Lubo SVD as you mentioned does avoid numerical problems as does other methods such as the conjugate gradient method. SVD minimizes on the residuals |A x - b| after solving the matrix equation A x = b.
I would like to point out however that errors obtained from the covariance matrix are an approximation. The idea of fixing parameters as in SVD when a singular value is encountered is also a little arbitrary as it requires the user setting a lower limit. The A matrix is formed at a point in parameter space; when there are strong correlations (as SVD would report) then that point in space changes from one refinement to another after modifying the parameter slightly. If derivatives are numerically calculated, as is the case for convolution parameters, then the A matrix becomes a function of how the derivative are calculated; forward difference approximation for example gives different derivatives than both forward and backwards if the step size in the derivative is appreciable. For most convolutions and numerical derivatives in general then it needs to be appreciable for good convergence. Rietveld people may want to look at the re-sampling technique known as the bootstrap method of error determination. It gives similar errors to the covariance matrix when the correlations are weak; the maths journals are full of details. It requires some more computing time but it actually gives the distribution. And yes TOPAS has the bootstrap method; other code writers may wish to investigate it. Cheers Alan -----Original Message----- From: Lubomir Smrcok [mailto:[EMAIL PROTECTED] Sent: Wednesday, 21 March 2007 5:50 PM To: rietveld_l@ill.fr Subject: Re: Problems using TOPAS R (Rietveld refinement) Gentlemen, I've been listening for a week or so and I am really wondering what do you want to get ... Actually you are setting up a "refinement", whose results will be, at least, inaccurate. I am always surprised by attempts to refine crystal structure of a disordered sheet silicate from powders, especially when it is known it hardly works with single crystal data. Yes, there are several models of disorder, but who has ever proved they are really good ? I do not mean here a graphical comparison of powder patterns with a calculated trace, but a comparison of structure factors or integrated intensities. (Which ones are to be selected is well described in the works of my colleague, S.Durovic and his co-workers.) As far as powders are concerned, all sheet silicates "suffer" from prefered orientation along 001. Until you have a pattern taken in a capillary or in transmission mode, this effect will be dominating and you can forget such noble problems like anisotropic broadening. Last but not least : quantitative phase analysis by "Rietveld" is (when only scale factors are "on") nothing else but multiple linear regression. There is a huge volume of literature on the topic, especially which variables must, which should and which could be a part of your model. I really wonder why the authors of program do not add one option called "QUAN", which could, upon convergence of highly sophisticated non-linear L-S, fix all parameters but scale factors and run standard tests or factor analysis. One more diagonalization is not very time consuming, is it ? To avoid numerical problems, I'd use SVD. This idea is free and if it helps people reporting 0.1% MgO (SiO2) in a mixture of 10 phases to think a little of the numbers they are getting, I would only be happy :-) Lubo P.S. Hereby I declare I have never used Topas and I am thus not familiar with all its advantages or disadvantages compared to other codes. On Wed, 21 Mar 2007, Reinhard Kleeberg wrote: > Dear Leandro Bravo, > some comments below: > > Leandro Bravo schrieb: > > > > > In the refinement of chlorite minerals with well defined disordering > > (layers shifting by exactly b/3 along the three pseudohexagonal Y > > axis), you separate the peaks into k = 3.n (relative sharp, less > > intensive peak) and k  3.n (broadened or disappeared > > reflections). How did you determined this value k = 3.n and n = > > 0,1,2,3..., right? > > > The occurence of stacking faults along the pseudohexagonal Y axes causes > broadening of all reflections hkl with k unequal 3n (for example 110, > 020, 111..) whereas the reflections with k equal 3n remain unaffected > (001, 131, 060, 331...). This is clear from geometric conditions, and > can be seen in single crystal XRD (oscillation photographs, Weissenberg > photographs) as well in selected area electron diffraction patterns. The > fact is known for a long time, and published and discussed in standard > textbooks, for example *Brindley, G.W., Brown, G.: Crystal Structures > of Clay Minerals and their X-ray Identification. Mineralogical Society, > London, 1980.* > > > First, the chlorite refinement. > > > > In the first refinement of chlorite you used no disordering models and > > used ´´cell parameters`` and ´´occupation of octahedra``. So you > > refined the lattice parameters and the occupancy of all atoms? > > Yes, the lattice parameters. > Only the occupation/substitution of atoms with significant difference in > scattering power can be refined in powder diffraction. In case of > chlorites, the substitution Fe-Mg at the 4 octahedral positions can be > refined. > > > > > In the second refinement, you use na anisotropic line broadening ´´in > > the traditional way``. So you used a simple ellipsoidal model and/or > > spherical harmonics? > > > Simple ellipsoidal model, assuming very thiny platy crystals. But it was > clear that this model must fail, see above the known fact of disorder in > layer stacking. And from microscopy it is clear that the "crystals" are > much too large to produce significant line broadening from size effects. > You can see this for a lot of clay minerals: If the "ellipsoidal > crystallite shape" model would be ok, the 00l reflections would have the > broadest lines, and the 110, 020 and so on should be the sharpest ones. > But this is not true in practice, mostly the hkl are terribly broadenend > and smeared, but the 00l are still sharp. > > > The last refinement, describing a real structure. You used for the > > reflections k  3.n (broadened peaks) a ´´rod-like intensity > > distribution``, with the rod being projected by the cosine of the > > direction on the diffractogram. You used also the lenghts of the rods > > as a parameter, so as the dimension of the rods for 0k0 with k > >  3.n. I would like to know how did you ´´project`` these rods > > and use them in the refinement. > > > > For the k = 3.n reflections, you used an anisotropic broadening model > > (aniso crystallyte size) and and isotropic broadening model > > (microstrain broadening). But you said that crystallite size is an > > isotropic line broadening in my kaolinite refinement and I should not > > use it. So I use or not the cry size? > > > Yes, we used an "additional" ellipsoidal broadening in order to describe > any potential "thinning" of the crystals. But this broadening model was > not significant because the broadening was dominated by the stacking > faults. A "microstrain" makes sense because of natural chlorits are > sometimes zoned in their chemical composition and a distribution of the > lattice constants may occur. > In one of your mails you mentioned "crysize gave reasonable numbers with > low error", and from that I assumed you looked only on the errors of the > isotropic crysize as defined in Topas. You must know what model you did > apply. But it is clear that any "crysize" model is inadequate to > describe the line broadening of kaolinite. > > > Now the kaolinite refinement. > > > > In the first refinement was used fixed atomic positions and a > > conventional anisotropic peak broadening. This conventional > > anisotropic peak broadening would be the simple ellipsoidal model > > and/or spherical harmonics?! > > Only ellipsoidal model, assuming a platy crystal shape, see above. Only > for comparision. > > > > > After that you use the introduced model of disorfering. Is this model > > the same of the chlorite (rods for k  3.n and microstrain > > broadening and anisotropic crystallite size? > > > Not exactly the same like in chlorite, because the disorder in kaolinite > is much more complicated like in chlorites. See also the textbook cited > above, and extensive works of Plancon and Tchoubar. Thus, most of the > natural kaolinites show stacking faults along b/3 as well as along a, > and additional random faults. Thus, more broadening parameters had to be > defined, and this is not completely perfect until now. See the > presentation I sent you last week. > > Best regards > > Reinhard Kleeberg >
