# Error bars for pair-correlation function

I have obtained some data from neutron diffraction for some material samples. The "rawest" form of the data is the structure function $S(Q)$.

We can choose a variety of different Q-maxes when Fourier transforming the data into $g(r)$, along with applying something like a Lorch function to smooth it out. Right now the whole process is done by eyeballing it and using whichever curve ends up looking the best.

Well, it turns out there may be some long-range order at high values of $r$, which — if this order really exists — will lead to a very exciting paper. However, I want to be absolutely certain that what we're seeing is not some Fourier transform artifact or some kind of systematic artifact. However, I'm a computational modeler, not an experimentalist, so I don't really know how to verify this mathematically.

Ideally, I would like the $g(r)$ curve to have an envelope around it, the "error envelope" that represents some confidence interval. But I don't know the process of getting it, and in fact, a literature search reveals no one may have even done this before (though you'd think all experimentally obtained data would have error bars nowadays).

Any ideas? Thanks for the help.

• Welcome to Physics Stack Exchange, Nick! To be clear, are you looking for guidance calculating the error introduced by the upper bound on the numerical Fourier transformation? You might find some useful information at our sister site Computational Science. (It might turn out that this question would be more appropriate at that site, in which case we can migrate it there, you don't have to repost it.) – David Z May 20 '13 at 5:18
• Yes, that is quite what I'm looking for! If you think this question is a better fit for that site, then feel free to move it there. Thanks! – Nick May 20 '13 at 5:53

In general, radial distribution functions $g(r)$ have not been computed with uncertainty bars attached to them. In principle, they do exist, since the $g(r)$ (computationally or through experimental observation) is effectively an average over many different measurements of the same system. Typically, the average would also be performed over all of the different particles of a given type. Consequently, the average is comprised of many different contributions, leading to a standard error in the mean that is quite small, relatively speaking. A larger error probably comes from discretization of the data to form the $g(r)$ in the first place.