Small-angle scattering analysis has never been easy for those working with oriented nanostructures (e.g. fibres, processed polymers, rolled metal alloys), whose structure may lead to anisotropic small-angle scattering. Upon the collection of such 2D scattering patterns, one can integrate thin pie-slices of the data to obtain 1D curves and analyse them in the same way as “normal”, isotropic scattering patterns. This way, however, important cross-correlation information is lost. Alternative full-pattern fitting methods have been developed (amongst others during my Ph.D. studies), but they are complicated to tune to the system at hand and can be quite unstable in least-squares optimisations.
It has been a long time in the making, but now the day has finally come where the 1D Monte Carlo method has been published! To top it off, the publication is open access (courtesy of my current institute: NIMS), and has a wicked showcase document as supplementary material. Feel (very) free to check it out here!
Good news for those of you on the hunt for a way to get polydispersity (size distribution) information from your scattering patterns. Two pieces of good news, to be precise! Firstly, the paper that describes my implementation of the method that does exactly this has just been accepted earlier this month for publication in J. Appl. Cryst, though it will probably not make it into the February issue. With a bit of luck, I will be able to make it open access, though! I have talked about the method before (e.g. here) so I will not spend more words on it. The second news is that the Python code with the fitting procedure is now available in an online repository here, thanks to Pawel Kwasniew at ESRF for his efforts in setting up the repository. The code comes complete with a quickstart guide with several pictures and some test data. If you are reasonably familiar with Python, why not grab a copy and try the method on your data? Reports from early testers have been positive, and everyone is encouraged to comment or send me an e-mail so it can be improved. License-wise, the code is released under a creative-commons-attribution-sharealike license. Lastly, if you want to contribute to the code you are more than welcome to. Currently, the code is being recoded in object-oriented form to improve flexibility, with the first release of the OO version expected later this month. Afterwards, a smearing function will be implemented for directly fitting slit-smeared data, and more shape functions should be included. As it is intended to be integrated in existing SAS analysis GUI’s (of which there are quite a few), there is no graphical user interface, and as such the focus is on getting the base functionality implemented right. As usual, drop me a line or leave a comment!
Dear readers, First of all, a merry end-of-year thingie (insert name here) to all of you. Last week, I presented a short 15-minute talk at the MRS-J conference in Yokohama. While it was a nice opportunity to present and meet people, it is a relatively small conference. With that in mind, I decided to re-record the presentation and post in online so you can take what you want from it. I was slightly ill when re-recording so the edit is a bit choppy (after removing the sneezes). It is available on Youtube here (and embedded below). Please leave comments and/or thumbs up if you like, and follow me on twitter @drheaddamage if you want to hear more ramblings from this side. Lastly, the next video definitely should be a custom one explaining the monte-carlo method we’ve developed referred to in the video.
edited on 2011-07-20 14:51 to add equation defining observability Of course I could not leave you hanging after the last post with that question “does the observability still scale with the radius squared for samples with more than two particles?”. Short answer: yes. Long answer: mostly, with some interesting lessons w.r.t. q-angles and limitations. Click “more” to read on…
Abstract of this post: In this post, I will show that for a spherical two-particle system, the observability of the smaller particle scales with the inverse squared radius, quite different from the scattering power, which scales with the radius to the sixth power (volume squared). This means that scattering data with 1% error can be used to distinguish at most particles 1/20th the size of the largest particle present, and data with 0.1% error can be used to observe single particles at most 1/50th of the size of the largest particle. Beyond this, only proportional numbers of smaller particles can be observed. Additionally the point of maximum observability is determined to be at q*Rg~2.6 for numeric examples, and q*Rg~2.45 for an algebraic approximation.
After browsing through a recent Journal of Applied Crystallography, I came across a paper by Deschamps. It indicates to me that there is a slight lack of information communication in some aspects of SAXS. Firstly, it mentions in the introduction that the advanced (Fourier-transform-based) SAXS analysis methods cannot “extract simultaneously the precipitate form factor and the precipitate size distribution”. Indeed they cannot, but neither can the classical methods (see for example page 147 of ). It is only when we assume a shape of the scatterer in the classical methods, that the number of possible size distributions reduces to a single solution. Inversely, if we assume a size distribution, there is only one general form factor which will match. This leads to the erroneous conclusion that the classical methods can result in a simultaneous determination of size and polydispersity. My main point, however, is the research on the behaviour of the Guinier method in polydisperse systems. I, too, have been looking at this and found out that others have tread this path before me. My results here are in agreement with the results of others that the radius of gyration of the Guinier method in polydisperse systems follows the volume-squared weighted radius of gyration . The limits of applicability shift accordingly. The work by Deschamps on the Porod behaviour in polydisperse systems is to my knowledge unique, but I have not looked in detail into this.  O. Glatter and O. Kratky, Small angle X-Ray Scattering, (Academic Press, 1982), available online  G. Beaucage, H. Kammler, and S. Pratsinis, Journal of Applied Crystallography 37, 523 (Jan 2004).
As promised, some more details which hopefully make the monte-carlo code (discussed here) more useful to you; the software documentation. softwaremanual_mcfit In the documentation are also some examples using the scattering pattern generators discussed before. Let me know if you canor cannot get it to work and post your findings here (or father, on the software page)!
…well, his famous SAXS analysis method. This documentGuinier_short, copyright Brian Pauw gives a short description and review of the applicability of the Guinier method to polydisperse systems. It also shows, through analysis of simulated data, what q-range should be measured for the Guinier method to be valid. In short, the rule of qmax=1.3/Rg still holds, but Rg in polydisperse systems is the volume-squared weighted Rg of the distribution. This then implies that the Guinier method for polydisperse systems quickly becomes unusable as the required qmax cannot be reached with anything but USAXS systems for polydisperse samples. This text (the linked PDF) is released under copyright (copyright by Brian R. Pauw, 2011) as I may want to include some of this in a later publication. I hope you understand…