Correction factors for a (2D) scattering pattern in the case of 95% in-plane (horizontal) polarization, such as found at beamlines.

Does it matter part 2: polarization factor and spherical corrections

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).

Figure 1: Correction factors for a (2D) scattering pattern in the case of 95% in-plane (horizontal) polarization, such as found at beamlines.

Figure 2: Correction factors for a (2D) scattering pattern in the case of no polarization, such as found at lab sources.

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.

 

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Does it matter? part 1: sample direction-dependent absorption

Figure 1: Scattered radiation traveling different distances through a sample

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:

Figure 2: Correction factor for different levels of absorption and different scattering angles

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!

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Live Fourier transform

During some recent presentations, I have used a small matlab program giving me a live Fourier transform of the laptop camera input. It can be used in combination with some printed “structures” to show what we would see on a SAXS or WAXS detector.

The idea is not mine, I heard that it was used by dr. Henrik Lemke for his Ph.D. defense to show the effects of lattice strain on the diffraction pattern.

It turns out to be quite popular with the audience so far, so I will post the code for Matlab running on a macbook Pro here. Feel fee to use the code. I will also make a short movie showing one example of how to use it. I am sure you can think of many other useful purposes for it!

The package is here: FT_cam_package

Since it is so specific in its application, there is no documentation. Also, run “FTcamdemo” and the rest should be quite self-explanatory in the GUI that comes up. This package uses (and comes with) camera image capture code (Java) by Kalle Kempe and Ikkjin Ahn.

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Notes on Guinier

…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…

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More Youtube videos

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:

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Teaching and how it applies to us.

We teach. Every one of us. If we have a classroom of students, it is obvious, but also when we talk to colleagues, we sometimes try to teach them something (even if it is only our point of view).

I spoke last time about the teaching horrors that modern textbooks have. Well, this video by Dan Meyer explains why the textbooks are absolutely not helping to teach by asking the questions wrong, and he proposes an alternative way to pose the questions. He also, by the way, has a nice blog full of examples of how to get a class of students to actually think.

This gets me thinking. How can we apply this to our presentations? Are we really engaging our audience by starting with a “talk outline”, or should we pose the final question as simple as possible and work from there? I will be experimenting with this and I will let you know how it goes!

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Dresden codak – detailed visual narration lessons from comics

There is a scathing review coming out on the topic of academia (particularly the US one). While the general tone of the dissertation is negative, they do indicate a strong need, or rather duty, of scientists to communicate our findings as clear and understandibly as possible. Personally, I completely agree with this. Clear communication sometimes hinges on explaining everything in cool graphics, be it in posters or in presentations. So how do we learn how to create cool graphics? Well, one avenue is comics! They have supplied us with concise stories, be it comical or educational, applying graphics to provide a visual narrative.

One of my favourite online comics is Dresden Codak. Witty, beautifully drawn and often requiring quite some thought to understand the story, it is everything I seek in a comic. The author/artist also provides us with excellent tools to learn about visualisations, as he keeps a superb weblog analysing various forms of visual narration. This post is my favourite, as it indicates very clearly how we can use space intelligently in our graphics to tell a story, and what pitfalls to avoid. I hope it will provide us all with good inspiration for our next talks! Good luck!

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