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| Power law in spectral scattering by natural dispersions: Finite range of the particle diameter | Parent topic |
Modification of Eq. 1 in Power law in spectral scattering by natural dispersions, caused by setting finite limits in the particle diameter, D, in a model of the attenuation coefficient spectrum (Eq. 4 in Mie theory: Integration of particle size-dependent patterns for a power-law PSD) can be qualitatively explained in the following alternative to the Morel's derivation of Eq. 1. Let us recall that model in a simplified form:
| c(λ) = const λ3 - sD ∫xmin(λ)xmax(λ) F(x, m) dx | (1) |
where λ is the wavelength of light in the dispersion medium, sD is the slope of the power-law PSD of the dispersion, xmin(λ) = π Dmin / λ, xmax(λ) = π Dmax / λ, and F denotes the integrand, a function of the relative particle size, x, and the refractive index, m, of the particles. In simplifying the model, we used the numerical equivalence of λn (nondimensional wavelength) and λ to obtain λ3 - sD.
If the particle diameter limits Dmin and Dmax are set respectively to 0 and ∞, then the integration limits become xmin = 0 and xmax = ∞, i.e. are independent of the wavelength. Thus, with an assumption that the refractive index, m, of the particles is constant, the integral evaluates to a value independent of the wavelength, making c(λ) proportional to λ3 - sD and leading to Eq. 1. Otherwise, as it follows from the above definitions of xmin and xmax, these limits remain functions of the wavelength, making the integral also wavelength-dependent. This modifies the spectral dependence of c from that expressed by λ3 - sD and an equation other than Eq. 1 in the parent topic (for example, Eq. 2) may be needed to describe a relationship between the slopes of the power-laws for the dispersion PSD and attenuation spectrum.
Incidentally, by arbitrarily setting xmin = a and xmin = b for the integral in Eq. D1, where a and b are finite constants, we can also make that integral independent of the wavelength. However, in such a case Eq. 1 ceases to represent the attenuation coefficient spectrum of a real dispersion. Indeed, at a wavelength, λ1, it represents a dispersion, or its subset, with the particle diameter in a range of Dmin 1 = a λ1 / π to Dmax 1 = b λ1 / π. At another wavelength, λ2, it represents a different particle diameter range, and so on.
| 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: 31-Aug-2006 Peer-reviewed: 13-Nov-2006 |
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