RE: Rietveld refinement and PDF refinement ?
do you really have the resolution even on HRPD to see the diffuse scattering between Bragg peaks at high Q ? No we don't, but this is not the main point (by the way, we don't use HRPD for PDF, it doesn't go to sufficiently short wavelengths). The main reason to go to high Q is to avoid truncation errors. If you truncate S(Q), all your G(r) peaks will be convoluted with the Fourier transform of a step function, which is a sinx/x function. The width of the central peak is roughly 1/Qmax. If you use a wavelength of 0.5 A, this corresponds to about 0.08 A, or an equivalent B of 0.5. This in itself can be a problem when you want to look at sharp correlation features. Even worse, the ripples will propagate to adjacent PDF peaks, generating unphysical features. There are ways to suppress the ripples by convoluting the data with an appropriate smooth function rather than truncating them (these are extensively used in disordered materials work), but they all tend to broaden the features. You can also fit a model including the ripples (as in PDFfit) but it is clearly better not to have them if you are trying to exploit the model independence of PDF. Going to high Q does not solve all the problems. If the high-Q data are noisy, your truncation function will have higher frequency but also higher (and random) amplitude in the ripples, so there is always a compromise Qmax, depending on statistics. Finally, very high-Q data are quite difficult to normalise, because of the epithermal background. You may get better temperature factors with high-Q PDF refinement, but you will also do that with high-Q Rietveld. Generally, all crystallographic parameters come out worse from PDF refinements than from Rietveld on the same data sets. I think this is because you are trying to fit an average structure to something that contains correlations, so the fit is bound to be worse. You could fit a correlated model, but then you would not get temperature factors in the usual sense. I also doubt that just because PDF uses data between the Bragg peaks, then it must be superior for seeing details not centered on atoms in real space in a crystal, eg the split atom sites in (In/Ga)As). You might do just as well with Bragg scattering if you use the result of Rietveld refinement to construct a Fourier map of the structure. Happily, a sampling of reciprocal space (Bragg peaks) is sufficient to re-construct the entire density of a periodic structure in real space, not just point atoms, to a resolution limited only by Q. You are right. PDF is not always superior. It is the interpretation of the Fourier density in terms of correlated displacements that emerges uniquely from PDF, although you can often guess it right from the Fourier map in the first place. The case of Jahn-Teller polarons in manganites (La,Ca)MnO3 is quite illuminating. Several groups noticed that the high-temperature phase (above the CMR transition) has large DW factors for O. We showed that this affects primarily the longitudinal component along the Mn-O bonds, and guessed that this was caused by an alternation of short and long Mn-O distances. Simon Billinge showed the same thing quite convincingly from PDF data. Only the latter can be considered direct evidence (with some caveats). But you do agree that in a PDF experiment you integrate over energy, so you only see an instantaneous snapshot of the structure... Yes, I agree with this and the fact that inelasticity corrections are an issue. Sometimes they are exploited to obtain additional information, and there is a claim that one can measure phonon dispersions with this method, but the issue is quite controversial. So while I am convinced of the interest of PDF for non-crystalline materials, with short or intermediate range order, I am not yet convinced that you gain much from PDF refinement of crystalline materials, where you can also apply Rietveld refinement. I agree completely. The directional information gained from phasing and the fact of locking in to specific Fourier components is a major asset of Rietveld analysis. PDF is useful when correlated disorder is important (and large), even if superimposed on an ordered structure. Paolo Radaelli
RE: Rietveld refinement and PDF refinement ?
Another reason which may preclude your self-convincing is the fact that all the very good PDF studies of materials... are not by using constant wavelength neutrons... You are right Armel :-) About the current advantage of SR and TOF for PDF, I mean. That is why I am interested in being convinced. So, this PDF advantage does not impress me a lot (like opening an already open door), EXAFS reveals the same. Open doors are good, and open minds are even better :-) Alan.
RE: Rietveld refinement and PDF refinement ?
Adding 2 cents to the discussion... But I will try to convince myself otherwise :-) Another reason which may preclude your self-convincing is the fact that all the very good PDF studies of materials that are not perfectly crystallized (producing diffuse scattering), though not being amorphous, are made by using synchrotron data or neutron data from spallation sources : not by using constant wavelength neutrons... In the past, I have studied a few glasses at ILL by using the D4 instrument, at 0.5 A wavelength, allowing to attain modestly high Q values. The fact that the instrument resolution was very poor was not a problem for amorphous materials, but it was a problem for crystalline or partially crystalline materials : they also look like amorphous in the reciprocal space, due to the quite large instrumental contribution to the peak broadening. Is that improved now ? If not, you would not be able to apply both the Rietveld and the PDF methods from the same data, raw or Fourier transformed. The raw data would be too bad for applying the Rietveld method. So, no PDF at ILL ?-). A different question is about size/strain effects, possibly anisotropic, which may have effects on the peak shapes and peaks broadening. We can more or less (progress are to be made), take account of these effects reflecting the deviation of the real sample from a model of infinite perfectly periodical structure. All that peak shape information is lost in the PDF... So, both PDF and Rietveld approaches may appear necessary and complementary, sometimes - for ill-crystallized compounds. However, is it really necessary to see on the PDF the difference between the Si-O and Al-O distances in respectively SiO4 and AlO4 ? This is already a well established fact (the same for any statistical substitution of atoms with different radii like In and Ga in (In/Ga)As). So, this PDF advantage does not impress me a lot (like opening an already open door), EXAFS reveals the same. Apart from being able to show such differences, I expect more from the PDF approach, but the more an ill-crystallized sample is close to be amorphous, and the more the structure hypothesis will be dubious... Armel
Re: Rietveld refinement and PDF refinement ?
Well, that is an old chestnut that Cooper and Rollet used to oppose to Rietveld refinement. I think Rollet eventually agreed that Rietveld was the better method. Has Bill really gone back on that ? The difference between the two approaches are just an interchange of the order of summations within a Rietveld program. Differences in esds should only arise through differences in accumulated rounding errors, assuming you don't apply any fudge factors. Since most people do apply fudge factors, the argument is really about which fudge factor you should apply. I will only comment that the conventional Rietveld approach (multiply the covariance matrix by chi^2) is often poor. As for the PDF versus Rietveld - you should get smaller esds on thermal factors if you were to write a program which treats the background as a part of the crystal structure and has no arbitrary degrees of freedom in modelling the background. This is just due to adding in more data points that are normally treated as background but which should help to determine the thermal parameters via the diffuse scattering. So, provided you were to remove the arbitrary background from the Rietveld program and compute the diffuse scattering the methods ought to be equivalent. Something like DIFFAX does this already for a subset of structures, but I think without refinement. The real difficulty arises with how to visualise the disordered component, decide what it is, and improve the fit - hence the use of the PDF. Although no one appears to have written such a program there does not seem to be any fundamental reason why it is not possible (compute the PDF to whatever Q limit you like, then transform the PDF and derivatives into reciprocal space). Biologists already manage to do this in order to use an FFT for refinement of large crystal structures! In practice a large percentage of the beamtime for these experiments at the synchrotron is used to measure data at very high Q which visually has relatively little information content - just so that a Fourier transform can be used to get the PDF. This is silly! The model can always be Fourier transformed up to an arbitrary Q limit and then compared whatever range of data you have. For things like InGaAs the diffuse scatter bumps should occur mainly on the length scale of the actual two bond distances. Wiggles on shorter length scales are going to be more and more dominated by the thermal motion of the atoms, and so don't really add as much to the picture (other than to allow an experimentalist to get some sleep!). In effect it is like the difference between measuring single crystal data to the high Q limit and then computing an origin removed Patterson function and doing a refinement against that Patterson as raw data. No one does the latter as you can trivially avoid the truncation effects by doing the refinement in reciprocal space. The question then is whether it is worth using up most of your beamtime to measure the way something tends toward a constant value very very precisely? Could the PDF still be reconstructed via maximum entropy techniques from a restricted range of data for help in designing the model? Currently the PDF approach beats crystallographic refinement by modelling the diffuse scattering. As soon as there is a Rietveld program which can model this too then one might expect the these experiments become more straightforward away from the ToF source. I'd be grateful if someone can correct me and show that most of the information is at the very high Q values. Visually these data contain very little compared to the oscillations at lower Q and seem to become progressively less interesting the further you go, as there is a larger and larger random component due to thermal motion. Measuring this just so you can do one transform of the data instead of transforms of the computed PDF and derivatives seems like a dubious use of resources? Since the ToF instruments get this data whether they like it or not, the one transform approach is entirely sensible there. For x-rays and CW neutrons, it seems there is a Rietveld program out there waiting to be modified. August is still with us, Happy Silly Season! Jon
RE: Rietveld refinement and PDF refinement ?
I would argue that the Bragg diffuse scattering both reflect the average instantaneous atomic structure. Yes. If you integrate over energy, the scattering function factors to a delta function in time, corresponding to an instantaneous snapshot of the spatial correlations. It is not a question of Bragg or or diffuse scattering. Your statement about integrating over energy is correct regardless of Bragg or diffuse scattering, but Brian's statement is not, at least in the context of the PDF/Bragg discussion, so it is and it isn't would have been a more appropriate statement on my part. It is true that in diffraction you measure the intermediate scattering function S(Q,t=0), but this is not the same thing as saying that you can then Fourier-transform any part of it you like to a G(x, t=0). To get a real-space function G(x) you have to integrate over the *whole* Q domain, and in doing so for Bragg scattering you set to zero everything that is outside the nodes of the RL. However, it is easy to see that this can be equivalent to setting an energy cut-off. This is because fluctuations in time and space are usually correlated, so by selecting an integration range in Q for you Bragg peaks, you also effectively select an integration range in energy. Your superstructure example shows it clearly: if you are far from the phase transition and the correlation length of your tilt fluctuation is 10 A, you would not see a Bragg peak there and you would get the time-average structure (without the superstructure). Clearly there is the limiting case of critical scattering very near the phase transition, where the fluctuating regions are so large that you effectively take a snapshot of each of them. There could even be a deeper point here to do with ergodicity, whereby you could show that coherent space average and coherent time average are effectively the same (I am not positive about this, though). The point I was trying to make is a different one. We are discussing about the difference between Bragg and PDF. If all the scattering is near the Bragg peaks, so that you integrate it all in crystallography, there is and there cannot be any difference between the two techniques. I am sure we are not discussing this case. The interesting case is when there is additional diffuse scattering. What I am saying is that if this diffuse scattering is inelastic, then PDF will reflect istantaneous correlations in a way that is missed by Bragg scattering. Let's look at the case of two bonded atoms again, with a bond length L, and lets this time assume that they vibrate harmonically in the transverse direction, and that the semi-axis of the thermal ellipsoid is a. The possible istantaneous bond lengths range from L to sqrt(L^2+4a^2) for an uncorrelated or anti-correlated motion, but is always L for a correlated motion. Bragg scattering will give you a distance of L between the two centres, which is only correct for correlated motion. It also provides information about the two ellipsoids, which is the same you would get by averaging the scattering density over time, regardless of correlations. Istantaneous correlations are only contained in the diffuse scattering, and are in principle accessible by PDF. Paolo Radaelli
Re: Rietveld refinement and PDF refinement ?
Jon others, Well, there is an attempt at this in GSAS - the diffuse scattering functions for fitting these contributions separate from the background functions. These things have three forms related to the Debye equations formulated for glasses. The possibly neat thing about them is that they separate the diffuse scattering component from the Bragg component unlike PDF analysis. As a test of them I can fit neutron TOF diffraction data from fused silica quite nicely. I'm sure others have tried them - we all might want to hear about their experience. Bob Von Dreele From: Jon Wright [mailto:[EMAIL PROTECTED] Sent: Sun 8/22/2004 6:13 AM To: [EMAIL PROTECTED] Well, that is an old chestnut that Cooper and Rollet used to oppose to Rietveld refinement. I think Rollet eventually agreed that Rietveld was the better method. Has Bill really gone back on that ? The difference between the two approaches are just an interchange of the order of summations within a Rietveld program. Differences in esds should only arise through differences in accumulated rounding errors, assuming you don't apply any fudge factors. Since most people do apply fudge factors, the argument is really about which fudge factor you should apply. I will only comment that the conventional Rietveld approach (multiply the covariance matrix by chi^2) is often poor. As for the PDF versus Rietveld - you should get smaller esds on thermal factors if you were to write a program which treats the background as a part of the crystal structure and has no arbitrary degrees of freedom in modelling the background. This is just due to adding in more data points that are normally treated as background but which should help to determine the thermal parameters via the diffuse scattering. So, provided you were to remove the arbitrary background from the Rietveld program and compute the diffuse scattering the methods ought to be equivalent. Something like DIFFAX does this already for a subset of structures, but I think without refinement. The real difficulty arises with how to visualise the disordered component, decide what it is, and improve the fit - hence the use of the PDF. Although no one appears to have written such a program there does not seem to be any fundamental reason why it is not possible (compute the PDF to whatever Q limit you like, then transform the PDF and derivatives into reciprocal space). Biologists already manage to do this in order to use an FFT for refinement of large crystal structures! In practice a large percentage of the beamtime for these experiments at the synchrotron is used to measure data at very high Q which visually has relatively little information content - just so that a Fourier transform can be used to get the PDF. This is silly! The model can always be Fourier transformed up to an arbitrary Q limit and then compared whatever range of data you have. For things like InGaAs the diffuse scatter bumps should occur mainly on the length scale of the actual two bond distances. Wiggles on shorter length scales are going to be more and more dominated by the thermal motion of the atoms, and so don't really add as much to the picture (other than to allow an experimentalist to get some sleep!). In effect it is like the difference between measuring single crystal data to the high Q limit and then computing an origin removed Patterson function and doing a refinement against that Patterson as raw data. No one does the latter as you can trivially avoid the truncation effects by doing the refinement in reciprocal space. The question then is whether it is worth using up most of your beamtime to measure the way something tends toward a constant value very very precisely? Could the PDF still be reconstructed via maximum entropy techniques from a restricted range of data for help in designing the model? Currently the PDF approach beats crystallographic refinement by modelling the diffuse scattering. As soon as there is a Rietveld program which can model this too then one might expect the these experiments become more straightforward away from the ToF source. I'd be grateful if someone can correct me and show that most of the information is at the very high Q values. Visually these data contain very little compared to the oscillations at lower Q and seem to become progressively less interesting the further you go, as there is a larger and larger random component due to thermal motion. Measuring this just so you can do one transform of the data instead of transforms of the computed PDF and derivatives seems like a dubious use of resources? Since the ToF instruments get this data whether they like it or not, the one transform approach is entirely sensible there. For x-rays and CW neutrons, it seems there is a Rietveld program out there waiting to be modified. August is still with us, Happy Silly Season! Jon
RE: Rietveld refinement and PDF refinement ?
Two very good points by Armel: all the very good PDF studies ...are made by using synchrotron data or neutron data from spallation sources This is because they are the only means to get to high Q (i.e., high resolution in real space) and sufficiently high resolution (in reciprocal space) simultaneously. The RMC method is somewhat similar and does not require such high Q, but it has the drawback of requiring a starting model (arguably, there is also a uniqueness issue with RMC). One nice feature of the latest generation of TOF instruments is that one does not have to choose in advance between PDF and crystallography, as long as one has appropriate references (empty can, empty instrument etc.), which are collected as a matter of course anyway. PDF analysis requires better statistics, but, in the context of a large phase diagram study, it is always possible to collect a few data point to PDF accuracy. So, this PDF advantage does not impress me a lot True, in most cases PDF=Rietveld + Common Sense. However, there are some exceptions. For some nice cases see the work of Simon Hibble et al. (e.g., Hibble SJ, Hannon AC, Cheyne SM Structure of AuCN determined from total neutron diffraction INORG CHEM 42 (15): 4724-4730 JUL 28 2003 and references cited therein) and that by Simon Billinge (e.g. Petkov V, Billinge SJL, Larson P, et al. Structure of nanocrystalline materials using atomic pair distribution function analysis: Study of LiMoS2 PHYS REV B 65 (9): art. no. 092105 MAR 1 2002 ). Particularly, Simon Billinge makes the point that the future of PDF is in the study of materials with short and intermediate-range order but no long-range order (nano-crystallography). It is an interesting point of view, although, at the moment, there are not very many examples of this in the literature. Paolo Radaelli
RE: Rietveld refinement and PDF refinement ?
Two very good points by Armel: all the very good PDF studies ...are made by using synchrotron data or neutron data from spallation sources And he is modest as well :-) But do you really have the resolution even on HRPD to see the diffuse scattering between Bragg peaks at high Q ? You may get better temperature factors with high-Q PDF refinement, but you will also do that with high-Q Rietveld. I also doubt that just because PDF uses data between the Bragg peaks, then it must be superior for seeing details not centered on atoms in real space in a crystal, eg the split atom sites in (In/Ga)As). You might do just as well with Bragg scattering if you use the result of Rietveld refinement to construct a Fourier map of the structure. Happily, a sampling of reciprocal space (Bragg peaks) is sufficient to re-construct the entire density of a periodic structure in real space, not just point atoms, to a resolution limited only by Q. What I am saying is that if this diffuse scattering is inelastic, then PDF will reflect istantaneous correlations in a way that is missed by Bragg scattering. But you do agree that in a PDF experiment you integrate over energy, so you only see an instantaneous snapshot of the structure, ie the spatial correlations. If it is a periodic structure, you also average over unit cells, even though the spatial correlations are different from one cell to the next. (BTW, talk of inelastic scattering raises the question of different inelastic cross-sections for the very different neutron energies used on your TOF machine). So while I am convinced of the interest of PDF for non-crystalline materials, with short or intermediate range order, I am not yet convinced that you gain much from PDF refinement of crystalline materials, where you can also apply Rietveld refinement. A Rietveld refinement probably has fewer correlations between parameters, and can be used to construct a Fourier map of the complete scattering density, while PDF is only a Patterson map of the pair correlations, and a lot more difficult to interpret. Alan.
Re: Rietveld refinement and PDF refinement ?
I'd only add that given the clue that the peak in GaInAs is split from the PDF then one should model it that way in a Rietveld refinement. It should agree. The thrown away info in a Rietveld refinement is also evident in the Bragg peak intensities - shows up as funny thermal parameters, low atom fractions, odd bond lengths, etc. in the results. I should also say that it isn't fair to compare truncated (in Q) Rietveld refinements with untruncated PDF refinements say the latter is better. Let's be even-steven about this. From: Alan Hewat [mailto:[EMAIL PROTECTED] Sent: Thu 8/19/2004 9:31 AM To: [EMAIL PROTECTED] Cc: [EMAIL PROTECTED] But if you refine the full data with the same model, can there really be any fundamental difference, if in one case you simply do a Fourier transform to real space ? Thanks to Stefan Bruehne for providing an obvious (in retrospect :-) answer i.e. that in Rietveld refinement we throw away the non-Bragg peak data, where-as with PDF all scattering is included. He quoted an example where PDF shows a split peak in Ga(1-x)In(x)As for Ga-As and In-As distances, where Rietveld refinement always gives the average (Ga,In)As structure. Alan. Alan Hewat, ILL Grenoble, FRANCE [EMAIL PROTECTED] fax (33) 4.76.20.76.48 (33) 4.76.20.72.13 (.26 Mme Guillermet) http://www.ill.fr/dif/AlanHewat.htm ___
RE: Rietveld refinement and PDF refinement ?
The only truly unique PDF information is about *correlations*. Let's say you have two bonded sites, both with anisotropic thermal ellipsoids along the bond, and let's assume that the motion is purely harmonic. A sharp PDF peak will indicate that the atoms move predominanly in-phase, a broad PDF peak that the atoms move predominantly out-of-phase. The two scenarios will give identical Fourier maps as reconstructed from the Bragg peaks, whatever the Qmax, so the additional width (or additional narrowness) of the peak with respect to an uncorrelated model arises purely from the non-Bragg scattering. You can make the same argument for static correlations. Ga(1-x)In(x)As is a typical case. It is not a split-site problem, in that both Ga/In and As will be slightly displaced locally depending on their surrounding, but the displacements are correlated in such a way as to give a shorter bond length for Ga-As and a longer one for In-As. Of course, PDF is also used to look at more general issues of static/dynamic disorder that could also be examined using Bragg scattering, and in many case it does quite well. PDF is not (yet) very good for structural refinements (so it is to be used only in desperate cases of highly disordered systems) and is pretty hopeless for weak ordered displacement patters, since the extra Bragg peaks in crystallography lock-in on the new modulation even in the presence of large unrelated displacements. For this very reason, PDF tends to miss phase transitions, particularly at higher temperatures, which led to some very wierd claims in the past literature. There is a lot of controversy about PDF being able to say something about weak disordered displacement patters (e.g., dynamic stripes), but I am personally very skeptical. PDF requires exquisite data and a true passion for data analysis. If you have a good problem, you can get (probably) the best PDF data worldwide almost routinely on my instrument GEM at the ISIS facility (see also the cited paper by Billinge). Paolo Radaelli
Re: Rietveld refinement and PDF refinement ?
Alan, But if you refine the full data with the same model, can there really be any fundamental difference, if in one case you simply do a Fourier transform to real space ? in Rietveld refinement we throw away the non-Bragg peak data, where-as with PDF all scattering is included. It is funny to switch roles and argue this from your side w/r to the papers we each published on the disorder in the Tl2Ba2CaC2O8 superconductor in the 80's (you worked on the problem with pretty much traditional methods while Takeshi Egami, Wojtek Dmowski and I developed methods for modeling the PDFs of crystals -- and we did come up with results in pretty good agreement). I would argue that the Bragg diffuse scattering both reflect the average instantaneous atomic structure. In the case where PDF shows split sites and the Rietveld gives the high symmetry site, this is really a failure of our crystallographic modeling techniques, as the split model should really do a better job with the Bragg-only data, too. As Paolo pointed out before I could finish this e-mail, the place where the PDF is different from crystallographic results is that the former will reflect correlation in interatomic distances. The other idea you raise, could one use the entire range of Q, up to 25 A-1 or even 50 A-1 in Rietveld to me raises a more profound question. In conventional use, the answer is probably no -- adding the gentle wiggles at high Q to a refinement provides almost nothing new. The reason for this is that Rietveld treats the background at high Q is an adjustable parameter. Thus, there are no termination errors in Rietveld, but the leverage of the high-Q data w/r to the ADPs is nearly zero once the peaks start to get quite broad due to extensive superposition -- where the computer (or user) draws the background curve is arbitrary. In total scattering the background is measured experimentally and fixed. Perhaps in the quest for fundamental parameters, someone should develop a Rietveld code that uses additional background empty container scans (as is done to obtain the PDF) so that instrumental background is derived rather than fit in Rietveld. Then we could then have ADPs on an true absolute scale (something more important IMHO than relating every blip in peak shape to something intrinsic to the instrument or sample.) Finally, in the shameless self-promotion department. Simon and I just published a paper in Acta A on error analysis with models fit to PDFs. The results: s.u. from models fit to PDFs (if done right) are equivalent to those of Rietveld (for the same reasons why Bill David et al can do a Pawley fit and then get the same esd's by fitting to the extracted I's). However, s.u.'s from a PDF model fit values should be smaller, since it typically uses more data. Brian
