Absorption coefficients, scattering media

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 ... 

Fig. 3.5-13 demonstrates the influence of sample and instrumental parameters on the intensities of the observed Raman lines, excited with the Nd YAG laser at 1064 nm. The lower abscissa shows the absolute wavenumber scale, the upper abscissa represents the Raman shift. The relative intensity of Raman radiation is calculated by Eq. 3.5-9, taking into account the absorption coefficient a cm ) of water (Fig. 3.5-3) and a constant Raman scattering coefficient (assumed to be equal to 1 cm ). The elastic scattering coefficients as parameters, r = 0, 10, 100, 500 cm , describe the properties of a liquid, a coarse, a medium, and a fine powder, respectively. The traces show the relative output voltage of a Germanium detector. 

The discrete ordinates method in a S4-approximation is used to solve the radiation transport equation. Since the intensity of radiation depends on absorption, emission and scattering characteristics of the medium passed through, a detailed representation of the radiative properties of a gas mixture would be very complex and currently beyond the scope of a 3D-code for the simulation of industrial combustion systems. Thus, contributing to the numerical efficiency, some simplifications are introduced, even at the loss of some accuracy. The absorption coefficient of the gas phase is assumed to have a constant value of 0.2/m. The wall emissivity was set to 0.65 for the ceramic walls and to a value of 0.15 for the glass pane inserted in one side wall for optical access. 

Here /,. is the irradiance of frequency between vand v + dv, r = kvpdz is the optical path length along inward normal z, p is the local density of the medium and kv is the effective absorption coefficient which includes radiative losses by absorption and scattering from the inclination to the z-axis, p - cos(0). The angular distribution of the scattered radiation is specified by the so called phase function p(0 p,p ) which is proportional to the rate at which light is being scattered from the direction p to p ... 

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... 

Here, k> and aK are the spectral absorption and scattering coefficients, respectively. The attenuation of radiation intensity by the medium is proportional to both of these coefficients. The spectral attenuation (or extinction) coefficient is expressed by the sum of the absorption and scattering coefficients, as... 

The Beer law is similar to the Lambert law with the exception that the absorption coefficient is expressed as the product of a unit absorption coefficient and the concentration of particles. In general, this definition is more fundamental and appropriate for application to dispersed media, where the concentration can be directly measured. If there are particles in the medium, depending on their size with respect to the wavelength of the incident radiation, they scatter as well as absorb the incident radiation. With increasing concentration, the multiple scattering effect becomes significant, and the Beer law deviates from the experimental measurements, especially if the size of the particles is comparable to the wavelength of radiation. Under these conditions, the complete radiative transfer equation should be solved. For solid materials, the Beer and Lambert laws are identical. 

Calculation of radiative properties through modeling of radiation absorption and reflection from the ceramic structure has not yet been attempted. Instead, the porous medium is treated as a homogeneous absorbing and scattering medium. For such a treatment, it is necessary to know the effective absorption and scattering coefficients of the medium as well as the scattering phase function. 

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]. 

It is important to rcali/c that the X-rays produced in the fluorescence process are generated not only from atoms at the surface of a sample bui also from atoms well below ihc surface. Thus, a part of both the incideiU radiation and the resuliiiigfluorescence iraverse a significani thickness of sample within which absorption and scattering can occur. T he extent either beam is attenuated depends on the mass absorption coefficient of the medium. which in turn i.s determined by the absorption coefficients of all of the elements in the sample. T here-... 

Absorption and scattering occur simultaneously because all molecules (and particles) both absorb and scatter. The attenuation of radiant energy in a medium is expressed by the extinction coefficient, which is the sum of absorption and scattering,... 

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