Peter et al., If the reason for their use is "historic and somewhat convenient, but their usual application is based on no theory whatsoever," why does their use continue? It is generally recognized that adding parameters to a least squares fitting process causes refinement to proceed to a lower residual (R-factor), even if the results are not physically meaningful. Would this not be the case when one adds U,V,W in Rietveld structure refinement? Frank May University of Missouri - St. Louis
________________________________ From: [EMAIL PROTECTED] [mailto:[EMAIL PROTECTED] Sent: Mon 6/25/2007 3:56 PM To: [email protected] Subject: Re: More Caglioti U V W parameters In my opinion, the short answer (regarding use of Caglioti parameters) is that their use is historic and somewhat convenient, but their usual application is based on no theory whatsoever, and they can be quite troublesome to apply. They came from a paper (Nuc. Instrum. & Methods, 1958) on the resolution of a neutron powder diffractometer using mosaic crystals and S\"{o}ller (that's an umlaut over the o; please, not solar) collimators, which gives precise expressions for U, V, and W in terms the various geometric parameters of the diffractometer. If (as was true of most samples on neutron powder diffractometers at the time) the instrument dominated the peak shape, they give a good representation of the observed linewidth. Maybe you could tweak them up a bit to account for sample broadening. Accordingly, they were ideally suited to Rietveld's method which was first developed for CW neutron powder diffractometers. Historically, they seem to have overstayed their welcome, I mean their theoretical justification. This is especially so for high resolution x-ray powder diffractometers at synchrotrons and elsewhere where the peak width is almost entirely from the sample, not the instrument. One problem with them is that for inappropriate choices of U, V, and W, the linewidth can become an imaginary number over a certain range of diffraction angles. This leads to some unpleasant instabilities in refinement programs that use them. The fundamental parameters approach would have you model the instrument and the sample separately, and for any other kind of diffractometer, U, V, and W are probably not a very good model of either. You can learn about fundamental parameters e.g., from the Bruker Topas documentation, or from Klug and Alexander, chapter 6. If you are not going to try to separately model instrument and sample, you can get a pretty good line through your data points and relative intensities suitable for Rietveld analysis with U, V, and W (and some of their extensions, such as Lorentzian X and Y in, e.g., GSAS) Toward that end note that if you forget V, the (Gaussian) FWHM is $(U \tan^2 \theta + W)^{1/2}$, which suggests that U is kind of like strain broadening and W is kind of like size broadening, coming together in quadrature. I have had generally OK luck leaving V set to zero and refining U and W. That has the advantage of being more robust than refining the three (or more) parameters. I guess once your refinement is pretty much under control, you could let V vary to see if the fit improves. Just be careful not to believe that the refined values of U, V, and W have any meaning in such a refinement. ^~^~^~^~^~^~^~^~^~^~^~^~^~^~^~^~^~^~^~^~^~^~^~^~ Peter W. Stephens Professor, Department of Physics and Astronomy Stony Brook University Stony Brook, NY 11794-3800 fax 631-632-8176
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