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In the case of a dispersion characterized by a power-law size distribution :
| f(D) = tDn -s | (1) |
written here in its frequency or differential form, where t [length-4] and s [nondimensional] are the concentration factor and slope respectively, and Dn is a nondimensional particle diameter, i.e. D/D0, where D0 = 1 in the length unit chosen to represent both the particle diameter, D, and the wavelength of light, λ, in a medium surrounding particles of the dispersion.
By substituting D = xλ / π, Eq. 1 may be expressed with the relative particle size, x, as an argument:
| f(x) | = t (π / λn ) s x -s | (2) |
where the nondimensional wavelength λn = λ / D0 is numerically identical to λ in the chosen wavelength unit. Thus, Eq. 2 in Mie theory: Integration of particle size-dependent patterns becomes:
| P(θ, λ) = t (π / λn )s (λ / π) ∫xmin(λ)xmax(λ) p[θ, x, m(λ)] x -s dx | (3) |
where xmin(λ) and xmin(λ) are defined in Eq. 3 and Eq. 4 of Mie theory: Integration of particle size-dependent patterns.
In reference to Eq. 2 and Eq. 3 of Mie theory: Integration of particle size-dependent patterns by using Mie sums, the attenuation coefficient, c, due to the particles of a dispersion with a power-law particle size distribution can be expressed as follows:
| c(λ) | = ( π t / 2 ) ( λ / π )3 ( π / λn ) s ∫xmin(λ)xmax(λ) Σc(x, m) x -s dx | |
| = ( π t D03 / 2 ) ( λn / π )3 - s ∫xmin(λ)xmax(λ) Σc(x, m) x -s dx | (4) |
It follows that the dimension of c is length-1. Elements, βij, of the matrix form of the scattering function of such dispersion can be expressed as follows:
| βij(θ, λ) = ( t D03 / 4 ) ( λn / π )3 - s ∫xmin(λ)xmax(λ) Mij(x, m) x -s dx | (5) |
where Σc is the Mie sum for the attenuation coefficient and Mij is the element of the scattering matrix.
Finally, the scattering phase matrix can be expressed as follows:
| pij(θ, λ) = |
|
(6) |
where Σb is the Mie sum for the scattering coefficient of the dispersion.
Note that Eq. 4 and Eq. 5 imply that the wavelength spectra of the attenuation coefficient c and the matrix form of the scattering function, βij, (including the volume scattering function) for a dispersion of homogeneous spheres with a power-law PSD each follow a power law with a slope of -(s - 3) when the integration limits are 0 to ∞ (see also Power law in spectral scattering by natural dispersions: Finite range of the particle diameter). It also follows from Eq. 6 that m, the scattering phase matrix (including the phase function) in such a case is independent of the wavelength of light in the first approximation, aside from the wavelength dependence implied by such dependence of the refractive index, m. It is important to note that these conclusions apply also to light scattering models other than Mie theory, because they result from the dependence of light scattering on the relative particle size rather than on the absolute particle size.
| CITATION: Jonasz M. 2006. Mie theory: Integration of particle size-dependent patterns for a power-law size distribution (www.tpdsci.com/Tpc/MiePtnSzIntgPwLw.php). In: Top. Part. Disp. Sci. (www.tpdsci.com). |
HISTORY: Published: 30-Jun-2006 Modified: 23-Sep-2006 Reviewed: PENDING |
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