On physical scattering density fluctuations of amorphous samples

TL;DR

This paper rigorously characterizes the conditions under which a function can be considered a physical scattering density fluctuation in amorphous samples, linking Fourier transform behavior to scattering intensity properties.

Contribution

It establishes mathematical conditions based on Wiener’s results for identifying physical scattering density fluctuations in amorphous materials.

Findings

Fourier transform modulus limit exists and obeys Porod invariant.

Conditions relate scattering intensity continuity to Fourier integral properties.

Numerical examples illustrate the theoretical conditions.

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.