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.
This series is part of a set to determine which corrections matter when. We all heard -or read- about corrections Small-angle Scatterers do not need to do, because they are supposedly negligible. Let’s look at some of them and determine if this is really true or not. The first looks at the sample direction-dependent absorption. This is the effect of scattered radiation travelling a slightly different distance through the sample (and therefore experiencing different levels of absorption) depending on the direction of scattering. Take, for example, the simplest case of a sample like a plate or sheet (c.f. Figure 1). From this figure, it can easily be seen that the radiation scattering to an angle has to travel longer through the sample than radiation passing straight through. Therefore, the scattered radiation suffers from more absorption. But how much more? In this document: plate_transmission, with the help of Samuel Tardif, we calculated the exact angle-dependent absorption of a plate-like sample. The short answer, for those who do not want to read the document: The sample absorption indeed accounts for less than a percent of deviation to a scattering angle of 10 degrees (degrees, as this effect is independent of the wavelength), if the sample absorption is lower than 75 % (which is already much more absorption than the max. 30% rule-of-thumb you were told by your senior scatterers). The full diagram is shown in Figure 2 (click to enlarge). So, on the grand scale of things, this correction is indeed negligible for most samples (watch out for the highly absorbing ones though!). However, given its simplicity, it should not be much trouble to implement for sheet-like samples at least. The correction is very straightforward to implement (see document) as you only need to know the transmission factor (which you knew anyway, because you measured it for the background subtraction, right?). This only applies to sheet-like or plate-like samples, the calculation for a capillary is a little bit more complicated. The capillaries also suffer from this, but this plate-like absorption is the maximum correction to apply for capillary geometry (in the direction parallel to the capillary axis). So also for capillaries, implementation is unnecessary unless you are hunting for the final fractions of a percent accuracy. Next time we discuss another one: the polarization correction!
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).
So, I could not do what I promised last time, the Monte-Carlo fitting works on perfect simulated scattering patterns but is as of yet unable to deal with the addition of a flat background. So I will have to take a raincheck. In the mean time, I have made two videos (part 1 and part 2) together with some colleagues during my time in Denmark. The video demonstrates small-angle scattering using laser light scattering on a hair. In part 2, the diameter of the hair is calculated. (I watched too many Carl Sagan videos and I am impressed and encouraged by them…) Check out the videos: Part one: Part two: