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Eq. 2 of Mie theory: Integration of particle size-dependent patterns and other integrals of such patterns can be simplified by using the Mie sum, Σp, for a property p. For example, in the case of the attenuation coefficient, c, that sum (see Eq. 5 in Mie theory: Overview) is defined as follows:
| Σc = ∑n=1 to ∞ (2n + 1) Re(an + bn) | (1) |
In this case, Eq. 2, referred to above, becomes:
| c(λ) | = (λ / π)∫xmin(λ)xmax(λ) c(x, m) f(x) dx | |
| = (λ / π)∫xmin(λ)xmax(λ) Qc(x, m) A(x) f(x) dx | ||
| = (λ / π)∫xmin(λ)xmax(λ) 2 Σc(x, m) / x2 (π / 4) (λ / π)2 x2 f(x) dx | ||
| = (π / 2) (λ / π)3 ∫xmin(λ)xmax(λ) Σc(x, m) f(x) dx | (2) |
where xmin(λ) and xmin(λ) are defined in Eq. 3 and Eq. 4 of Mie theory: Integration of particle size-dependent patterns, Qc = 2 Σc / x2 is the Mie attenuation efficiency and A = (π / 4) (λ / π)2 x2 is simply the geometric cross section of a microsphere with a relative size of x.
An equivalent equation for the scattering coefficient can be obtained from Eq. 2 by substituting an appropriate Mie sum (see Eq. 6 in Mie theory: Overview). Equations for other optical properties of a dispersion can be obtained accordingly. For example, the corresponding equation for an element βij of the scattering function in the matrix form, Eq. 3 in Scattering matrix of a dispersion ..) would read
| βij = ¼ (λ / π)3 ∫xmin(λ)xmax(λ) Mij[x, m(λ)] f(x) dx | (3) |
because the efficiencies, Qβ, ij, for the elements, βij, of the scattering function in the matrix form, β, can be expressed as follows
| Qβ, ij = [1 / (π x2 ) ] Mij | (4) |
where Mij is the scattering matrix element [i, j]. The above equation results from the definition of the efficiency of an optical property of a microsphere as a ratio of that optical property to the geometric cross section area of the microsphere and also from a relationship between the scattering function in the matrix form and the scattering matrix (Eq. 3 in Scattering matrix of a dispersion ..).
| CITATION: Jonasz M. 2006. Mie theory: Integration of particle size-dependent patterns by using Mie sums (www.tpdsci.com/Tpc/MiePtnSzIntgMieSum.php). In: Top. Part. Disp. Sci. (www.tpdsci.com). |
HISTORY: Published: 30-Jun-2006 Modified: 29-Aug-2006 Reviewed: PENDING |
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