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The scattering matrix of a volume of a dispersion of particles that are randomly distributed in space and sufficiently far from each other to avoid multiple scattering is a sum of the scattering matrices of all particles in that volume. As it will be shown shortly, the scattering matrix of a dispersion is, up to constant, equivalent to a matrix form of the volume scattering function. Random distribution of the particles assures that, when the scattered waves are summed over the scattering volume, the cross-interference terms vanish because the randomness of the relative phases of these waves assures an equal probability of phases: φ and φ + π for all scattering angles.
With the scattering matrix representing light scattering properties of a single particle in a dispersion of identical and identically oriented particles, the top-left element, M11, of that matrix is related to the scattering function, β, of that dispersion as follows (see also Eq. 11 in Penndorf 1962a):
| β(ξ) = n M11(ξ) / k2 | (1) |
where ξ is the unit vector of a direction in space, which, for an axisymmetric scattering medium can be represented by the scattering angle, n is the number of particles per unit volume of the dispersion, and k is the wavenumber (k = 2π / λ). This equation follows from Eq. 1 in Scattering matrix and the definition of the scattering function (Eq. 1 in Volume scattering function). In the case of a polydispersion of particles, Eq. 1 assumes the form:
| β(ξ) = (1 / k2) ∑i = 1 .. N ni M11, i(ξ) | (2) |
where index i numbers each of N various types of the particles. The above relationship can be generalized to include other elements of the scattering matrix (as discussed in Relationship between the scattering matrix and function). Such a generalization defines a matrix form of the volume scattering function, β, a 4x4 operator that converts the incident beam irradiance into an angular distribution of intensity of the scattered electromagnetic radiation (ER):
| β(ξ) | = (1 / k2) ∑i = 1 .. N ni M(ξ) | (3) |
If the dispersion contains many particles with a wide and densely populated range of particle size, the sums in the above equations can be replaced by integrals.
| CITATION: Jonasz M. 2006. Scattering matrix for a dispersion and a matrix form of the VSF (www.tpdsci.com/Tpc/ScaMtxDsp.php). In: Top. Part. Disp. Sci. (www.tpdsci.com). |
HISTORY: Published: 18-Mar-2006 Modified: 19-Sep-2006 Peer-reviewed: PENDING |
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