauchet et al., 2015), as discussed in Section 3.6.1. 

Figure 10 shows representative streamline patterns for oblates, prolates, and spheres in Newtonian and shear-thinning fluids similar results (not shown here) are obtained for dilatant fluids. The streamline patterns for sphere match with the literature predictions for example see Clift et al. (1978) for Newtonian fluids and Adachi et al. (1973) for power-law fluids (n< 1). The effect of the flow behavior index on streamline patterns for a sphere is found to be negligible, except the fact that the wake formation is somewhat delayed. For prolate spheroids (E = 5), no wake formation occurs even at Re = 100, whereas for oblates, a visible wake is formed even at Re= 10 for = 0.2. To recap, the flow patterns appear to be much more sensitive to the... 

Figure 10. Normalized light scattering intensity distribution with scattering angle for two different cases absorptive and non-absorptive extra-eellular medium. Cell parameters are the same as for the spheroid shape cell illustrated in Fig. 8. The values of the imaginary part of the refractive index of the extra-cellular medium...

Problem 5.6. Estimate the maximum SERS intensity enhancement factor for A = 820 nm at a rough silver surface based on the simulation of surface roughness by (a) small spheres (b) small spheroids with the depolarization factor A = 0.1. The index of refraction of silver at the given wavelength is fl = 0.04 + / 5.73. For the estimate, neglect the variation of the dielectric function over the vibrational frequency shift. 

The nonlinearly optically active medium is an array of spheroidal alkali clusters with index of refraction n, which gives rise to a DFWM signal intensity (Fisher 1983) ... 

In the next example, we show results computed for a perfectly conducting spheroid of size parameter h a = 10, aspect ratio ajh = 2, and Euler angles of rotation Op = / p = 45°. The perfectly conducting spheroid is simulated from the dielectric spheroid by using a very high value of the relative refractive index (rur = l.e- -30), and the version of the code devoted to the analysis of perfectly conducting particles is taken as reference. For this application, the... 

The results plotted in Fig. 3.70 are computed with the TMULT routine and show the differential scattering cross-sections for the two-spheroid system. The particles are placed in a medium with a refractive index of 1.2, and the... 

---