On physical scattering density fluctuations of amorphous samples

Abstract

Using some rigorous results by Wiener [(1930). {\em Acta Math.} {\bf 30}, 118-242] on the Fourier integral of a bounded function and the condition that small-angle scattering intensities of amorphous samples are almost everywhere continuous, we obtain the conditions that must be obeyed by a function $\eta(\br)$ for this may be considered a physical scattering density fluctuation. It turns out that these conditions can be recast in the form that the $V\to\infty$ limit of the modulus of the Fourier transform of $\eta(\br)$, evaluated over a cubic box of volume $V$ and divided by $\sqrt{V}$, exists and that its square obeys the Porod invariant relation. Some examples of one-dimensional scattering density functions, obeying the aforesaid condition, are also numerically illustrated.