Diffuse reflectance Kubelka-Munk equation

The nature and the distribution of different types of Fe species in calcined (C) and steamed (S) samples were investigated by means of UV-vis spectroscopy. UV-vis spectra of Fe species were monitored on UV-vis spectrometer GBS CINTRA 303 equipped with a diffuse reflectance attachment with an integrating sphere coated with BaS04 and BaS04 as a reference. The absorption intensity was expressed using the Schuster-Kubelka-Munk equation. 

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... 

Diffuse reflectance R is a function of the ratio K/S and proportional to the addition of the absorbing species in the reflecting sample medium. In NIR practice, absolute reflectance R is replaced by the ratio of the intensity of radiation reflected from the sample and the intensity of that reflected from a reference material, that is, a ceramic disk. Thus, R depends on the analyte concentration. The assumption that the diffuse reflectance of an incident beam of radiation is directly proportional to the quantity of absorbing species interacting with the incident beam is based on these relationships. Like Beer s law, the Kubelka-Munk equation is limited to weak absorptions, such as those observed in the NIR range. However, in practice there is no need to assume a linear relationship between NIRS data and the constituent concentration, as data transformations or pretreatments are used to linearize the reflectance data. The most used linear transforms include log HR and Kubelka-Munk as mathemati-... 

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. 

Diffuse reflection from powder sample is a complex combination of transmission, internal and external reflections, and scattering. It is dependent on the particle size, absorption and refractive indices of the studied material. The case of proper prepared powder diffuse reflection R carries the information primarily about the transmission spectrum of the sample (Willey 1976 Fuller and Griffiths 1978). The traditional method of the absorption spectra (K) calculation on the base of the diffuse reflection R is the Kubelka-Munk equation K = (1 - R)2S/2Rc, where S is the scattering coefficient, concentration of the studied material is c = 1 in our case. 

Is it a real absorption It is well known that analysis on the base of the Kubelka-Munk equation is applicable at diffuse reflection R not much less than R 30%. The case of low diffuse reflection the deviations from linearity should be taken into account. We have R is near 1%. So, we should be careful The case of strongly absorbing samples it is possible to dilute them in nonabsorbent powder, for example in KBr powder. We have not used this traditional method because were afraid of possible chemical reactions at nigh temperature treatment of the mixture of the hydrogenated SWNTs with KBr. 

Fig. 11.6 Absorption spectra K restored from diffuse reflection R by using the Kubelka-Munk equation. Spectra 1, 2, and 3 are initial nanostructures, hydrogenated and annealed at 700°C during 6 h, respectively, (a) Spectra of NFs, (b) SWNTs. Dashed curves are Drude approximation of the absorption spectra, K = A. v0 5...

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 ... 

Loyolka, S.K. Riggs, C.A. Inverse problem in diffuse reflectance spectroscopy accuracy of the kubelka-munk equations. Appl. Spectrosc. 1995, 49, 1107-1110. 

Ground-state absorption studies of the probe molecules adsorbed within zeolites were performed by UV-visible diffuse reflectance spectroscopy. Cationic form of the zeolite without probe molecule was used as a comparison sample. The remission function was calculated by using the Kubelka-Munk equation. Steady-state photoluminescence studies were carried out with the Hilger spectrofluorimeter. The spectra were recorded at room temperature and 77 K, respectived. 

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 obtained chitosan carriers and catalytic systems on their base were studied by transmission and diffuse-reflectance FTIR spectroscopy. IR-spectra were obtained in Nicolet Impact 410 equipment. To record the diffuse-reflectance spectra the samples were evacuated at 100°C for 2 h. The quantitatively spectrum analyses were performed using Kubelka-Munk equation according to the program OMNIC [5]. 

The crystal structure was checked by X-Ray diffraction using a Philips PW 1877 automated powder diffractometer with CuKa radiation. AAS and ICP-AES measurements were carried out with Perkin Elmer instruments type 1100, 5000 Z and Plasma-40. UV/VIS measurements were carried out on a Varian Cary spectrometer. Diffuse reflectance spectra were recorded against a barium sulphate reference on the Varian Cary-3 and reflectance spectra were converted according to the Kubelka-Munk equation. 

Kubelka and Munk developed a theory describing the diffuse reflectance process for powdered samples, which relates the sample concentration to the scattered radiation intensity. The Kubelka-Munk equation is as follows ... 

The diffuse reflectance experiment requires that the incident beam penetrate into the sample, but the path length is not well defined. The path length varies inversely with the sample absorptivity. The resulting spectrum is distorted from a fixed path absorbance spectmm and is not useful for quantitative analysis. Application of the Kubelka-Munk equation is a common way of making the spectral response linear with concentration. 

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 ... 

In diffuse reflectance spectroscopy, there is no linear relation between the reflected light intensity (band intensity) and concentration, in contrast to traditional transmission spectroscopy in which the band intensity is directly proportional to concentration. Therefore, quantitative analyzes by DRIFTS are rather complicated. The empirical Kubelka -Munk equation relates the intensity of the reflected radiation to the concentration that can be used for quantitative evaluation. The Kubelka-Munk equation is defined as ... 

Scattering coefficient, Kubelka-Munk Multiple (diffuse) scattering coefficient for a unit thickness and concentration of scattering material in a medium of different refractive index as used in the Kubelka-Munk equation. It is the rate of increase of reflectance of a layer over black as thickness is increased. Hence, the assumption... 

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 ... 

In the case of the reflection mode, the incident light also illuminates perpendicular to the sample surface. Then NIR lights propagate in a sample with a series of absorption, scattering, diffraction, and transmission. And, finally, diffuse reflected light radiates from the sample surface. In this case, the sample should be opaque (for example, a powdered sample). The Kubelka-Munk equation is applicable for this mode where both absorption and scattering coefficients are important factors to explain the variation of NIR spectra. Normally, the incident light could not reach a... 

The use of the Kubelka-Munk equation for quantitative analysis by diffuse reflectance spectroscopy is common for measurements in the visible, mid-IR and far-IR regions of the spectrum, but not in the near-IR region. As has been pointed out [187, 188], almost all near-IR diffuse reflectance spectra have been converted to log(l// ) R = reflectance of the sample relative to that of a non-absorbing sample). The use of log(l// ) instead of the K-M function provides a more linear relationship between reflectance and concentration. Olinger et al. [189] explain this behaviour by the effective penetration depth of the beam, which is very short, when absorption is strong. For many of the algorithms developed to achieve multicomponent determinations from the diffuse reflectance spectra of powdered samples, a linear dependence of band intensity on analyte concentration is not absolutely mandatory for an analytical result to be obtained. 

Diffuse reflectance IR spectroscopy has become an attractive alternative to mulls with the introduction of DRIFT cell by Griffiths,29 later modified by Yang.30 Since materials are dispersed in a nonabsorbing medium and not subjected to thermal or mechanical energy during sample preparation, DRIFT spectroscopy is especially suitable for the qualitative/quantitative analysis for polymorphs, which are prone to solid-state transformations. The Kubelka-Munk (K-M) equation,31 which is analogous to Beer s law for transmission measurements, is used to quantitatively describe diffusely-reflected radiation ... 

The most widely accepted theory for quantitative analysis of simple diffuse reflectance is that developed by Kubelka and Munk, who established the equation ... 

The measurement of diffuse reflectance effectively involves focusing the infrared source beam onto the surface of a powder sample and using an integrating sphere to collect the scattered infrared radiation.59 The technique requires careful attention to sample preparation, and often one must dilute the analyte with KBr powder to reduce the occurrence of anomalous effects.60 In practice, one obtains the spectrum of the finely ground KBr dispersant, and then ratios this to the spectrum of KBr containing the analyte. The relative reflectance spectrum is converted into Kubelka-Munk units using standard equations,61 thus obtaining a diffuse reflectance spectrum that resembles a conventional IR absorption spectrum. 

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 ... 

Figure 3.9 shows the reflectance spectra of unbleached and peroxide-bleached TMP from black spruce. Reflectance is the ratio of the intensity of reflected to incident light, and is thus mathematically analogous to the transmittance of the Beer-Lambert Law. Unlike transmittance, however, reflectance is not easily rendered to a quantity proportional to chromophore concentration. Diffuse reflectance is related to chromophore concentration by the Kubelka-Munk remission function, Equation... 

The left-hand side of the equation is commonly called the remission function or the Kubelka-Munk function and is frequently denoted by /IjR )- Experimentally, one seldom measures the absolute diffuse reflecting power of a sample, but rather, the relative reflecting power of the sample compared to a suitable white standard. In that case, k = 0 in the spectral region of interest, JRocgtd = 1 [from equation 9.4)], and one determines the ratio... 

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... 

The samples were characterized by UV-visible reflectance spectroscopy using a Varian-Carry-5 spectrometer equipped with a double monochromator. Diffuse reflectance spectra were recorded in air at room temperature in the range 200-800 nm against alumina as reference. Spectra are presented indicating the frequency of the Schulz-Munk-Kubelka equation as function of the wavelength. 

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