Schuster absorption 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 ... 

UV-VIS-NIR diffuse reflectance (DR) spectra were measured using a Perkin-Elmer UV-VIS-NIR spectrometer Lambda 19 equipped with a diffuse reflectance attachment with an integrating sphere coated by BaS04. Spectra of sample in 5 mm thick silica cell were recorded in a differential mode with the parent zeolite treated at the same conditions as a reference. For details see Ref. [5], The absorption intensity was calculated from the Schuster-Kubelka-Munk equation F(R ,) = (l-R< )2/2Roo, where R is the diffuse reflectance from a semi-infinite layer and F(R00) is proportional to the absorption coefficient. 

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... 

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. 

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