The Importance of Optical Model

In the beginning there was the Fraunhofer Approximation and it was good. This
model, which was popular in older laser diffraction instruments, makes certain
assumptions (hence the approximation) to simplify the calculation. Particles are
assumed…

- to be spherical
- to be opaque
- to scatter equivalently at wide angles as narrow angles
- to interact with light in a different manner than the medium

Practically, these restrictions render the Fraunhofer Approximation a very poor
choice for particle size analysis as measurement accuracy below roughly 20
microns is compromised. The Mie scattering theory overcomes these limitations. Gustav Mie developed a closed form solution (not approximation) to Maxwell’s electromagnetic equations for scattering from spheres; this solution exceeds Fraunhofer to include sensitivity to smaller sizes (wide angle scatter), a wide range of opacity (i.e. light absorption), and the user need only provide the refractive index of particle and dispersing medium. Accounting for light that refracts through the particle (a.k.a. secondary scatter) allows for accurate measurement even in cases of significant transparency. The Mie theory likewise makes certain assumptions that the particle…

- is spherical
- ensemble is homogeneous
- refractive index of particle and surrounding medium is known

These figures show a graphical representation of Fraunhofer and Mie models using
scattering intensity, scattering angle, and particle size (ref. 13). The two models
begin to diverge around 20 microns and these differences become pronounced below 10 microns. Put simply, the Fraunhofer Approximation contributes a magnitude of error for micronized particles that is typically unacceptable to the user. A measurement of spherical glass beads is shown in Figure 19 and calculated using the Mie (red) and Fraunhofer (blue) models. The Mie result meets the material specification while the Fraunhofer result fails the specification and splits the peak. The over-reporting of small particles (where Fraunhofer error is significant)is a typical comparison result.