Schuster scattering coefficient

Complex processes are involved in transmittance and reflectance of scattered radiation, which are theoretically described by Schuster [4]. In an ideal scattering medium all fluxes of light can be summed up as components of two vectors. Vector I stands for the light flux in the direction of the incident light, and the vector J describes the light intensity in the antiparallel direction. With k, the absorption coefficient, and 5", the scattering coefficient, the two Schuster equations are as follows ... 

Equations (2) and (3) arc formally identical with the earlier Kubelka s hyperbolic solutions of differential equations for forward and backward fluxes [3], although the two theories start from different assumptions and employ different definitions of constants characterizing the scattering and absorption properties of the medium. The constants a, b and Y are related to what has become known as the Schuster-Kubelka-Munk (SKM) absorption K and scattering S coefficients as K/S = a 1 and SbZ = Y. In Chandrasekhar s theory, the true absorption coefficient av = Kvp( 1 - mo) and true scattering coefficient oy = Kvp mo. There are simple relations between the Chandrasekhar and the SKM coefficients... 

The reflectance spectra were recorded in the geometry R0,d (4) in the frequency range 5000-40,000 cm-1 in a previously described apparatus (4, 6). The absorbance is represented by the logarithm of the Schuster-Kubelka-Munk (SKM) function, F(RX) = (1 — RX)2/2RX, where Rx is the reflectance measured against a white standard. Since the comparison of the reflectance spectra with the transmission spectra of complexes in molecular sieves revealed that the scattering coefficient is a constant independent of the wavelength (4), the logarithm of the SKM function is, but for an additional constant, a correct representation of absorbance. 

Numerous researchers have developed their own simplified solutions to the radiation transfer equation. The first solution were Schuster s equations (3), in which, for simplification, the radiation field was divided into two opposing radiation fluxes (+z and -z directions). The radiation flux in the +z direction, perpendicular to the plane, is represented by /, and the radiation flux in the -z direction, resulting from scattering, is represented by J. The same approximation was used by Kubelka and Munk in their equations, in the exponential (4) as well as in the hyperbolic solution (5). In the exponential solution by Kubelka-Munk, a flat layer of thickness z, which scatters and absorbs radiation, is irradiated in the -z direction with monochromatic diffuse radiation of flux I. In an infinitesimal layer of thickness dz, the radiation fluxes are going in the + direction J and in the -direction I. The average absorption in layer on path length dz is named K the scattering coefficient is S. Two fundamental equations follow directly ... 

Unlike Schuster, Kubelka envisioned using the solution for dense systems. While Schuster defined the two constants k and i in terms of the absorption and scattering coefficients for single scattering, Kubelka simply defines K and S in the equations as absorption and scattering coefficients for the densely packed sample as a whole. A tabulation of the variables that are used in their derivation is found in Table 3.1. Figure 3.3 shows a schematic representation of the type of system for which Kubelka and Munk derived their solution. 

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. 

The symbol s used by Schuster is identical to the albedo coq for single scattering. It is relevant to discuss a coefficient of single scattering for a continuum model in that these coefficients relate to the reflectance measured as if the particles in the model were exploded apart so that only single scattering could occur. 

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