Representative layer theory

The concept of the representative layer came about though the work of Donald Dahm, with the assistance of Kevin Dahm and Karl Norris, to describe spectroscopic properties of particulate samples, especially as applied to near-infrared spectroscopy. A representative layer has the same void fraction as the sample it represents and each particle type in the sample has the same volume fraction and surface area fraction as does the sample as a whole. The spectroscopic properties of a representative layer can be derived from the spectroscopic properties of particles, which may be determined by a wide variety of ways. While a representative layer could be used in any theory that relies on the mathematics of plane parallel layers, there is a set of definitions and mathematics, some old and some new, which have become part of representative layer theory.

Representative layer theory can be used to determine the spectroscopic properties of an assembly of particles from those of the individual particles in the assembly. The sample is modeled as a series of layers, each of which is parallel to each other and perpendicular to the incident beam. The mathematics of plane parallel layers is then used to extract the desired properties from the data, most notably that of the linear absorption coefficient which behaves in the manner of the coefficient in Beer’s law. The representative layer theory gives a way of performing the calculations for new sample properties by changing the properties of a single layer of the particles, which doesn’t require reworking the mathematics for a sample as a whole.

History

The first attempt to account for transmission and reflection of a layered material was carried out by Stokes in about 1860 and led to some very useful relationships. Lord Rayleigh and Mie developed the theory of single scatter to a high degree, but Schuster was the first to consider multiple scatter. He was concerned with the cloudy atmospheres of stars, and developed a plane-parallel layer model in which the radiation field was divided into forward and backward components. This same model was used much later by Kubelka and Munk, whose names are usually attached to it by spectroscopists.

Following WWII, the field of Reflectance Spectroscopy was heavily researched, both theoretically and experimentally.  The remission function, $F(R_{\infty })$, following Kubelka-Munk theory, was the leading contender as the metric of absorption analogous to the absorbance function in transmission absorption spectroscopy. 

The form of the K-M solution originally was: $F(R_{\infty })\equiv {\frac {(1-R_{\infty })^{2}}{2R_{\infty }}}={\frac {a_{0}}{r_{0}}}$, but it was rewritten in terms of linear coefficients by some authors, becoming $F(R_{\infty })\equiv {\frac {(1-R_{\infty })^{2}}{2R_{\infty }}}={\frac {k}{s}}$ , taking $k$ and $s$ as being equivalent to the linear absorption and scattering coefficients as they appear in the Bouguer-Lambert law, even though sources who derived the equations preferred the symbolism ${\frac {K}{S}}$ and usually emphasized that $K=2k$ and $S$ was a remission or backscattering parameter, which for the case of diffuse scatter should properly be taken as an integral.

In 1966, in a book entitled Reflectance Spectroscopy, Harry Hecht had pointed out that the formulation $F(R_{\infty })={\frac {k}{s}}$ led to $\log F(R_{\infty })=log(k)-log(s)$, which enabled plotting $F(R_{\infty })$ "against the wavelength or wavenumber for a particular sample" giving a curve corresponding "to the real absorption determined by transmission measurements, except for a displacement by $-log(s)$ in the ordinate direction." However, in data presented, "the marked deviation in the remission function ... in the region of large extinction is obvious." He listed various reasons given by other authors for this "failure ... to remain valid in strongly absorbing materials", including: "incomplete diffusion in the scattering process"; failure to use "diffuse illumination; "increased proportion of regular reflection"; but concluded that "notwithstanding the above mentioned difficulties, ... the remission function should be a linear function of the concentration at a given wavelength for a constant particle size" though stating that "this discussion has been restricted entirely to the reflectance of homogeneous powder layers" though "equation systems for combination of inhomogeneous layers cannot be solved for the scattering and absorbing properties even in the simple case of a dual combination of sublayers. ... This means that the (Kubelka-Munk) theory fails to include, in an explicit manner, any dependence of reflection on particle size or shaper or refractive index".

The field of Near infrared spectroscopy (NIR) got its start in 1968, when Karl Norris and co-workers with the Instrumentation Research Lab of the U.S. Department of Agriculture first applied the technology to agricultural products. The USDA discovered how to use NIR empirically, based on available sources, gratings, and detector materials. Even the wavelength range of NIR was empirically set based on the operational range of a PbS detector. Consequently, it was not seen as a rigorous science: it had not evolved in the usual way from research institutions to general usage. Even though the Kubelka-Munk theory provided a remission function that could have been used as the absorption metric, Norris selected $log(1/R_{\infty })$ for convenience. He believed that the problem of non-linearity between the metric and concentration was due to particle size (a theoretical concern) and stray light (an instrumental effect).

In 1976, Hecht published an exhaustive evaluation of the various theories which were considered to be fairly general. In it, he presented his derivation of the Hecht finite difference formula by replacing the fundamental differential equations of the Kubelka-Munk theory by the finite difference equations, and obtained: $F(R_{\infty })=a{\biggl (}{\frac {1}{R}}-1{\biggr )}-{\frac {a^{2}}{2r}}$, where $a$ and $r$ are the fractions of light absorbed and remitted by a single layer.  He noted "it is well known that a plot of $F(R_{\infty })$versus $K$ deviates from linearity for high values of $K$, and it appears that (this equation) can be used to explain the deviations in part", and "does represent an improvement in the range of validity and shows the need to consider the particulate nature of scattering media in developing a more precise theory by which absolute absorptivities can be determined."