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.
In this series of posts, we take a quick look at some uncommon corrections you can do to your scattering patterns and we evaluate whether they are worththe trouble or not. The goal is to arrive at intensities which are within 1% of their correct values. In the previous post, we looked at the sample self-absorption behaviour. This turned out only to significantly affect the scattering patterns at very high absorptions and large scattering angles. Additionally, its complexity makes it difficult to implement for all types of samples (it was only derived for the simple case of a sheet-like sample, not for the more common capillary- or cylindrical shape). Thus, this correction should only be considered necessary to apply for highly absorbing samples scattering to angles above 5 degrees. Two other corrections to consider are the polarization correction and the spherical correction. The spherical correction, correcting for the differences in angular coverage by the detector pixels, was also treated in the “how to do a perfect measurement”-document of a few posts ago. This is a straightforward correction requiring just some geometrical parameters, and correcting the small-angle scattering intensity to a few tens of percent. While not big, its simplicity means it is not worth neglecting. The polarization correction is a correction whose magnitude depends on the degree and direction of polarization of the incident radiation and the direction of the scattered radiation. Even for unpolarized beams, the polarization correction has to be applied. While this correction is also detailed in the aforementioned document, no information on the magnitude of this correction was given. We will consider two cases, one where the beam is highly polarized in the horizontal plane (95 % as commonly found at SAXS beamlines at synchrotrons), and the case where the incoming radiation is not polarized (as on your laboratory equipment). Implementing the equation as given in the aforementioned document, we arrive at the correction factors for the 95% polarization case shown in Figure 1 for up to 5 degrees scattering angle. While the scattering in the vertical direction does not need any significant correction, the horizontal scattering correction is in the tenths of percents. For unpolarized radiation, the correction factors are isotropic, again on the order of tenths of percents. While these results show that the polarization factor correction is small, this (like the spherical correction) is a straightforward correction to implement, not requiring much effort. Therefore it should also be applied in the search for intensities accurate to 99%. What should be apparent by now from these results, is that collecting (or rather correcting) a scattering pattern to much less than 1% error is quite a bit more difficult than trying to correct them to 1% error. But what do we gain from all these corrections? Read the next entry to find out what 1% error (or 99% accurate intensities) means for your results.
I apologise for my low activity in the last few weeks. Those of you who are following my Twats (Twitter messages) will probably be able to link this to our newborn baby. Now growing strongly for two weeks, having to wake up every four hours in what seems like the longest beamtime ever have left me feeling like a pinball at night. Despite that, the research continues, and I am happy to say that getting the Perfect measurement (or an approximation thereof at least, as described in the document of a few posts ago) will become maybe a little easier with some new software I wrote. The software draft is written in Matlab, but I am trying to learn Python to recode the essentials in free software (interested in helping? Let me know!). Why all this focus on the perfect measurement, you may ask? Well, it turns out that for most of the SAXS analyses to give you the correct answer (as opposed to just an answer), your data should be correct to within 1%. Most beamlines will not give you this accuracy out of the box (and some beamlines have more problems that will come to light if you run a standard sample), and thus advanced data correction is necessary. This software will, after entering the right information, do most of the corrections necessary to get good data. This means background subtraction, corrections for flatfield, distortion, spherical correction, polarisation and darkcurrent corrections, and corrections for variables such as incoming flux, transmission and scaling to absolute intensity. Furthermore, it will do error calculation and propagation and integration/binning of the data. The end result is ascii file with columns for q, I and error in I, ready for fitting in one of the existing software packages. For anisotropic patterns, a 2D binning method will be implemented shortly, until then the corrected images can be used. So that was just to wet your appetite, more in the near future after more testing and debugging. Bye!
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!
Hi all. For a work-related project, I have been developing some software and writing some documentation. This documentation is still a work in progress (as is the software), and it lacks much graphics. However, I think it could serve as a good introduction to those who want to do a good SAXS measurement, or those who will join on an expedition to a SAXS beamline. Let me know what you think! The appendix is a chapter from my thesis. imp_userguide