TPDSci Mie theory Small particle limit

Mie theory: Small-particle limit

In the limit of (|m|x << 1), Mie theory reduces to Rayleigh theory of light scattering (for example, Bohren and Huffman 1983). The term Rayleigh scattering has several meanings, see Young (1981) for an enlightening discussion. Here it means a theory of light scattering by a particle much smaller than the wavelength of light in the surrounding medium. In that case, also referred to as the electrostatic approximation (Bohren and Huffman 1983), the entire particle experiences nearly the same electric field and behaves as a single electric dipole. This is ensured by the appearance of the refractive index in the above condition, implying that the phase of the light wave is not appreciably delayed inside the particle.

For moderate and real refractive index, m, this typically implies that the particle diameter, 2a must be less than ~0.1λ, where λ is the wavelength of light in the medium surrounding the particle (Kerker et al 1978, Kerker and Cooke 1976, Heller 1965). In particular, Heller (1965) points out that the error of the Rayleigh approximation for a given particle size decreases with increasing real refractive index, m, as long as the latter is below ~1.8. This makes it possible to apply that approximation to the attenuation of light by spheres as large as ~0.3 of the wavelength of light in the surrounding medium in a range of m = ~1.6 to ~1.8 as long as one can tolerate errors on the order of 2% in the particle volume concentration-specific attenuation coefficient c' = c/φ, where φ is the particle volume concentration.

Kerker and Cooke (1976) point out that the range of validity of the Rayleigh approximation to the scattering function depends on the scattering angle. They show that, in a range of the refractive index of m = 1.33 to 2.10 of the particle material (relative to that of the surrounding medium), the maximum relative particle size, x_max(ε, θ), where ε is the Rayleigh approximation error, peaks in the scattering angle, θ, range of about 60 to 120°. The x_max decreases with increasing refractive index, m, for scattering angles < ~90° and increases with m at much larger angles.

See also equations relevant to the small-particle limit of Mie theory.

CITATION:
Jonasz M. 2006. Mie theory: Small-particle limit (www.tpdsci.com/Tpc/MieTheSmall.php). In: Top. Part. Disp. Sci. (www.tpdsci.com).

HISTORY:
Published: 03-Mar-2006
Modified: 19-Apr-2006
Reviewed: PENDING

Topics in Particle and Dispersion Science