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| Power law in spectral scattering by natural dispersions: Derivation | Parent topic |
In his derivation of the power-law dependence of the scattering coefficient spectrum for a dispersion of particles characterized by a power-law PSD of slope sD, Morel A 1973a considers a size class of particles within a diameter range dD centered at D.
At a wavelength of light, λ, these particles contribute to the scattering coefficient a fraction, db(λ), that is a product of the average scattering efficiency Qb, average geometric cross section, A, and the number, dN = tD-sDdD, of particles in that class, where t is the magnitude factor of the power-law PSD. By neglecting effects of the particle shape and orientation, the scattering efficiency, Qb, is made to depend on the particle size, D, and wavelength of light, λ, only through the relative particle size, x = πD/λ, and refractive index, m, of the particle material. By limiting the wavelength to a range in which the refractive index can be assumed constant, the only dependence of Qb on the wavelength is through x.
When the wavelength of light changes from λ0 to λ = kλ0, then the former value of Qb becomes representative of another size class of particles, with the same x (at the new wavelength), but a different diameter, D = kD, and span, dD = kdD. This changes dN and A by factors of k 1-sD and k 2 respectively. Hence, the contribution, db, to the scattering coefficient must change by a factor of k 3-sD. Since this argument applies to all particle size classes and since k = λ / λ0, it follows that the spectrum of the scattering coefficient of the dispersion follows a power law:
| b(λ) = const (λ / λ0) 3-sD. | (D1) |
Thus, as also implied by Eq. 4 in Mie theory: Integration of particle size-dependent patterns for a power-law PSD (by using the numerical identity of λ and λn, where λn is a non-dimensional wavelength of light), this form of the scattering coefficient spectrum is a direct consequence of the power-law PSD for a dispersion, and a limited exchangeability of the effects of the particle size and wavelength in light scattering.
This derivation assumes a semi-infinite particle size range, from 0 to ∞ in which the PSD of a dispersion has a power-law form. Indeed, as discussed above, for any particle size class, a change in the wavelength must transform this class into another particle size class which is only possible under the above condition.
Furthermore, by neglecting the particle-shape and orientation effects on light scattering, one limits the range of strict applicability of this derivation to certain dispersions, most notably to those of spherical particles. Nevertheless, Boss E et al 2001 found that the effect of particle non-sphericity, as approximated by considering particles shaped as oblate and prolate spheroids, is relatively small.
| CITATION: Twardowski M., Jonasz M. 2006. Power law in spectral scattering and attenuation by natural dispersions (www.tpdsci.com/Tpc/ScaCfSptPwLwNatDsp.php). In: Top. Part. Disp. Sci. (www.tpdsci.com). |
HISTORY: Published: 07-Sep-2006 Modified: 22-Aug-2006 Peer-reviewed: 13-Nov-2006 |
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