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:

where Ø is the scattering angle, a_k and b_k are complex functions of light wavelength, particle diameter and complex refractive indices of particle and medium, and π_k and _k are functions of cos(Ø). The formulations for a_k, b_k, _k, and π_k can be found elsewhere.

Scattering Angle (degree)

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.

Footnotes 

¹XU, R. Particle Characterization: Light Scattering Methods, Kluwer Academic Publishers, Dordrecht (2000)