Mie and Fraunhofer Diffraction Theories

Laser diffraction uses the Mie theory of light scattering to calculate particle size.

With laser diffraction, particle size is specified based on light intensity distribution patterns. For this, the Fraunhofer and Mie theories are used to calculate what kind of light intensity distribution patterns are produced by particles of various sizes. As this data is stored in advance as a parameter table, it doesn’t affect measurement time.

Mie theory requires knowledge of the optical properties (refractive index and imaginary component) for the sample being measured and the refractive index of the dispersant. Typically, dispersant optical properties are fairly easy to find and many modern instruments have integrated databases with common dispersant values. For samples with unknown optical properties, users can measure or estimate them using an iterative approach based on the fit between the modeled and actual data collected for the sample.

If you don’t know the refractive index of your material or if the size of the particles is above 50 µm, you can use the Fraunhofer approximation. In other cases, such as when working with relatively transparent particles or particles below 50 µm, the Mie theory will produce more accurate results.

 

The Mie Theory

 

The Mie theory is a rigorous solution for the scattering intensity from a spherical, homogeneous, isotropic and non-magnetic particle of any diameter d in a non-absorbing medium. The mathematical formulation for the scattering pattern from a spherical particle illuminated by vertically and horizontally polarized incident light predicted by the Mie theory is very complex:


Figure 1. Schematics of scattering pattern for spheres.

where Ø is the scattering angle, ak and bk are complex functions of light wavelength, particle diameter and complex refractive indices of particle and medium, and πk and k are functions of cos(Ø). The formulations for ak, bk, k, and πk can be found elsewhere.


Figure 1. Schematics of scattering pattern for spheres.

The above figure shows the scattering patterns from two spherical particles of different sizes. They are symmetric with respect to the axis of incident light, i.e., the scattering pattern is the same for the same absolute value of the scattering angle. In these patterns there are scattering minima and maxima at different locations depending on the properties of particle. The general characteristics are that the location of the first intensity minimum is closer to the axis and the peak intensity is greater for a large particle (the solid line in Figure 1) as compared with that of a smaller particle (the dashed line in Figure 1).

Because of the complicity of the formulation, it was impossible to apply the Mie theory in laser diffraction experiments before computers and microelectronics had enough computation power and speed. Prior to the era of Pentium computers, often used was Fraunhofer diffraction approximation in retrieving particle size distribution from laser diffraction measurements.