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# Spacing Distributions

There are two classifications of spacing equations which will be discussed here. Some equations estimate the proximity of the paste to the voids, and others estimate the proximity of the voids to one another. Although this may seem a subtle distinction, it will be shown that the mathematical relationships that characterize these concepts have different behaviors.

Also, any reasonable concept of "spacing" should address the fact that there must exist a distribution of distances which characterize the spacing. Clearly, some regions of the paste are closer to an air void than other regions, and some voids have nearer neighbors than others. This characteristic can be represented by a distribution of distances, as depicted in Fig. 1 for a distance s. In this figure, the probability density function (PDF) is a normalized function with unit area under its curve. This function represents the fraction of spacings found in the interval [s,s+ds] for some differential element ds. The associated cumulative distribution function (CDF) is the integral of the PDF. This function increases monotonically from zero to unity and represents the fraction of spacings less than s.

Figure 1: An idealized representation of a spacing cumulative distribution function (CDF) and the associated probability density function (PDF) for some distance s. The dashed lines demonstrate how to determine the 50 th and the 95 th percentiles from CDF data.

An illustration of using the CDF is also shown in Fig. 1. Two horizontal dashed lines intercept the ordinate axis at the 50 th and 95 th percentiles. These lines intercept the CDF at s values of 1.95 and 3.1, respectively; 50 % of the spacings are less than 1.95, and 95 % are less than 3.1. In theory, the CDF only asymptotes to unity, and to capture all of the spacings, s must increase to infinity. In practice, however, the quantity s can only increase to the size of the system. Therefore, the concept of a maximum spacing is an ill-defined quantity. In this experiment, the 50 th and the 95 th percentiles of the spacing distributions will be used to characterize both the measured and the estimated values since these percentiles are intuitive to one's concept of spacing and protected paste.

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