Geometric Corrections for Calculating the Pair Correlation Function

The pair correlation function of inhomogeneities in samples has been actively studied using small-angle scattering methods. Recently, it has become possible to determine this function from atom probe tomography (APT) data. This study has examined the effect of the finite size and shape of a sample on the pair correlation function of inhomogeneities derived from APT data. The number of impurities near the boundary of a large cubic sample with dimensions in all directions significantly exceeding the characteristic correlation radius, can be considered much smaller than that in the bulk. If this assumption is not fulfilled, there arises a geometric factor for which a general expression is derived. The geometric meaning of this factor represents the probability to find the specific interpoint distance within the sample. For the case in which the sample is an elongated rectangular parallelepiped, an analytical expression for the geometric factor in terms of elementary functions has been obtained. The following model systems have been considered: a completely uncorrelated distribution of centers, a simple cubic lattice, and a densely packed system of polydisperse hard spheres. These systems have been selected owing to their differing degrees of spatial order. It has been shown that accounting for the geometric factor has led to the correct pair correlation function for the selected model systems of inhomogeneities.