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Simulating the Particulate Spectra: Mie Theory

Mie theory describes the way in which spherical, homogeneous particles interact with electromagnetic radiation. Certain assumptions are made about these particles in order to simplify the situation. These assumptions are:

1) Only the interactions of a single particle with light of arbitrary wavelength are considered.

2) The optical properties of the particle are completely described by frequency-dependent optical constants (i.e. the complex refractive index or the complex dielectric function).

3) Scattering is elastic, i.e. the frequency of the scattered light is identical to that of the incident light.

4) Only single-scattering occurs. That is, each individual particle is acted on by an external field (due to the incident radiation) in isolation from the other particles. The total scattered field is merely the sum of the fields scattered by each particle (i.e. the particles do not affect each other).

5) For a collection of particles, the number of particles is large and their separations are random, so that the waves scattered by the individual particles have no systematic phase relation.

6) The medium in which the particles are embedded is considered to be: a) linear, b) homogeneous and c) isotropic.

The theory used to find the transmission spectra of small particles from the optical constants is known as Mie Theory. This calculates the Q-factors, which are the efficiency factors for extinction, scattering and absorption. There is no simple explicit formula for the Q-factors, however if 2tex2html_wrap_inline241a / tex2html_wrap_inline237, (where a is the radius of a spherical particle and tex2html_wrap_inline237 is the wavelength of the incident light), then for a spherical particle:

  Qsca = (8/3)(2tex2html_wrap_inline241a/tex2html_wrap_inline237)4 Re[{(n2 - 1)/(n2+2)}2]

  Qabs = 8tex2html_wrap_inline241a / tex2html_wrap_inline237 Im[(n2 - 1)/(n2+2)]

where n = n - ik (Bohren & Huffman 1983).

Given the close relation between the complex refractive index and the complex dielectric function, we can also write these equations in terms of the dielectric function:

  Qsca = (8/3)(2tex2html_wrap_inline241a/tex2html_wrap_inline237)4 Re[{(epsilon - 1)/(epsilon + 2)}2]

  Qabs = 8tex2html_wrap_inline241a / tex2html_wrap_inline237 Im[(epsilon - 1)/(epsilon + 2)]

where epsilon epsilon = tex2html_wrap_inline1117 - itex2html_wrap_inline1119.

There are computer programs that use Mie Theory to evaluate the Q-factors and thus produce the relevant spectra. These programs require the input of:
1) A dimensionless length 2 8tex2html_wrap_inline241a / tex2html_wrap_inline237
2) The real and imaginary parts of the refractive index of the materials n(tex2html_wrap_inline237) = n - ik, or the real and imaginary parts of the dielectric function of the material epsilon epsilon = tex2html_wrap_inline1117 - itex2html_wrap_inline1119.

Next: Experimental Determination of Optical Up: The complex refractive indices Previous: Scattering

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