I recently stumbled across this great article by Paul Ewald on using and understanding the Fourier transform (FT) for crystallography. It’s not been downloaded all that much, which for a foundational crystallography paper is very surprising! There are more surprising things lurking in there though…
For crystallography as well as scattering (which are the same thing, after all), understanding FTs is near essential. The 1940s paper discusses how the FT is inherently tied to crystallography and how it can be used to calculate the effects of, for example, finite crystallite particles and various imperfections (such as thermal effects).
There are also honorable mentions in it of the early SAXS work by André Guinier, who was around 30 years old back then. During this time period, SAXS was used a lot for metallurgy, hence the appearance of Guinier’s name in the “Guinier-Preston zones” that mark the early precipitation stages in alloys.
Guinier’s investigations were noted by Ewald, who highlighted that SAXS is, in essence, the study of the peak broadening of the (000) reflection. Ewald mentions that this should be the same broadening as observed around any (hkl) reflection, with the added benefit of the (000) reflection being that it is much stronger than the others.
At this point, Ewald may have overlooked (or omitted) that the peak broadening of this (000) reflection carries information on the lengths of *all* contrasting matter in the sample, whereas the peak broadening of (hkl) reflections carry information on the dimensions of the crystalline phase only. However, in this, even I am not sure, so perhaps it is time for some more in-depth Fourier Transformation investigations in the near future.
In any case, the short article is very easy to read, and has some interesting insights not only on Fourier transforms, but also on the early days of crystallography and scattering. Why not give it a try?
Apropos, around the same time, a similar (and ostensibly less obscure) treatise was published by Arthur Patterson. Interestingly, Patterson included some form factors as well for the Parallelepipedon, Octahedron, Tetrahedron, Ellipsoid, Rhombic Dodecahedron, and Elliptic Cylinder.
Shamefully, neither of these two articles are open access yet.