# Accession Number:

## ADA232715

# Title:

## Optimal Scaling of the Inverse Fraunhofer Diffraction Particle Sizing Problem: Analytic Eigenfunction Expansions

# Descriptive Note:

# Corporate Author:

## ARIZONA STATE UNIV TEMPE DEPT OF MECHANICAL AND AEROSPACE ENGINEERING

# Personal Author(s):

# Report Date:

## 1989-04-01

# Pagination or Media Count:

## 10.0

# Abstract:

There are many possible strategies for sampling the near forward scattering pattern produced by a field of large particles and for subsequently solving the inverse problem in order to obtain an estimate of the particle size distribution. In a previous paper an optimally scaled formulation of the problem was derived based on consideration of condition numbers of the linear system obtained through numerical quadrature of the governing Fredholm integral equation. Here we consider scaling of the problem to involve selection of the parameters under control of the instrument designer e.g. the number, angular positions, and aperture geometries of the detectors and the number, positions, and widths or weighting functions of the discrete size classes. Since many numericalanalytical schemes for solving for the size distribution given a finite number of scattering measurements are fundamentally very similar, optimal scaling of the problem will improve the performance of the instrument regardless of the selected computational inversion algorithm. This paper considers the analytic eigenfunction expansion method of solving the inverse Fraunhofer diffraction problem, and in particular how the scaling strategy affects the inversions. The eigenfunctions and associated eigenvalues for the diffraction problem assuming infinite support are derived in terms of two variable scaling parameters which describe the detector geometry and the size class configurations.

# Descriptors:

- *ALGORITHMS
- CONTROL
- MEASUREMENT
- PARTICLE SIZE
- SCATTERING
- OPTIMIZATION
- COMPUTATIONS
- DETECTORS
- STRATEGY
- SIZES(DIMENSIONS)
- DISTRIBUTION
- PARAMETERS
- FORMULATIONS
- NUMERICAL ANALYSIS
- EIGENVECTORS
- EIGENVALUES
- WEIGHTING FUNCTIONS
- PARTICLES
- DIFFRACTION
- MATHEMATICAL ANALYSIS
- CONFIGURATIONS
- SCALING FACTOR
- SAMPLING
- PATTERNS
- INSTRUMENTATION
- GEOMETRY
- INVERSION
- EXPANSION
- NUMBERS
- INTEGRAL EQUATIONS
- SOLAR SPECTRUM
- NUMERICAL QUADRATURE
- FORWARD SCATTERING
- LINEAR SYSTEMS

# Subject Categories:

- Atomic and Molecular Physics and Spectroscopy