# Single Crystal Diffuse Scattering and Pair Distribution Function: Some Kind of Comparison

Here begins a technicalish, science-y post.

This post is all about a paper we recently published in IUCrJ, here is the link: http://dx.doi.org/10.1107/S2052252515018722.

When X-rays or neutrons scatter off a sample of crystalline powder, the result is a powder diffraction pattern. Usually the intensity of the scatting is measured as a function of the angle of scattering for radiation of a fixed wavelength. The angle can be converted to the more universal ‘scattering vector’:

Now, when analysing a pattern like this, the most common method is Rietveld refinement, in which a possible unit cell is posited, and its diffraction pattern calculated and compared to the observed.

Now, this is very useful indeed, but there are a couple of issues. The first is that this sort of analysis only uses the strong Bragg reflections in the pattern — the big sharp peaks. Mathematically, this means it finds the *single body average* which is to say that it can show what is going on on each atomic site but not how one site relates to another. For example, it might say that a site has a 50% chance of having an atom of type **A** on it and 50% of type **B**, but it can’t say how this influences a neighbouring site. Do **A** atoms cluster? Do they like to stay apart? This information, if we can get it, tells of the short-range order (SRO) in a crystalline material, where the Bragg peaks tell of the long-range order. SRO is important, interesting, and rather difficult to get a handle on.

Now, the flat, broad (‘diffuse‘) scattering between the Bragg peaks — stuff that looks rather like background, and is often mixed up with background — contains two body information. If the non-sample scattering is carefully removed, then what is left is all the scattering from the sample, and only scattering from the sample. This is called the Total Scattering. This can then be analysed to try to understand what it going on. The most common way of doing that is to calculate the pair distribution function (PDF) from the TS. This essentially shows the probabilities of finding scatterers at different separations — a two-body probability, which helps us ‘get inside’ the average structure that we get from Bragg peak (Rietveld) analysis.

Now, this is all talking about powders. The main issue is that a powder is a collection of randomly oriented crystallites/grains which means the pattern is averaged. Ideally, it would be nice to have a single crystal, to measure the total scattering in a way that is not averaged by random orientation. This is Single Crystal Diffuse Scattering, SCDS. It is (in my opinion) rather a gold standard in structural studies, but is pretty tricky to do…

What the paper we have just published in IUCrJ does is to take a system we have studied using SCDS, and then study it using PDF to show what things the PDF can reasonably be expected to reveal and what features are hidden from it (but apparent in the SCDS). We did this because we felt that PDF, powerful as it is, was perhaps being over-interpreted, and treated as more definitive than it is, and in many cases it is the only viable technique, so it is hard to gauge when it is being over-interpreted. Hence we look at in for a case when it is *not* the only available method.

What we found was that PDF is very good for showing the magnitudes of the spacings between atoms, and for showing the population of the spacings between atoms, but is not good for showing how these spacings might be correlated (ie, are the closely spaced atoms clustering together?). Similarly, it was not good at showing up the ordering of atoms (…ABABA… vs …AAABBBB… for example).

The PDF is in real space — it is a plot of probability against separation, separation is measured in metres, like distances are measured in the world we experience. The SCDS and the TS exist in reciprocal space, where distances are measured in inverse metres (m^{-1}). Some atomic orderings give rise to features that are highly localised in reciprocal space, so are best explored in that space. Also, if the ordering in question only affects a small section of reciprocal space, and that is getting smeared out by the powder averaging, then it won’t show up very well in TS or then in PDF.

For example, above is a cut of SCDS calculated from an analysis of the PDF, whereas below is our model for the SCDS. Clearly the latter *should* be a lot better — and it is. No surprise. Now this is not making the PDF fight with one hand tied behind its back, and not setting up a straw man, either. The point it not to show that SCDS is a more definitive measurement, the point is to show what PDF *can* be expected to tell us, so that when we are studying the many systems that we cannot do with SCDS because we cannot get a single crystal, we know when we are stretching the data too far.

Mission accomplished.