As I have said before countless time, one should not lose sight of the objective of Rietveld refinement, that it is to refine a sensible crystallographic structure. One can reduce R factors in all sorts of ways by playing with the peak shape functions (even by using lower symmetry and increasing the number of refinable parameters!) but in the end what matters is: does the structure make sense? My own experience is that by judicious use of methods like bond valence calculations, studies of the bond lengths etc one can rule out unlikely refinements better than by concentrating on R factors. Many times one can reduce R factors by playing with the diffraction geometry terms, but with little obvious improvement of the structural results.
-----Original Message----- From: Alan Hewat [mailto:he...@ill.fr] Sent: 20 March 2009 07:13 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 ______________________________________________
