Mie Scattering & Mie Theory Explanation on the use of refractive index in laser particle size measurement

Because the surface of a particle produces an electromagnetic field due to the presence of electrons and since light represents an electromagnetic radiation, it can interact to produce a phenomenon that is described as mie scattering or diffraction. Mie scattering as well as the according Mie theory are named after German physicist Gustav Mie (1868-1957), who calculated the phenomenon for the first time at the beginning of the 20th century.

Mie scattering, at some distance from the particle in the direction of the incident light, is a pattern that will develop depending on the size of the particle and the wavelength of the incident light. From this Mie scattering pattern information related to the size distribution of the material can be obtained.

Some materials do not transmit light and absorb the energy. In these cases, the substance can be assumed to have an extremely high refractive index as well as a large imaginary component (see Transparent particles below). Under these conditions the calculations can be those described by Fraunhofer theory.

Light can also be reflected from the surface of a substance and the use of such data for size measurements would be the subject of a different issue.

A third occurrence of interaction is a special case that occurs when the material is somewhat transparent. In this case light pass through the particle much as it would through a diamond. In the case of a diamond, it is refracted and produces the well-known glitter; however, when passing through a particle it may add to the Mie Scattering / diffraction pattern. This effect will be discussed below.

Mie scattering & Mie theory in Microtrac particle analyzers

Different Microtrac analyzers use Mie scattering & Mie Theory to analyze particles.

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Mie Scattering & Mie Theory What is each of the above dependent upon and how do they interact?

As mentioned above, diffraction / Mie scattering is solely dependent upon the size of the particle. Reflection has no effect on diffraction but may affect refraction if the surface is sufficiently reflective. The effect on refraction would be to limit the amount of light entering the particle and thus reduce the effect of refraction on a diffraction pattern.

Refraction can have considerable impact on a diffraction / Mie scattering pattern, but the magnitude of the effect is highly dependent upon the size and shape of the material.

A sphere will transmit the same refraction pattern regardless of its orientation. In a measuring system where the spherical particle is constantly changing orientation with respect to the incident light, the pattern is always identical and can give rise to well-defined, reinforced extraneous information that can distort or interfere with the computation of particle size from the diffraction pattern. (Figure 1)

The impact of the refraction is also affected strongly by the shape of the particle. Non-spherical ones can also refract light and may produce a scattering pattern that is superimposed on the diffraction pattern as a spherical particle may. However, the effect is somewhat different.

Remember that the particles are in motion and will tumble as a result of the motion. Each change of orientation will provide a new and different surface for the light to enter and be refracted. Upon exit a new refraction pattern emerges which is superimposed on the diffraction pattern.

The reinforcing effects observed with a spherical particle do not occur. The refracted pattern is spread across the diffraction pattern as a somewhat constant pattern and affects the diffraction pattern to much less an extent than a spherical particle. (Figure 2)

Figure 1

Scattered light is concentrated at one place. Tumbling has no effect.

$Mie Scattering & Mie Theory - Secondary peak is Interference (combination) of patterns resulting from light refracted through the sphere and diffracted off the surface.$

Secondary peak is Interference (combination) of patterns resulting from light refracted through the sphere and diffracted off the surface.

$Mie Scattering & Mie Theory - Scattered light is spread across and is not concentrated at any one place. Therefore, the effect of refractive index of irregularly shaped particles is far less than with spherical ones and corrections are much less.$

Figure 2

Scattered light is spread across and is not concentrated at any one place. Therefore, the effect of refractive index of irregularly shaped particles is far less than with spherical ones and corrections are much less.

Mie Scattering & Mie Theory How does one correct for potential errors in the diffraction pattern that refraction can cause?

Mie Theory required: For spherical particles, the well-accepted concepts embodied in a theory developed by Gustav Mie may be used. This compensation is popularly termed “Mie Theory” which describes the effect of spherical shapes on light. Mie Theory includes the aspects of refractive index of the particle in relation to the refractive index of the surrounding medium as well as the scattering efficiency of the transparent material. Scattering efficiency can be understood as the relative capability of a material to scatter light. According to Mie theory the amount of scattering will vary non-linearly with size.

Mie Theory not required: If a material is not transparent (such as in the case of carbon black), Mie theory compensation for refraction is not required while calculation for scattering efficiency must be included. For Microtrac instruments, materials such as dark pigments, carbon black, and metals are considered to be light absorbing (non-transparent). An appropriate selection in Microtrac software addresses this situation where Fraunhofer theory calculations can be used.

Mie Scattering & Mie Theory Light scattering of transparent particles.

Consider only the case of transparent particles. Also consider that the refractive index has two terms that might be considered to act independently of one another to some degree. These two are recognized by the names real component and imaginary component of refractive index. Each has a particular effect on the compensation in combination with scattering efficiency according to “Mie Theory”. The assumption that refractive index has no effect on light scattering (true in the case of carbon black) will reduce Mie Theory to the well-known Fraunhofer diffraction theory. Errors can occur in the determination of size distribution if Fraunhofer diffraction is applied in situations where the particles are transparent and thus require Mie Theory for spherical particles or other compensation for non-spherical particles.

N= m-ik where N is the total refractive index which is a combination of the real component (m) for a substance compared to a vacuum and the imaginary component (ik). The terminology extends from the study of complex numbers. In the case of size measurement with particles suspended in a fluid, the value k represents the extinction co-efficient (related to the absorption coefficient of the material and the wavelength), i is √-1 and m is the relative refractive index (RI sample/ RI fluid each of which has been measured compared to a vacuum). To summarize, pure diffracted light is the desirable information that should be used for size measurements. The relative refractive index defines where the exiting light will focus and spread while the imaginary component is an indication of the intensity of the refracted light. If the imaginary component is very low, the intensity of refraction will be high.

Thus, for alumina the equation would be N= 1.76/1.33 - ik. The equation can be fulfilled by knowing the value for ik. Unfortunately, such values are NOT readily available in the literature and are difficult to obtain experimentally. Another consideration of the use of the imaginary component is evaluating its influence in calculation of N and the Mie compensation.

Since this discussion is a non-mathematical, explanatory, conceptual approach, mathematical proof of the following is not provided, but the reader is encouraged to study the area as it is fully developed from Maxwell’s treatment. Summary of the effects of an RI value and its corresponding imaginary component for a particle is presented below.

Mie Scattering & Mie Theory Spherical, Transparent Particles (Figure 1)

When particles are smaller than 1 micron, transparent and not highly absorbing (e.g. low index glass), the light path through them is very short - absorption does not occur and the imaginary term can be assumed to be zero. Remaining is the relative refractive index (ratio of RI values), which continues to have an effect on the scattering pattern resulting by light refracting through the material.

When particles are larger than approximately 10-30 microns, the amount of light transmitted is very low and refraction in general has a very small effect. At sizes much larger than this, the Fraunhofer theory approximation equations can be used for calculation.

Within the range of approximately 1 - 10 microns, there can be effects resulting from absorption, but only if the value for k is of the order 0.5 - 1.0 (high imaginary values). Values considered to be high would include carbon black (0.66i), and metals (imaginary component can be very high, m is very low due to high reflectance: thus no correction for refraction required and can be treated as Fraunhofer diffraction).

Mie Scattering & Mie Theory Non-spherical, Transparent Particles (Figure 2)

Mie Scattering & Mie Theory - Scattered light is spread across and is not concentrated at any one place. Therefore, the effect of refractive index of irregularly shaped particles is far less than with spherical ones and corrections are much less. — Figure 2

If the particles are not spherical but are transparent, the compensation (calculation) is not the same as for spherical particles. For NON-SPHERICAL shapes, orientation is constantly changing (Figure 2). The refracted components then produce a combined refraction pattern due to the many orientations presented to the incident light.

The resulting pattern has little definition when combined with the diffraction pattern but still requires some compensation. Since the imaginary component is a minor correction to the relative (real) component, its effect is negligible and can be disregarded. This is shown in Figure 3 where three cases are considered: transparent spherical, transparent non-spherical and absorbing.

At the size considered in the diagram, note that a strong resonance feature exists for spherical particles. In comparison, the transparent non- spherical one having the same size shows extensive reduction of the resonance to the extent that it approaches a completely absorbing particle. In this case, the strict Mie theory (spherical) calculations should not be used which explains the use of Modified Mie Theory calculations in Microtrac instruments.

Also, the refractive component is much less important (but not completely). Since the imaginary component is usually a weak secondary effect compared to the real component, the imaginary component for materials having non-spherical shape, has negligible or insignificant importance.

Mie Scattering & Mie Theory With what has been discussed, how can the compensation be performed?

From the foregoing, several approaches could be developed regarding the issue of refractive index. In one, the concept can be completely disregarded and Fraunhofer diffraction theory can be used exclusively, but this may result in extraneous refracted light at wider scatter angles which may in turn result in erroneous reporting of distribution tails particularly at the finer particle portion. Mie scattering for spherical particles may be used in combination with relative and imaginary refractive index values, if both are known. This could be applied to both spherical and non-spherical particles (as supported by Figure 3, this may be an unwise choice of calculation options for both types of shapes).

Usually the imaginary component is not known and selection of the “correct” value is made empirically by choosing compensation values (both components of refractive index) based upon the “operator’s opinion” of the “correct” light scattering particle size distribution. The same empirical approach may be used in the instance when both values are unknown. These latter two approaches exhibit undesirable science and provides opportunities for large errors if the particle size should change, even slightly, because the incorrect (unscientific) selection for the values can lead to under- or over- compensation; particularly in regards to the presence or absence of small amount of distribution fines.

$Mie Scattering & Mie Theory - Figure 3 above shows the response of light to a 6 micron particle having Refractive Index =1.54. Major peaks are purposely drawn displaced because the patterns are identical for size indication. Note on the right side that non-spherical transparent particle refraction is more similar to an absorbing response than to spherical curve. Microtrac developed special calculations that are used to take this effect of non-spheres into consideration.$

Figure 3

Figure 3 above shows the response of light to a 6 micron particle having Refractive Index =1.54. Major peaks are purposely drawn displaced because the patterns are identical for size indication. Note on the right side that non-spherical transparent particle refraction is more similar to an absorbing response than to spherical curve. Microtrac developed special calculations that are used to take this effect of non-spheres into consideration.

Mie Scattering & Mie Theory How does Microtrac treat the issue of refractive index?

In consideration of all the above information Microtrac laser diffraction instruments use the following approach described here and shown in Figure 4. For spherical, transparent material, a requirement exists for the index of refraction of the suspending fluid and the particles. The imaginary component does not require consideration because of the above discussion.

In the case of NON-SPHERICAL particles, consideration for refraction is made by selection of the RI sample and RI fluid, which determine the proper compensation to make in the calculations (Microtrac proprietary modified-Mie Theory calculations) according to proprietary research and development data.

A third option is available for materials that are highly absorbing such as carbon black and toners.

Figure 4

Selection of suitable parameters for calculating size distributions with Microtrac analyzers.

Mie Scattering & Mie Theory Summary

The imaginary component of the total refractive index demonstrates little effect on the refraction of light through a particle except in the 1-10 micron region. Even in this region of sizes, the effect is important when the imaginary component is of the order of carbon black (0.66i) or higher (reflective metals).

In the case of non-spherical particles refractive index has generally less impact on the calculated size distribution but still requires minor compensation from semi-empirically determined data. Under this condition, the imaginary component is of no consequence and can be ignored. In general, the imaginary component can be described as having negligible effect on diffraction light scattering particle measurements except in highly specific cases, which are rarely encountered.

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Mie Scattering & Mie Theory References:

Principles of Optics: Electromagnetic theory of propagation, interference and diffraction of light. Max Born and Emil Wolf, 6th Ed. Pergammon Press
Encyclopedic Dictionary of Physics. J Thesis. The Macmillan Company.
Vibrations, Waves, and Diffraction. H.J.J. Braddock, B.A., Ph.D. McGraw-Hill Book Company.
The Practicing Scientists Handbook. Alfred J. Moses. Van Nostrand Reinhold Company