Home | Survey | Topics | Index | References | Dictionary | Contributing | Gallery | Community
| Mie theory: Integration of particle size-dependent patterns | Prev topic | Next topic |
Let us denote a particle size-dependent Mie pattern for a single microsphere by p[θ, D, m(λ)], where θ is either the scattering angle, (for example, in the case of the Mij elements of the scattering matrix) or nothing (for example, in the case of the attenuation coefficient), D is the microsphere diameter, m(λ) is its refractive index relative to that of the surrounding medium, and λ is the wavelength of light in that medium. A corresponding optical property of a polydispersion (in that medium) of particles that are size-distributed according to a PSD of f(D) in a diameter range, Dmin to Dmax, is then represented by:
| P(θ, λ) = ∫DminDmax p[θ, D, m(λ)] f(D)dD | (1) |
The particle diameter, D, range is generally limited to a finite range of the allowable range of 0 to ∞, although this may not be required for the convergence of the integral in the above equation for certain forms of the particle size distribution, f(D). This limitation may simply be implied by a limited range of particles sizes that can be counted by a particle counter or can affect the measurement of the optical property, P, of a dispersion.
This equation can be rephrased by using the relative particle size, x, as follows:
| P(θ, λ) = (λ / π)∫xmin(λ)xmax(λ) p[θ, x, m(λ)] f(x) dx | (2) |
where λ is the wavelength of light in the medium surrounding the microspheres:
| xmin(λ) = π Dmin / λ | (3) |
and
| xmax(λ) = π Dmax / λ | (4) |
Note that a fixed particle size range, D = Dmin to Dmax takes priority in defining a finite integral in Eq. 2. This means that when evaluating that integral, the limits of that integral "float" according to Eq. 3 and Eq. 4 in respect of the wavelength, λ, at which the optical property is evaluated.
| CITATION: Jonasz M. 2006. Mie theory: Integration of particle size-dependent patterns (www.tpdsci.com/Tpc/MiePtnSzIntg.php). In: Top. Part. Disp. Sci. (www.tpdsci.com). |
HISTORY: Published: 30-Jun-2006 Modified: 29-Aug-2006 Reviewed: PENDING |
| Copyright 2005-2008 MJC Optical Technology. All rights reserved. | Terms of use | Menu |