Abstract
This paper presents a new proposition on how to derive mathematical formulas that describe an unknown Probability Density Function (PDF3) of the spherical radii (r3) of particles randomly placed in non-transparent materials. We have presented two attempts here, both of which are based on data collected from a random planar cross-section passed through space containing three-dimensional nodules. The first attempt uses a Probability Density Function (PDF2) the form of which is experimentally obtained on the basis of a set containing two-dimensional radii (r2). These radii are produced by an intersection of the space by a random plane. In turn, the second solution also uses an experimentally obtained Probability Density Function (PDF1). But the form of PDF1 has been created on the basis of a set containing chord lengths collected from a cross-section.The most important finding presented in this paper is the conclusion that if the PDF1 has proportional scopes, the PDF3 must have a constant value in these scopes. This fact allows stating that there are no nodules in the sample space that have particular radii belonging to the proportional ranges the PDF1.
Highlights
The problem of mapping an unknown probability distribution of spherical particles size, randomly placed in non-transparent materials, is one of the classical stereological tasks
According to the obvious relation in Equation (6) the function f3(x) takes the value of 0 everywhere where F3(x) is constant. This allows us to claim that each proportional scope of the PDF1 has no nodules with radii which belong to this range
The data is collected from a two-dimensional cross section and has a form of sets containing mark radii and random chords
Summary
The problem of mapping an unknown probability distribution of spherical particles size, randomly placed in non-transparent materials, is one of the classical stereological tasks. The data (a set of mark radii, an area of intersections or chord lengths) used in such an analysis, is obtained from an examination of flat cross-sections (random cutting planes). A mean value of mark radii, an area of intersections or chord lengths differs from the mean value of the linear sizes of nodules, 2. A random cutting plane intersects bigger particles more often than smaller ones, rather than this results from the true count of the particles To solve this stereological problem, many attempts have been proposed and those solutions can be categorised into three groups. The methods from the first group are currently most frequently used for an analysis of the graphite size distribution in ductile iron [8,9,10]. A list of variables used in the text can be found at the end of the article
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