Componente Contenedor de Controles

Many of the algorithms that have been developed to achieve multi-component determination of a variety of samples have been based upon diffuse reflection. Of these the most accurate have had a dependence upon the linear

relationship between band intensity and analyte concentration. The band intensity is described as log 1/R where R is the reflectance of the sample compared to a non-absorbing standard (e.g. a ceramic disc). A brief summary of some of the major diffuse reflectance theories is given here. Ciurczak has given a more expanded explanation of the mathematics behind these

theories^®.

1.7.1 Kubelka - Munk theory

Kubelka and Munk^^ made the assumption that radiant energy is transferred from the incident NIR beam continuously and is converted to thermal vibrational energy of molecules. The decrease in intensity of the reflected light is related to the absorption coefficient of the reflectance material (i.e. the sample). The reflectance is described as being related to the absorption coefficient (K) and scattering coefficient (S) by

K/S = (1 - R f/2 R = f(R)

Equation 12 Kubelka - Munk Function

where R, the diffuse reflectance is a fraction of K/S and is proportional to addition of absorbing species to the sample medium.

On this basis is the assumption that the diffuse reflectance of the incident beam is directly proportional to the quantity of the absorbing species that interacts with the incident beam. Therefore, for quantitative determinations the

reflectance is dependant upon analyte concentration. Practically, in NIR terms a relative reflectance is measured rather than the absolute diffuse reflectance. The relative reflectance is equal to the ratio of intensity of radiation reflected from the sample/ intensity of radiation from a reference material.

The reference material used in the project is a ceramic disc. The absolute reflectance is described as:

M o

Equation 13 The Absolute Reflectance

where Is is the radiation intensity reflected from the sample and lo is the intensity of the incident radiation

Generally it is accepted that the Kubelka - Munk equation is a limiting equation and should only be applied for weakly absorbing bands when absorptivity and concentration are low, due to vibrational overtones and combination bands. The absorptivities of these bands are much weaker than the respective

fundamentals. Therefore, most analysis in the NIR region is considered weakly absorbing without dilution, however, many analytes are not isolated but

surrounded by a matrix of other components which are strongly absorbing of the incident radiation at analytical wavelengths. Therefore, unless a proper referencing method is used, the absorption matrix may cause deviations from the Kubelka/ Munk theory.

1.7.2 Lambert Cosine law

Lambert in 1760 sought to demonstrate that the reflected radiation from a perfect reflector (or as near as was physically possible to one) was of the same intensity regardless of the angle of observation or the angle of incidence^®.

dl^ !d f ^ CS,

dû) 71 cos cos (9 = Bcosd

Equation 14 Lambert Cosine law

where Ir is the remitted radiation flux, / is the surface area of the reflector in is the solid angle in steradians (sr), a is the angle of incidence, »9 is the angle of observation. So is the irradiation intensity in W / cm^, C is a constant representing the fraction of the incident that is remitted (<1 as some radiation is always absorbed) and B is the surface brightness in W / cm^ sr. There was some debate as to whether this was only possible where the

reflector was in fact a black body radiator acting as the ideal diffuse reflector (as used by Lambert), otherwise the angle of distribution of the reflected/

remitted radiation was dependant upon the angle of incidence. It was, however, argued that such a black body could not be deemed an ideal diffuse reflector as it would absorb a significant amount of the incident radiation anyway. As the ideal diffuse reflector has not been found in practice there will be deviations to the law.

1.7.3 Mie Scattering

This theory was developed by Mie in the early 1900s and deals with the

scattering of radiation light by isolated particles^^. Mie described the scattering of plane polarised radiation by a particle that can be both non-conductive and absorbing. The particle was spherical and had no size limit. He showed that the distribution of such scattered radiation was not uniform in all directions.

Equation 15 Mie Scattering

where l^s is the scattered intensity at a distance R from the sphere’s centre, Iq is the intensity of the incident radiation, À is the wavelength of the incident radiation and ii, /2 are functions of the angle of scattered radiation

Functions of the angle include harmonics or their derivatives with respect to the cosine of the angle of scattered radiation, the refractive index of the sphere and its surrounding medium and the ratio of particle circumference to wavelength. The basic equation (Equation 15) is for a non-conductive non-absorbing particle and for non-polarised incident radiation. If the particle does absorb this must be taken account of in ii and /2. Mie theory encompasses spherical particles of any size but is only applicable to a single incidence of scattering. For example, scattering by a system with well separated molecules. It is applicable where the particle size is smaller than the incident wavelength of radiation.

However, in most NIR applications the scattering will only take place by

molecules that are close together (e.g. solids) therefore multiple scattering will take place.

Scattering order was described by Theissing as the number of times a photon is scattered and described how, as the order of scattering increased, Mie scatter decreased and the angular distribution of scattered radiation tended to become isotropic^^. The conclusion was that for a sufficiently thick sample and for a sufficiently large number of particles, multiple scattering will occur for the majority of samples in NIR reflectance analysis so that an isotropic distribution of radiation should be reached. There was no definition of sufficient number of particles or sufficient thickness, it is difficult therefore to describe the multiple scattering within a densely packed medium in relation to a change in analyte concentration.