Using the rigorous results obtained by Wiener [Acta Math. (1930), 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, the conditions that must be obeyed by a function η(r) for this to be considered a physical scattering density fluctuation are obtained. These conditions can be recast in the following form: the V→∞ limit of the modulus of the Fourier transform of η(r), evaluated over a cubic box of volume V and divided by V 1/2 , exists and its square obeys the Porod invariant relation. Some examples of one dimensional scattering density functions obeying the aforesaid condition are numerically illustrated.

On physical scattering density fluctuations of amorphous samples

Riello, Pietro;Benedetti, Alvise
2018

Abstract

Using the rigorous results obtained by Wiener [Acta Math. (1930), 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, the conditions that must be obeyed by a function η(r) for this to be considered a physical scattering density fluctuation are obtained. These conditions can be recast in the following form: the V→∞ limit of the modulus of the Fourier transform of η(r), evaluated over a cubic box of volume V and divided by V 1/2 , exists and its square obeys the Porod invariant relation. Some examples of one dimensional scattering density functions obeying the aforesaid condition are numerically illustrated.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10278/3711790
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