Theory of Diffuse Reflection

Diffuse reflection (DR) spectra result from the radiation incident on a powdered sample that is absorbed as it refracts through each particle and is scattered by the combined process of reflection, refraction, and diffraction. That fraction of the incident radiation that reemerges from the upper surface of the sample is said to be diffusely reflected. Because DR spectra result from an absorption process, they have the appearance of transmission spectra (i.e., bands appear in absorption), unlike the case for Fresnel reflection spectra of bulk samples (see Chapter 13). When DR spectra are acquired on Fourier transform spectrometers, the singlebeam spectra of the sample and a nonabsorbing reference are measured separately and ratioed to produce the reflectance spectrum, Rfy). 

Like transmission spectra, DR spectra must be converted to a different form in order to convert R(v) to a parameter that varies linearly with concentration. By analogy to transmission spectrometry, most practitioners of DR near-infrared (NIR) spectrometry convert R(v) to log]o[l/R(v)]. Plots of logio[l/R(v)] versus concentration are not linear over wide concentration ranges but are perfectly adequate for multicomponent quantitative analysis provided a) that the concentration of each analyte does not vary by much more than a factor of 2, and (h) that the absorptiv-ities of the analytical bands are fairly low. These criteria are often obeyed for the determination of the components of commodities by DR near-infrared spectrometry, but are not usually valid for mid-infrared spectra, where absorptivities are one or two orders of magnitude higher than in the near infrared. 

The theories of diffuse reflection spectroscopy have been summarized by Griffiths and Dahm [1]. None of these theories allows R(y) to be converted to a parameter that varies linearly with concentration over a wide range, but the Kubelka-Munk theory has provided the simplest and most useful parameter for practical measurements in the mid-infrared. Kubelka and Munk [2,3] derived a relationship between the reflectance of a sample at infinite depth, Roo(v), and its absorption coefficient, fe(v), and scattering coefficient, (v). (A sample whose DR spectrum does not change as it is made thicker is regarded as being at infinite 

Fourier Transform Infrared Spectrometry, Second Edition, by Peter R. Griffiths and James A. de Haseth Copyright 2007 John Wiley Sons, Inc. 

Kubelka and Munk derived a parameter,/(/ oo). that is simply the ratio of the absorption coefficient and the scattering coefficient. This parameter is given at any wavenumber, v, by 

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There are other sources of noise, whose behavior cannot be described analytically. They are often principally due to the sample. A premier example is the variability of the measured reflectance of powdered solids. Since we do not have a rigorous ab initio theory of diffuse reflectance, we cannot create analytic expressions that describe the variation of the reflectance. Situations where the sample is unavoidably inhomogeneous will also fall into this category. In all such cases the nature of the noise will be unique to each situation and would have to be dealt with on a case-by-case basis. 

J.M. Olinger, PR. Griffiths and T. Burger, Theory of diffuse reflectance in the NIR region. In Handbook of Near-Infrared Analysis, 2nd edition, D. Burns and E.W. Ciurczak (eds), Marcel Dekker, New York, 19-52, 2001. 

The generally accepted theory of diffuse reflectance was developed originally by Kubelka and Munk [43,44] for application to infinitely thick, opaque layers. 

The Kubelka-Munk theory of diffuse reflectance is a good description of the optical properties of paper. The two parameters of the theory, absorption and scattering coefficient, are purely phenomenological, but are closely related to basic properties of paper. The absorption coefficient is approximately a linear function of the chrcmgphore concentration in the paper. The scattering coefficient is related to the nonbonded fiber surface area in the paper, or the area "not in optical contact," and the Fresnel reflectivity of that surface. 

Morris, R. V., Neely, S. C. Mendell, W. W. (1982) Application of Kebulka-Munk theory of diffuse reflectance to geologic problems The role of scattering. Geophys. Res. Lett., 9, 113-16. 

Simmons (1975) compared various theories of diffuse reflectance. He introduced a modified remission function, which explains deviations from linearity when F(p) is plotted versus k. He also concluded that the Kubelka-Munk function is proportional to the absorption coefficient k as obtained from transmission measurements for "weakly absorbing samples." Unfortunately, most literature is vague in that "weak" or "strong" absorption is not specified. One value given for "weak" is F(p) < 1 (Kellermann, 1979). 

JM Obnger, PR Griffiths, and T Burger. Theory of Diffuse Reflection in the NIR Region. In DA Burns and EW Ciurczak, eds. Handbook of Near Infrared Analysis. New York Marcel Dekker, 2001, pp. 19-52. 

P. R. Griffiths and D. J. Dahm, Continuum and Discontinuum Theories of Diffuse Reflections, in Handbook of Near-Infrared Analysis, 3rd ed., ed. D. A. Burns and E. W. Ciurczak, Taylor and Francis, Boca Raton, FL, 2008. 

What for the case of small particles is called backward and forward scatter is, for the case of particles with large, flat surfaces, the sum of reflection and transmission. For infinitesimally small particles, continuum theories of diffuse reflection may be applied. As particles get larger, it becomes more likely that the terms for geometrical optics will be applied and discontinuum theories are more relevant. 

In summary, therefore, although detailed investigations of the theory of diffuse reflection spectrometry by many workers have been carried out during the last century, they have not resulted in a single metric that is proportional to analyte coneentration (in the same manner as absorbance in transmission spectroscopy). Fortunately, simple conversion of reflectance values to log(l/R ) appears to be effective for many powdered samples being analyzed by NIR diffuse reflection spectrometry. 

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