Restoration of Three-Dimensional Correlation Function and Structure Factor from Two-Dimensional Observations of Sliced Volume

Abstract

The static pair correlation (distribution) function and the structure factor of particle distributions in three-dimensional homogeneous isotropic systems are explicitly restored from two-dimensional counterparts observed in a thin slab sliced out from original systems. While two-dimensional values for given thickness of the slab are readily calculated from three-dimensional functions, one would like to make the reverse in usual experiments by, for example, scanning the slab perpendicularly. Such a reconstruction is also possible even without scanning and three-dimensional pair correlation function and the structure factor are expressed by two-dimensional ones in the form of an expansion with respect to the thickness of the slab. As an application, the behavior of the structure factor corresponding to the critical fluctuation is discussed. These results are expected to be useful when three-dimensional systems are observed by the illumination of thin planar lasers.