Restoring the three-dimensional correlation function and structure factor from two-dimensional data

In uniform and isotropic systems the three-dimensional pair distribution function and the structure factor can be restored from two-dimensional observations of the sliced volume with finite thickness without scanning along the third dimension. By numerically simulating such observations of a known system, it is explicitly shown that proposed formulas for such restorations are applicable, at least when the thickness of the sliced volume is smaller than or comparable to the mean distance between particles. It is also pointed out that one cannot simply guess the behavior of the long-wavelength limit of the three-dimensional structure factor from the two-dimensional observation: Even when the (true) three-dimensional structure factor vanishes, the observed two-dimensional structure factor goes to a finite value which depends on the thickness. The correct behavior of the three-dimensional structure factor is given by the restoration formula.