Light diffraction instruments are based on three basic assumptions:

(a) The particles scattering the light are spherical in nature,

(b) There is little to no interaction between the light scattered from different particles (i.e., no multiple scattering phenomena), and 

(c) The scattering pattern at the detectors is the sum of the individual scattering patterns generated by each particle interacting with the incident beam in the sample volume.

Deviations from these assumptions will introduce some degree of error due to the inability of the mathematical algorithms for the deconvolution and inversion procedures to account for the deviations. The assumption of spherical particle shape is particularly important as most algorithms in commercial instruments use the mathematical solution for Mie, Fraunhofer and Rayleigh scattering from spherical particles. At present even though mathematical solutions for cylindrical systems are available, these solutions have not been incorporated into commercially available programs. However, those interested can attempt to develop such algorithms based on published mathematical solutions.

Similarly, the assumption of the lack of multiple scattering is critical due to the  nature of the mathematical models which are used to deconvolute the resultant diffraction patterns. Most algorithms and mathematical models do not take into consideration the effects of multiple scattering on the resultant diffraction patterns and thus are unable to account for such interactions. The need to prevent multiple scattering also places an inherent limitation on the concentration of particles present in the sample volume.

Light diffraction instruments comprise of a light source, typically a low power (approximately 10 mW Helium-Neon, in the region of 632 nm wavelength) laser source, optical elements to process the incident beam, a sample cell within which the sample is introduced. Some sample cells have built-in ultrasonicators or agitators to keep the specimen powders dispersed and to prevent agglomeration. Sample cells also possess pumping systems to keep the specimen circulating. Light diffraction instruments lack the ability to distinguish between well-dispersed powders and agglomerates, and thus, prevention of agglomeration is a key factor in ensuring reliability and reproducibility. Light scattered from the sample is then focussed on to a detection system, that can be a multi-element array or numerous detectors placed at discrete locations. The detectors convert the scattered light intensity incident upon them into electrical signals that are then processed to obtain information about the particle size and size distribution. Figure 1 is a schematic diagram of the different components. Conversion of a scattering pattern into size distribution information requires the use of optical models and inversion procedures.

Figure 1: Schematic Diagram of Components in a Typical Laser Diffraction Instrument

The need for optical models arose in order to explain the scattering of light from spherical particles. Gustav Mie published a theoretical analysis for scattering of light from spherical particles in 1908. Lorenz is credited with having derived a solution for the same problem independently in 1890. Solutions to scattering of light by spherical particles of varying sizes can be thought to fit into the following categories:

(i) When the particle diameter (d) » wavelength of incident light (1), the Fraunhofer model may be used, and this represents one limiting case of the Mie theory.

(ii) When d « X, the solutions for scattering are best represented by Rayleigh scattering models.

(iii) When d < X, the Rayleigh-Gans optical model best represents the solutions to the scattering of light by the particles.

While the Fraunhofer model is independent of the material properties, the Rayleigh and Rayleigh-Gans models, which can be derived from the Mie scattering theory, are very much dependent on the optical properties, and thus require the use of both the real and imaginary components of the material's refractive index. The value of the real component of the refractive index can be found with relative ease, but determining the value of the imaginary component can be far more difficult.

The above distinctions represent solutions to general scattering problems of light by spheres under varying boundary conditions. Numerous texts detail the mathematical operations involved to obtain these solutions.

Mie Scattering: The Mie scattering theory is a comprehensive mathematical solution to scattering of incident light by spherical particles. The theory can be extended to consider scattering from particles with different shapes and aspect ratios. This theory indicates the necessity for a precise knowledge of the real and imaginary components of the refractive index of the material being analyzed, to determine particle size and size distribution. Most modern instruments based on laser light diffraction for particle size determination utilize this theory (with some proprietary variation) for modeling the diffraction patterns that are formed. The Mie theory is applicable when the particle size is equal to or smaller than, the wavelength of the incident light. 

Fraunhofer Approximation: The Fraunhofer approximation (also referred to as the Fraunhofer theory) is applicable when the diameter of the particle scattering the incident light is larger than the wavelength of the radiation. The Fraunhofer theory can be derived from the Mie scattering theory, or can be derived independently considering simple diffraction of light from two points giving rise to a phase difference between the two diffracted beams. The resultant diffraction pattern is marked by a series of maxima and minima. The spatial separation between these can be used to calculate the size of the particle. In the case of diffraction from numerous particles, the resultant diffraction pattern is the summation of the intensities of each pattern corresponding to the particle giving rise to that pattern. By its very nature, this model does not need any information about the refractive index of the particle and so is extremely useful for analysis of powders coarser than about lum to 2 um.

Rayleigh and Rayleigh-Gans Scattering: These two scattering models are based on the Rayleigh scattering model that is applicable when the size of the particle is much smaller than the wavelength of the incident light. The Rayleigh-Gans theory, however, is applicable when the particle diameter is not significantly smaller than the wavelength of the incident light. Like the Mie theory, both these models require the use of both the real and imaginary component of the material's refractive index.

Role of Refractive Index: An important point to be made here is the need for a precise knowledge of the optical constants of the material in certain instances. As is observed from equations 10-13, 17 and 19 presented in the Appendix, the solutions by the Mie. Rayleigh and Rayleigh-Gans theories all require the definition of the complex refractive index (m = n -in where n is the real component of the refractive index and n' the imaginary component of the same). However, equation 15, which represents the Fraunhofer solution, does not include the term m, and indicates no dependence and, hence, there is no need for a knowledge of the optical property. It is for this reason that optical models based on the Fraunhofer solution do not require the user to specify the refractive index of the material being studied, but those based on the Mie models do require the user to specify the values of both the real and imaginary components. Thus, the Fraunhofer model is preferred when analyzing powder systems containing mixtures of different materials, or when analyzing particles with inherent heterogeneity in density distribution. However, applicability of this model with the appropriate size criterion also has to be met.

Though there are various sources for obtaining the values of the refractive indices of materials, caution should be exercised while using them. The wavelength at which these values have been determined and reported may not be the same wavelength at which the instrument is operating and may thus introduce systematic errors. For materials that occur in different stable phases, the value used should correspond to that particular phase. For instance, it has been reported that the real component of the refractive index of a-AI2O3, occurring as natural corundum is 1.77, while that y-AI2O3 is 1.70. In most instances, it is relatively easy to find the value of the real component of the refractive index. It is however, far more difficult to obtain the value of the imaginary component of the refractive index of the material. Some tables exist for determination of these values, but often trial and error procedures of size determination using a microscopy-based technique and an instrument using laser diffraction have to be utilized. Such a procedure could introduce significant errors in the value determined and may be biased by the algorithm incorporated in the laser diffraction instrument. Furthermore, the imaginary component is affected by factors such as the surface roughness of the particle, density heterogenities in the particle. To enable repeatability and reproducibility of experiments, it is a good practice to report the wavelength of the incident radiation used when reporting refractive index values or size values based  upon the use of the refractive index information of a material.