Theory Fraunhofer

The Mie theory (actually Mie s solution to Maxwell s equations for spheres) can be applied to spherical particles that are smaller than, similar in size to, and larger than the wavelength of light used (Mie, 1908). With particles much larger than the wavelength, the Mie theory can be simplified to the Fraunhofer theory. The mathematics of scattering is complicated for other than spherical shapes, and that is why the assumption that particles are spherical is often made. 

The more complex Mie Theory (lj must be invoked to analyze particles with dimensions near the wavelength of light. Fraunhofer theory is an interference phenomenon, and is described in various optics text books (.2,3.). It is adequate for most particle sizing applications and will be discussed in detail. Mie Theory requires a knowledge c the refractive index of the material. A unique use of polarized side scatter at several wavelengths is employed to obtain particle size channels in the submicron region. 

Refractive index consists of two parts, a real part that describes the refracted light and an imaginary (complex) part that describes absorption. In the case of very small particles, or particles where the complex part of the refractive index is near zero, light is transmitted through the partieles and interferes with the diffracted radiation. The interaction between the transmitted and diffracted radiation results in anomolous scattering and can be catered for in the Fraunhofer theory if both parts of the refractive index are known. 

Jones discussed ISO 13320 [178] and stated that application of this guideline ensured high reproducibility and highlighted the weakness of Fraunhofer theory as opposed to the Mie theory [179]. 

The specimens were dispersed in silicon oil and the Fraunhofer theory was used to calculate size distributions... 

The Fraunhofer theory considers only scattering at the contour of the particle and the near forward direction. No pr nowledge of the refractive index is required, and 1(0) simplifies to... 

In accordance with the Fraunhofer theory (which was introduced by Fraunhofer over 100 years ago), the special intensity distribution is given by. 

Several factors can affect the accuracy of Fraunhofer diffraction (i) particles smaller than the lower limit of Fraunhofer theory (ii) nonexistent ghost particles in particle size distribution obtained by Fraunhofer diffraction applied to systems containing particles with edges, or a large fraction of small particles (below 10 pm) (iii) computer algorithms that are unknown to the user and vary with the manufacturer s software version (iv) the composition-dependent optical properties of the particles and dispersion medium and (v) if the density of all particles is not the same, the result may be inaccurate. 

Fraunhofer theory is a simplification that ctmsiders the particles to he irnnsparenl. spherical, and much larger than the wavelength of the incident beam. Absorption and interference effects arc not considered as they are in Mic theory. Thus, the particle behaves like a circular... 

A second type of instrument determines particle size distribution of either dry powders or hquid dispersions by using Mie laser hght-scattering theory. Mie theory, unlike Fraunhofer theory, allows consideration of particle refractive index, required for rehable results on particles < 10 pm. Capabihties may range from 0.05 to 900 pm for wet analysis and 0.5 to 900 pm for dry samples. This type of instrument is particularly useful for obtaining information on very fine particle polymers. 

The Fraunhofer theory was the basis for the first (approximating) optical model for particle size measurement. For particles with a diameter dp larger than the wavelength A, Fraunhofer diffraction is often assumed [105]. However, only scattering by opaque particles or particles with a large real refractive index ratio m, i.e., the ratio of the refractive index of scattering particles to that of the fluid. 

The advantage of the application of Fraunhofer theory is that it is relatively simple and that it requires no knowledge of optical properties of the particulate material. This theory also enables calculations of the scattering patterns for non-spherical objects [136]. The drawback is that it yields biased results for (ultra) fine transparent particles. 

In the early days of particle size measurement, the advantage of this relatively simple theory was that it usually describes the scattering patterns of transparent particles of a few micrometers in size in liquid media better than Fraunhofer theory. Similar to the Fraunhofer theory, the anomalous diffraction theory requires no exact knowledge of the refractive index, but it should not be used for opaque particles. 

Eq. 2.19 is derived assuming that a particle scatters as if it were an opaque circular disc of the same projected area positioned normally to the axis of the beam. Because of this assumption, in the Fraunhofer theory the refractive index of material is irrelevant. Fraunhofer diffraction applies to particles 1) whose... 

In this case, f(m,0) has strong angular dependence which diminishes only when m is larger than approximately 1.2 as shown in Figure 2.14. For m > 1.2, f(m,0) is a constant and the angular scattering pattern can be treated using the ordinary Fraunhofer theory. 

Figure 3,3S. Comparison of particle sizes obtained from measurements of PSL s in aqueous suspensions using different refractive index values. The relative deviations are based on results from the correct m value (m = 1.20) for polystyrene in water. Squares m = 1.05 Triangles m = 2.25. Diamonds represent results of the Fraunhofer theory.

Figure 3.35 is from a series of laser diffraction measurements of monodisperse PSL s of various diameters suspended in water. Different sizes are obtained when using Fraunhofer theory and three Mie models generated with different relative refractive index values. When compared with the vendor s specified nominal values (which are determined by EM measurement) Fraunhofer theoiy... 

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