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 ______________________________________________
