Kubelka-Munk equation, infinite

The Kubelka-Munk theory treats the diffuse reflectance of infinitely thick opaque layers [4], a situation achieved in practice for UV/VIS spectroscopy through the use of powder path lengths of at least several millimeters. In this instance, the Kubelka-Munk equation has the form... 

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. 

This equation gives the ratio of absorption and scattering coefficients in the case of R diffuse reflectance in an infinitely thick, opaque layer. In the presence of a sample with A molar absorptivity and c molar concentration, the Kubelka-Munk equation takes the following form ... 

Roo is the reflectance of an infinitely thick sample (in the near-infrared, this means an approximate 5-mm thickness and more). The theory was recently revisited by Loyalka and Riggs, ° who reinvestigated the accuracy of the Kubelka-Munk equations. They found that the coefficient k must be replaced by k = 2a with the absorption coefficient a = In(lO) ec, as derivable from Beer s law for the latter equation In(lO) = 2.303, e the molar absorptivity, and c the molar concentration. Such a dependency for k was stated earlier by other researchers when comparing more refined radiation transport theories for biomedical applications, e.g., Ref.[ l... 

The Kubelka-Munk equation is only valid when R corresponds to the diffuse reflectance of an opaque layer of infinite thickness, so that the background is no longer visible (i.e., the difference in intensity between the incident and reflected beams is independent of the thickness of the structure). 

The UV/vis spectra were recorded on a Perkin-Elmer Lambda 900 UV/vis spectrometer equipped with a diffuse reflectance and transmittance accessory (PELA-1000). The accessory is essentially an optical bench that includes double-beam transfer optics and a six-inch integrating sphere. Background corrections were recorded using a Labsphere SRS-99-020 standard. The reflectance data from were converted to k/s values by using the Kubelka-Munk theory (1931). The Kubelka-Munk equation describes the infinite reflectance as a function of absorption and scattering ... 

This function has become the fundamental law of diffuse reflectance spectroscopy. It relates the diffuse reflectance R of an infinitely thick, opaque layer and the ratio of the absorption and scattering coefficients K/S. Since the scattering coefficient is virtually invariable in the presence of a chromatographic band, the Kubelka-Munk equation can be written in the form ... 

The most important and widely used quantity derived from the Kubelka-Munk theory is the reflectance of an opaque (infinitely thick) film that is described by a very simple equation ... 

The DR spectrum of a dilute sample of "infinite depth" (i.e., up to 3 mm) is usually calculated with reference to the diffuse reflectance of the pure diluent to yield the reflectance, Ri. RiA is related to the concentration of the sample, c, by the Kubelka-Munk (K-M) equation ... 

The optical measmements of diffuse reflectance are dependent on the composition of the system. Several theoretical models have been proposed for diffuse reflectance, which are based on the radiative transfer theory, and all models consider that the incident hght is scattered by particles within the medium. The most widely used theory in photometric sensors is the Kubelka-Munk theory, in which it is assumed that the scattering layer is infinitively thick, which may, in practice, be the case with the chemical transducers utilized in photometric sensors. The absolute value of the reflectance R is related to the absorption coefficient K and the scattering coefficient S by the equation... 

Treatment of diffuse reflectance IR data normally involves the Kubelka-Munk (KM) Equation (3.13) that takes account of scattering and absorption characteristics [45]. The constants s and k are used to describe the scattering and absorption properties, respectively, and R is the reflectivity of an infinitely thick sample ... 

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