Error estimation in stereological determination of particle size distribution

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Abstract

Stereological determination of particle size distribution normally amounts to solving a Volterra's integral equation of the first kind. Computationally speaking, this integral equation must be discretized and converted into a set of n algebraic equations in n unknowns. As the value of n increases the accuracy of estimation also increases. Here is proposed a technique of replacing the distribution density f(r) obtained from randomly intersecting planes with cumulative distribution density φ(r). Here φ(r) is the number of cross-sections with radii equal to or smaller than r per unit area. Similarly Φ(R) is defined as the number of spheres with radii equal to or smaller than R per unit volume. The integral equation is thus reduced to an Abel's type. Finally, the error involved in discrete approximation with large n is obtained and the best formula to minimize the error is derived.

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Over : Mohsen Shahinpoor