### Laser Diffraction

Through laser diffraction measurements, one obtains information about particle size distribution through measurements of scattering intensity as a function of the scattering angle and the wavelength and polarization of light based on applicable scattering models. This is an absolute method without the need to calibrate instrument. In the past two decades, laser diffraction has to a large extent replaced conventional methods, such as sieving and sedimentation in sizing particles smaller than a few millimeters, and has taken the place of optical and electron microscopy for particles larger than some tens of nanometers. This is mainly due to the advantages of the technology: its ease of use and fast operation; its high reproducibility; and an extremely broad dynamic size range, spanning almost five orders of magnitude, from nanometers to millimeters.

Due to both the immaturity of the technology and limited computing power, at first particle sizing by laser diffraction was restricted to the use of only the Fraunhofer diffraction theory, even when particles were much smaller than some tens of a micrometer, far outside the applicable range of the theory. For this reason, this technology is historically known throughout industry as laser diffraction. However, the term laser diffraction no longer reflects the current state of the art. Firstly, laser diffraction analyzers are no longer limited to just simple diffraction effects. More general approaches based on the Mie theory and the measurement of scattering intensity over a wide scattering angular range is employed. Secondly, non-laser light sources are often used to complement the main laser source in order to gain additional characteristic information about submicron-size particles. The figure below shows the generic setup of a laser diffraction instrument and the major functions of each element.

The whole process starts with a light source that generates a monochromatic beam. After passing through several optical components, the raw beam is conditioned to create an expanded, collimated beam which illuminates the particles in the scattering volume. The particles scatter light, generating unique angular scattering patterns.

These patterns (I(Ø)) are then Fourier transformed into a spatial intensity pattern (I(r)), which is detected by a multi-element photodetector array. The photocurrent from the detectors is subsequently processed and digitized creating an intensity flux pattern (f(Ø)). Computer software utilizing appropriate scattering theories then converts the set of flux values into a particle size distribution (q(d)).

Most industrial particles closely resemble spheres and the scattering effects from the corners and edges of these particles are smoothed out due to the tumbling and rotational motion in sample circulation during the measurement. Therefore, we can apply either Mie theory or Fraunhofer theory to most practical systems with one parameter: diameter. However, such treatment only yields apparent values. One should always keep in mind that the “size” obtained from most particle sizing technologies (no exception for laser diffraction), may differ from the real dimension. To date, the spherical modeling approach is the only feasible choice for a commercial instrument designed to be used for a broad range of samples, no matter what the real particle shapes are.

### 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.