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Single scattering, i.e. scattering by a particle independent of other particles in a dispersion, occurs in the dispersion at distances, z, adjusted for the scattering anisotropy of the dispersion (as expressed by the average cosine, g, of the scattering angle) that are much smaller than the mean free pathlength of photons for scattering, zfb (for example, Bohren 1987a):
| z(1 - g) << zfb | (1) |
where zfb, also referred to as the scattering length, is the inverse of the scattering coefficient, b, of the dispersion:
| zfb = 1 / b | (2) |
Equations 1 and 2 can be combined resulting in the following, single-scattering criterion:
| τ << (1 - g) -1 | (3) |
where τ is the optical thickness or depth (here due to the scattering of light). The average cosine, g, of the scattering angle characterizes the asymmetry of the scattering function of the dispersion. For a dispersion with the scattering function being symmetrical about the scattering angle of 90°, g ~ 0. This implies that particles of the dispersion are each much smaller than the wavelength of light in the medium surrounding them, i.e. the case of Rayleigh scattering (for example, Bohren CF and Huffman 1983, Ch.5). Hence, multiple scattering can be neglected for such medium at z << 1 / b, i.e. at distances smaller than the scattering length (1/b) of the dispersion. For particles much greater than the wavelength of light, g is close to unity. Hence, in this case the single-scattering range is extended as compared to the case of the Rayleigh scattering.
Applicability of the single-scattering approximation has been recently studied in terms of the particle concentration and inter-particle distance by Mishchenko MI et al (2007c) using T-matrix method for particles with a relative size of 4. These authors found that for this approximation to apply, the particle volume concentration should be smaller than 1% and a condition:
| (2π /λm) < d > > 30 | (4) |
should be fulfilled, where λm is the wavelength in the medium surrounding the particles and < d > is the average distance between the particle centers. This is consistent with results of an earlier study of the effect of inter-particle distance on light scattering by bi-spheres (Mishchenko MI et al 1995b).
When single scattering prevails, photons are unlikely to be scattered more than once. This implies that the concentration of particles in the dispersion is sufficiently small, so that a photon is not likely to encounter a particle when travelling a distance small as compared to the mean free pathlength. At the larger optical thicknesses, the opposite is true and multiple scattering prevails
| CITATION: Jonasz M. 2006. Single and multiple scattering (www.tpdsci.com/Tpc/SngMulSca.php). In: Top. Part. Disp. Sci. (www.tpdsci.com). |
HISTORY: Published: 18-Apr-2006 Modified: 06-Mar-2008 Reviewed: PENDING |
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