Isotropic scattering coefficient

It should be noted here that Kubelka and Munk define scattering differently than does Mie (or Schuster). Mie defines scattering as radiation traveling in any direction after interaction with a particle. Kubelka and Munk defined scattered radiation as only that component of the radiation that is backward reflected into the hemisphere bounded by the plane of the sample s surface. In effect, the defining of S as equal to 2s makes S an isotropic scattering coefficient, with scatter equal in both the forward and backward directions. 

However, when dealing with molecular rotations for linear molecules, it is very desirable to express AA in the language of spherical harmonics [73]. For linear centrosymmetric molecules all nonzero spherical harmonics coefficients of Eq. (56) for anisotropic scattering (k = 2) are assembled in the Appendix given in Ref. 16 and for isotropic scattering (k = 0), in Ref. 15. 

Fig. 2-20. Photodissociation coefficients for N02, HCHO, CHjCHO, and 0( D) produced from ozone, for ground-level, clear-sky conditions. Nitrogen dioxide diamonds are measurements of Madronich et al. (1983), solid points are data of Marx et al. (1984), the solid line represents calculations of Madronich et al. (1983) using the method of Isaksen et al. (1977), and the dashed line is for isotropic scattering. Formaldehyde open points are measurements of Marx el al. (1984), and the solid line represents calculations of Calvert (1980). Acetaldehyde triangles are measurements of Marx et al. (1984), and the solid line represents calculations of Meyrahn et al. (1982). Ozone solid points are measurements of Dickerson et al. (1979) and Bahe et al. (1980) normalized to 325 Dobson units of total ozone overhead, and the solid line represents calculations of Dickerson et al. (1979).

Isotropic scattering indicates that the radiant energy incident on a volume element is uniformly distributed to all directions. For an isotropically scattering medium, all a, coefficients of the phase function are zero, except a0. If only a0 and the first coefficient a, are considered, then one obtains the linearly anisotropic phase function, which means that the phase function is a linear function of cos 0 (or, in the case of an azimuthally symmetric medium, a linear function of p = cos 9). 

Hale and Bohn [252] measured the scattered radiation from a finite sample of reticulated alumina from an incident laser beam at 488 nm. They then matched Monte Carlo predictions of the scattered radiation calculated from various values of extinction coefficient and scattering albedo and chose the values that best matched the experimental data for reticulated alumina samples of 10, 20, 30, and 65 ppi. A scattering albedo of 0.999 and an assumed isotropic scattering phase function reproduced the measured data for all pore sizes. The large reported albedo value indicates that alumina is very highly scattering and that radiative absorption is extremely small for this material. 

Mital et al. [256] have measured the radiative extinction coefficient and scattering albedo for five different porous ceramics in the temperature range from 1200-1400 K, assuming a gray isotropically scattering medium. Doermann and Sacadura [257] have proposed a method for predicting the radiative absorption and scattering coefficients and the phase function of open-celled materials based on the structure of the solid. Recently these authors have presented a comprehensive review of the subject [258]. 

The coefficients which correspond to this particular scattering process are easily computed if we recall that the frequency function for the change of lethargy in the case of isotropic scattering in (C) was given hy [cf. Eq. (4.50)1... 

For the special case of isotropic scattering in (C), we can compute the coefficients of the scattering function from (7.139). We require here only the first two ... 

As noted above, the phenomenological two-fiux theories that have been developed on the basis of the radiation transfer equation can be considered continuum theories. Continuum theories consider the absorption and scattering coefficients as properties of an irradiated isotropic layer of infinitesimal thickness. On the other hand, discontinuum theories consider layers containing a collection of particles. Consequently, the thickness of a layer is dictated by the size of the scattering and absorbing particles. Optical constants can then be determined from the scattering and absorption properties of these particles. 

Lateral density fluctuations are mostly confined to the adsorbed water layer. The lateral density distributions are conveniently characterized by scatter plots of oxygen coordinates in the surface plane. Fig. 6 shows such scatter plots of water molecules in the first (left) and second layer (right) near the Hg(l 11) surface. Here, a dot is plotted at the oxygen atom position at intervals of 0.1 ps. In the first layer, the oxygen distribution clearly shows the structure of the substrate lattice. In the second layer, the distribution is almost isotropic. In the first layer, the oxygen motion is predominantly oscillatory rather than diffusive. The self-diffusion coefficient in the adsorbate layer is strongly reduced compared to the second or third layer [127]. The data in Fig. 6 are qualitatively similar to those obtained in the group of Berkowitz and coworkers [62,128-130]. These authors compared the structure near Pt(lOO) and Pt(lll) in detail and also noted that the motion of water in the first layer is oscillatory about equilibrium positions and thus characteristic of a solid phase, while the motion in the second layer has more... 

The rotational diffusion coefficient Dr of a rodlike polymer in isotropic solutions can be measured by electric, flow, and magnetic birefringence, dynamic light scattering, and dielectric dispersion. However, if the polymer has some flexibility, its internal motion makes it difficult to extract Dr for the end-over-end rotation of the chain from data of these measurements. In other words, Dr can be measured only for nearly rodlike polymers. 

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