Hello Kengo,

thank you for sharing this interesting technical question with the RG community. Although we published more that ca. 1,000+ molecular and crystal structures in the past 40 years, I'm absolutely not a proven expert in X-ray crystallography (we work in the field of organometallic chemistry). The reason is that we always relied on the valuable collaboration with expert crystallographers.

However, quite often useful literature references can easily be found and accessed by searching the "Publications" section of RG. For example, please have a look at the following potentially useful article which is freely accessible as public full text on RG:

Article The uncertainty in the grain size calculation from X-ray dif...

Also please check:

Method for Determining Crystal Grain Size by X-Ray Diffraction

Article Method for Determining Crystal Grain Size by X-Ray Diffraction

This article has not been posted as public full text. However, one of the authors has an RG profile, so that there is a good chance that your can request the full text directly from him via RG.

Also in this context, it might be worth a try watching some YouTube tutorials about this research topic:

Crystallites (grain) size from XRD data using Scherrer equation

https://www.youtube.com/watch?v=m8L-B4A75Ec

and

How to calculate the crystallite size from XRD data using origin

https://www.youtube.com/watch?v=lZ1VESKBxKY

Please note that these are just two typical examples out of various closely related tutorial videos.

I hope you will find this information useful. Good luck with your research work and best wishes, Frank Edelmann

Dear Frank T. Edelmann,

Thank you very much for the informative instructions.

The idea of searching Youtube had never occurred to me, but it was beneficial to have a concrete idea of what to do before reading the literatures.

Ref. 1 by He et al. in 2018 includes the derivation of the equation. I found out that the size information is only for the direction perpendicular to the plane of incidence. It is inappropriate when the system broadening is more significant than the physical broadening (e.g., large particles). k=0.89 is the de facto standard. The use of either Lorentzian and Gaussian was not clear for me.

Ref. 2 by Tomaszewski in 2013 shows that the mean particle size can vary with different procedures and models. It is unclear which value is correct, so I need to describe the conditions and methods.

Dear Steven Van Petegem,

Thank you for your comment. Silica aerogel is chemically amorphous silica; the grain diameter is expected to be a few nanometers. If you have any useful information, I would be happy to hear from you.

Dear Steven Van Petegem,

Thank you for giving the negative but important information for me. I considered XRD over TEM because the former is five times more expensive for me than the latter. However, SAXS is so complex, and I should use TEM for the grain size of silica aerogels.

Kengo Shibuya, hump in the aerogel diffractogram is typical of an amorphous pattern. You can't use the Scherrer formulator to get the domain /crystallite size by its FWHM.

Only SAXS could give a cluster/particle/network information or PDF for the short-range order information you (probably) mentioned.

I think TEM and its SAED is the best option.