Dear All: Indeed, despite some more advanced approaches to modeling diffraction line shapes, the good, old Cagliotti function is still in use probably for historical reasons (as it is the case with many other things in sciences). However, to be fair to (practically all major) Rietveld programs, the original Cagliotti function (Gaussian term) was a long time ago amended by a Lorentzian contribution (linear in FWHM as opposed to a quadratic Gaussian term). This term immensely helps to accurately model high-resolution measurements, both x-ray and neutron (see, for instance examples of ESRF and ISIS data in Size-Strain Line-Broadening Analysis of the Ceria Round-Robin Sample, Journal of Applied Crystallography 37 (2004) 911-924--article and data available at http://www.du.edu/~balzar/s-s_rr.htm).
As discussed in this thread, the original Cagliotti function often gets into trouble because of the square root of a negative number. Bill David described a much better function. Thus, the general approach described by Bill and in the article cited above is to refine coefficient of a function used on a pattern obtained from a suitable "standard", such as LaB6, and then fix them (unfortunately, not always possible for all instruments, because some instrumental parameters depend on the angle in the same way as the strain term in the Bragg-Brentano geometry). Moreover, for high-resolution data it might be helpful to add a Lorentzian FWHM to that expression and then post it to Wikipedia or perhaps publish a paper... :-) Davor > -----Original Message----- > From: simon.billi...@gmail.com > [mailto:simon.billi...@gmail.com] On Behalf Of Simon Billinge > Sent: Sunday, March 22, 2009 2:50 PM > To: rietveld_l > Subject: Cagliotti and Other Issues > > Dear Rietvelders.... > > What is the most complete and authoritative source for issues such as > profile function definitions, what is their scientific basis, and when > are they appropriate to use, etc.? I am guessing there is not > one-stop-shop solution (Young's book? GSAS manual? Rietveld list > archive?) but advice on this would be helpful. > > I wonder if we should, as a community, put some of this stuff on > wikipedia, or another such place. In other words, distill the > community's collective knowledge in a single place that can be updated > in the future, and also curated for correctness also by the community. > What are people's thoughts on this? Rietveldipedia? > > S > > 2009/3/20 May, Frank <frank.l....@umsl.edu>: > > Back to basics and First Principles.... > > > > As Alan says, the [use of the Cagliotti function is > appropriate for the neutron case], "but not really for X-ray > and other geometries." > > > > My recollection is the Cagliotti function was adapted to > the x-ray case when we had low resolution x-ray instruments > and slow (or no) computers. Now that we have high resolution > instruments and fast computers, why does this inappropriate > function continue to be used? > > > > On another note, the world is venturing into the infinitely > small realm of "nano-particles." The classical rules for > crystallography work very well for ordered structures in the > macro-world (particles of the order of micron-sizes). However, as the particles become smaller, does one not need to > address the contribution of the "surface" of the particles? > The volume of the "surface" becomes much greater relative to > the volume of the "bulk" of the crystal. Models today > account for "stress" and "strain" in the macro-world. As the > relative fraction of the "bulk" becomes smaller, both the > physical structure as well as the mathematics used to > describe the bulk suffer from termination-of-series effect, > do they not? Does any of this make sense? Any thoughts? > > > > Frank May > > St. Louis, Missouri U.S.A. > > > > ________________________________ > > > > From: Alan Hewat [mailto:he...@ill.fr] > > Sent: Fri 3/20/2009 2:13 AM > > To: rietveld_l@ill.fr > > Subject: RE: UVW - how to avoid negative widths? > > > > > > > > matthew.row...@csiro.au said: > >> From what I've read of Cagliotti's paper, the V term > should always be > >> negative; or am I reading it wrong? > > > > That's right. If > > FWHM^2 = U.tan^2(T) + V.tan(T) + W > > then the W term is just the Full Width at Half-Maximum > (FWHM) squared at > > zero scattering angle (2T). FWHM^2 is then assumed to > decrease linearly > > with tan(T) so V is necessarily negative, but at higher > angles a quadratic > > term (+ve W) produces a rapid increase with tan^2(T). > > > > Cagliotti's formula assumes a minimum in FWHM^2, but if > that minimum is > > not well defined, U,V,W will be highly correlated and > refinement may even > > give negative FWHM. In that case you can reasonably constrain V by > > assuming the minimum is at a certain angle 2Tm, which may > be close to the > > monochromator angle for some geometries. So setting the > differential of > > Cagliotti's equation with respect to tan(T) to zero at that > minimum gives: > > 2U.tan(T) + V =0 at T=Tm or V = -2U.tan(Tm) > > this approximate constraint removes the correlation and > allows refinement. > > > > Cagliotti's formula simply describes the purely geometrical > divergence of > > a collimated white neutron beam hitting a monochromator, > passing through a > > second collimator, then scattered by a powder sample into a > collimated > > detector. It takes no account of other geometrical effects > (eg vertical > > divergence) or sample line broadening etc. This geometry is > appropriate > > for classical neutron powder diffractometers, but not > really for X-ray and > > other geometries. Still, such a quadratic expression with a > well defined > > minimum in FWHM, may be a good first approximation in many > other cases, > > requiring only a few parameters, hence its success. There > are many more > > ambitious descriptions of FWHM for various scattering geometries and > > sample line broadening, usually allowing more parameters to > be refined to > > produce lower R-factors :-) > > > > Alan > > ______________________________________________ > > Dr Alan Hewat, NeutronOptics, Grenoble, FRANCE > > <alan.he...@neutronoptics.com> +33.476.98.41.68 > > http://www.NeutronOptics.com/hewat > > ______________________________________________ > > > > > > > > > > > > > > -- > Prof. Simon Billinge > Applied Physics & Applied Mathematics > Columbia University > 500 West 120th Street > Room 200 Mudd, MC 4701 > New York, NY 10027 > Tel: (212)-854-2918 (o) 851-7428 (lab) > > Condensed Matter and Materials Science > Brookhaven National Laboratory > P.O. Box 5000 > Upton, NY 11973-5000 > (631)-344-5387 > > email: sb2896 at columbia dot edu > home: http://nirt.pa.msu.edu/ > >
