Realistic Q-spacing resolution in PDF data

[우기누기]

Hello. I'm new user of PDFgetX3.

Recently, I obtained PDF data of amorphous carbon using synchrotron X-ray. (~0.70A wavelength), and try to analyze it by PDFgetX3.

However, I noticed that the difficult part of PDF analysis is not only converting it correctly, but also interpreting it correctly.

My question is, if I used X-ray of 0.7000A, and Qmax is ~15, in this case what is maxium resolution in Q-space?

For example, can I distinguish 1.45A and 1.48A in Q space of G(r) function? 

Could someone tell me Q-resolution in this case? (0.7000A X-ray, Qmax = ~15)

Thanks.

[Wanuk Choi]

I found that "the spatial resolution is given by  Δr =3.78/Qmax" in a paper. ("On the short range atomic structure of non-crystalline carbon, Jounal of Non-Crystalline Solid 47 (1982) 391-402)

If so, when Qmax=~15, Δr = 0.252.

However, in the graph of G(r) function, still I can see the peaks closer than 0.252A. What does it mean?

2022년 9월 17일 토요일 오전 12시 49분 11초 UTC+9에 Wanuk Choi님이 작성:

[Mikkel Juelsholt]

Hi 

I think you are asking for the resolution in real space and not Q-space. The Q-resolution is your resolution of the diffraction data. 

In a PDF the resolution is given by Pi/Qmax, see this paper: https://journals.aps.org/prb/abstract/10.1103/PhysRevB.84.134105. So if you have a Qmax of 15 Å-1 peaks in the PDF need to be more than 0.21 Å apart. Any peaks closer than this is either a result of noise, termination ripple, or some other non-structural component in the diffraction data. 

Hope this answers your question

[Wanuk Choi]

Thanks to your reply. I clearly understant about 'resolution' in PDF analysis. However, I noticed that what I don't know is "significant figure". Even Q-resolution is ~0.20A, still can we discuss about the difference between 1.42A and 1.45A? For graphite, 1st coordination distance is 1.42A, but amorphous carbon 1.43~1.48A. 

2022년 9월 19일 월요일 오전 7시 10분 23초 UTC+9에 Mikkel Juelsholt님이 작성:

[Mikkel Juelsholt]

Sorry for the late reply. 

No, because the PDF peak position will also be influenced by experimental factors such as sample-to-detector distance. You need an internal standard such as LaB6 or Si to absolutely say if your peak is at 1.42 or 1.43 Å.

In your case, if you want to distinguish between graphite and amorphous carbon, you do not need to look at the interatomic distances. Just look at the correlation length. Graphite is crystalline so will extend to much higher r-values

Cheers Mikkel

[Simon Billinge]

Dear Wanuk and Mikkel,

Thanks for this question and thanks for the excellent responses Mikkel (as always!).

I may add a few general thoughts about "resolution"  We often talk about resolution and Mikkel correctly pointed out some mathematical expressions that govern resolution when it comes to looking at things in Fourier space (our resolution increases in real-space when our Q-range increases in reciprocal space). 

These things are all correct.  But when we talk about "resolution" it is important to understand what we mean by that concept.   Strictly, it is not a fully well defined concept, but roughly it means "our ability to resolve two points separated by some distance delta-r".  What does this even mean?  Clearly if the peaks have a half-width that is smaller than 2-times the separation they will appear largely speaking as two separate peaks (their tails will overlap) and no-one would argue that they are not resolved.  As the two points, and therefore their (let's say to be concrete, Gaussian) peaks get closer, it is still apparent that there are two peaks there because there is something tha looks like a minimum or saddle point in between.  At some point, before the points are exactly on top of each other so there are still definitely two points there, the minimum between the two peaks disappear and it looks like a single peak.  Argualy, at this point, we have lost our ability to resolve the two points, and this is the condition that is addressed by the resolution limit that Mikkel mentioned.  However, the peak will not have a Gaussian shape and so there is still information in the data that tells us that there is actually two points there, and we may get a better fit with two Gaussians than one, but we have to take to modeling.   As the points get closer and closer together the resulting unresolved double peak gets closer and closer to a pure Gaussian and our ability to distinguish it from a single broad Gaussian gets less and  less, but we may still infer that it is two sharp Gaussians rather than a single broad Gaussian if we have prior knowledge to that effect.  We can't make that determination from the PDF alone.   And so on.   So with modeling and prior knowledge,  you may be able to infer super-resolution effects such as the presence of two different bond lengths that are different by less than the resolution.  In the PDF, modeling also takes into account peaks at higher-r which also have to be consistent with the model.     Just be careful.  Since 1.48 - 1.45 = 0.03 which is probably 3-4x smaller than even the intrinsic width of that PDF peak, our ability to differentiate those distances without a long-range model (fit to long-range data) is quite dubious, and with Qmax = 16 there is really no hope.

S

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Simon Billinge

Professor, Columbia University

Physicist, Brookhaven National Laboratory