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Size-dependent patterns of optical properties of homogeneous spheres resulting from Mie theory are discussed here by using the optical efficiencies as representative examples. First, the efficiencies of the particles with small relative size, x, i.e. x << 1, converge to the Rayleigh scattering limit: ~x4 for the scattering efficiency, Qb , (and attenuation efficiency, Qc , for non-absorbing particles) and ~x for the absorption efficiency, Qa (Fig. 1).
Second, beyond the first maximum, i.e. in the intermediate to large particle size range, the efficiencies exhibit decaying oscillations (Fig. 2) with an asymptotic period of limx → ∞ Δx = π / |m - 1|, where m is the refractive index, which results from limρ → ∞ Δρ = 2π (van de Hulst 1981, p. 178, Penndorf 1957).
In the case of the scattering efficiency, Qb , the first maximum occurs at the phase shift parameter, ρ = ~ 4 (for example, Sorensen 2002). At the large ρ, the maxima occurr at about ρ = 2π(j + ¾), while the minima occurr at about ρ = 2π(j + ¼), where j is an integer (van de Hulst 1981, pp. 178-179).
Third, with the increasing relative particle size parameter, x, the attenuation efficiency tends to the large-particle limit of 2 (for example, Bohren and Huffman 1983, p. 110 or van de Hulst 1981, p. 107), a result that is referred to as the extinction paradox. In addition to the low-frequency oscillations, the size dependent patterns feature high-frequency oscillations (ripple), and - in certain cases - sharp peaks, referred to as optical resonances.
Fourth, strong absorption of light, such as that by carbon, suppresses all, except perhaps the first oscillation in the efficiencies (Fig. 2).
When the size-dependent patterns are plotted as functions of the phase shift parameter, certain regularity emerges: dominant oscillations of the integral efficiency patterns (attenuation efficiency, etc.) for various refractive indices become aligned (Fig. 3). Note that for strongly-absorbing materials, only the first maximum may be present, beyond which the pattern smoothly decays to an asymptotic value. Oscillations of scattering angle-dependent patterns for various refractive indecs are not aligned by expressing these patterns as functions of the phase shift parameter (Fig. 4).
All results presented in the figures referred to here were obtained with the MJC Light Scattering Calculator for Homogeneous Spheres. This Windows-based program utilizes the downward recursion for the An function with the initial value calculated by using the Lentz algorithm (Lentz 1976). See Algorithms for functions of the refractive index and/or size of the sphere and the program's help file for more detail.
| CITATION: Jonasz M. 2006. Mie theory: Particle size-dependent patterns (www.tpdsci.com/Tpc/MiePtnSz.php). In: Top. Part. Disp. Sci. (www.tpdsci.com). |
HISTORY: Published: 03-Mar-2006 Modified: 31-May-2007 Peer-reviewed: PENDING |
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