Optical Particle Sizing pp 159-175 | Cite as

# Modeling of Multiple Scattering Effects in Fraunhofer Diffraction Particle Size Analysis

## Abstract

A model for the direct problem of calculating the forward scattering signature of a multiple scattering medium is presented. The new formulation is optimized for integration into schemes for reconstructing the particle size distribution from laser diffraction (forward scattering) signatures obtained from optically thick media. The analysis is valid for media where the particle sizes and interparticle spacings are large (relative to the wavelength and the particle size, respectively) such that Fraunhofer diffraction theory adequately describes the properties of the forward scattered light from individual scattering events. The simulated performance of laser diffraction particle sizing instruments was then studied using predictions of the scattered light signatures which would be measured by laser diffraction instrument under multiple scattering conditions. The results were compared with experimental data and theoretical calculations based on other models.

## Keywords

Optical Depth Multiple Scattering Laser Diffraction Discrete Ordinate Successive Order## Abbreviations

## Nomenclature

- a
albedo, ratio of the scattering cross-section to the total extinction cross-section of a particle, i.e. the fraction of the incident energy intercepted by a particle which is scattered rather than absorbed

- a
_{f} forward scattering albedo, ratio of forward scattering cross-section to total extinction cross-section for a particle, a

_{f}=0.5 in the geometric optics case, independent of particle composition- f
_{n} probability that a photon will be scattered (in the forward direction) exactly n times while passing through a medium

- h
scattering redistribution function

- C
_{abs} optical absorption cross-section of a particle (m

^{2}/particle)- C
_{ext} optical extinction cross-section of a particle (m

^{2}/particle)- b
optical depth (dimensionless)

- C
_{sct} optical scattering cross-section of a particle (m

^{2}/particle)- L
scattering phase function which is the discrete angular distribution function for scattered light normalized to 1.0

- n
the number of particles in a finite volume

- <n>
the expected number of particles in a finite volume

- P
_{n} the probability that exactly n particles are in a finite volume

- T
transmittance of a medium, the probability that a photon will traverse a medium without being scattered or absorbed

## Subscripts

- det, i
refers to the i

_{th}detector- fwd
forward scattering

- inc
incident, for radiation incident on a particle

- sct
scattered

- x
refers to x component in cartesian coordinate system

- y
refers to y component in cartesian coordinate system

- z
refers to z component in cartesian coordinate system

## Superscripts

- /
the prime superscript indicates quantity is in local light scattering coordinate system rather than inertial system

## Greek

- γ
direction cosines of scattered rays

- ℓ
the length of the medium (m)

- φ
azimuthal scattering angle in local coordinate system

- Φ
azimuthal scattering angle in inertial coordinate system

- ρ
particle number density (particles/m

^{3})- θ
scattering angle in local coordinate system

- Θ
scattering angle in inertial coordinate system

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