chapter  3
44 Pages

Continuum and Discontinuum Theories of Diffuse Reflection

WithPeter R. Griffiths, Donald J. Dahm

Scatter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.5 Discontinuum Theories of Diffuse Reflection∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.5.1 Theory for an Assembly of Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.5.2 Theory for Sheets and an Assembly Thereof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.5.2.1 The Stokes’ Formulas for an Assembly of Sheets . . . . . . . . . . . . . . . . . . . . . . . 44 3.5.2.2 The Dahm Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.5.3 The Representative Layer Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.5.3.1 Model for a Layer Representative of Particulate Solids . . . . . . . . . . . . . . . . 46 3.5.3.2 Absorption and Remission of the Representative Layer . . . . . . . . . . . . . . . . 46 3.5.3.3 Mathematical Expression of Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.6 Application of Theory to Model Systems∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.6.1 Example 1: Graphite in NaCl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.6.2 Example 2: Carbazole in a Matrix of Varying Absorption . . . . . . . . . . . . . . . . . . . . . . . . 52 3.6.3 Example 3: Mixture of Wheat and Rape Seed Meal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.7 Experimental Considerations for Reflection Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.7.1 Depth of Penetration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.7.2 Effect of Relative Reflectance and Matrix Referencing . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

DBURNS: “7393_c003” — 2007/7/27 — 17:23 — page 22 — #2

The use of near-infrared (NIR) diffuse reflection for the quantitative analysis of products and commodities is now widely accepted. For many of the algorithms developed to achieve multicomponent determinations from the diffuse reflection spectra of powdered samples, a linear dependence of band intensity on analyte concentration is not absolutely mandatory for an analytical result to be obtained. Nonetheless it is probably true to say that all of these algorithms yield the most accurate estimates of concentration when the intensity of each spectral feature is linearly proportional to the analyte concentration. In an analogous manner to transmission spectrometry, reflectance is input to most of these algorithms as log(1/R′), where R′ is the reflectance of the sample relative to that of a nonabsorbing standard, such as a ceramic disk. The use of log(1/R′) as the preferred ordinate is contrary to what most physical scientists would consider appropriate for a diffuse reflection measurement on an optically thick sample. Thus an understanding of the theories of diffuse reflection and the validity of the assumptions for each theory should be helpful in understanding the strengths and limitations of NIR diffuse reflection. Perhaps, it will even help to explain why accurate analyses may be made when band intensities are expressed as log(1/R′).