Laboratory 9aCalculation of an Absolute Size for a Distribution of Second Phase ParticlesMatSE 471Laboratory 9aCalculation of an Absolute Size for a Distribution of Second Phase Particles1. ObjectiveTo calculate the particle radius that would most commonly be observed in the micrograph of a material which contains spherical particles of an absolute (3-d) radius of r.2. BackgroundMeasuring the absolute size of features using the reflected light microscope is not a simple task, asthis example illustrates. Consider a sample which consists of monosized spherical second phase particles in a continuous matrix. As shown by Figure 1, only those particles whose centers are located in the sectioning plane (sample surface) will yield a measured diameter which is the true diameter (2r).For all other particles, the measured radii (ri) will be less than r, and for particles whose centers are located at a distance r above, or below, the sectioning plane, the measured value is zero (Figure 1). Hence, a distribution of measured radii (ri) is obtained from the monosized particles. The problem is; what is this distribution?If the spherical particles are distributed at random, the measured radii (ri) can be obtained by considering a single particle (of radius r) and allowing the sectioning plane to "sweep through" the particle (Figure 2). In addition, there will be an equal probability of the sectioning plane intersecting thesphere at all values of hi (Figure 2) from -r to +r. Now, the equation of a circle is: Thus, for the geometry in Figure 2Setting r to unity, we have2/1222)1(1iiiihrhr−=−=3. ProcedureUsing the last equation above, write a short computer program (the language is up to you) or use a spreadsheet to determine the frequency with which a measured ratio of ri/r is obtained as a function of ri/r. This can be done by calculating ri as a function of hi, with hi varying from -r to +r in equal increments. The results should be in the form of a histogram (frequency of a particular value of ri/r vs. the value of
View Full Document