Dear Leonid, See coments below.
> > Dear Nicolae, > > This arithmetic is clear, thanks, but since you did not specify this > exact way of <R> calculation in the paper it was not evident. There are > several other ways of deriving <R>, for instance: to calculate Dv from > the inverse integral breadth and then use eq. (12) or (17) etc. Not only arithmetic, I think is clear that both <R> and c were refined in a whole pattern least square fitting. A private program, not a popular Rietveld program because no one has inplemented the size profile caused by the lognormal distribution. > Besides, you did not refine <R> for simulated data in chapter 6 - it > was "fixed". When you apply this formalism to real data you refine both > <R> and c, they may correlate and the result of such correlation is not > apparent. Let's clarify this point. What you call "simulated data in chapter 6" are in fact the exact function PHIbar(x) given in (15b). This can be calculated only by numerical integration and this function, as you can see from (15b), has only one parameter, c. This was the clue to use the parameters <R> and c=sigma(R)**2/<R>**2 in place of the original <R> and sigma(R). It is improper to say that "<R> was fixed", because <R> is contained in the argument x=2*pi*s*<R> of PHIbar(x). > > But the most important disadvantage is the necessity to choose the > exact type of size distribution. For Sample 1 (which, obviously, have > certain distribution with certain <R> and c) you got quite different > values of <R> and c for lognorm and gamma models, but the values of Dv > and Da were nearly the same. Don't you feel that Dv and Da values > "contain" more reliable information about <R> and c than those > elaborate approximations described in chapter 6? Well, this is the general feature of the least square method. In the least square you must firstly to choose a parametrised model for something that you wish to fit. Do you know another posibility with the least square than to priory choose the model? Without model is only the deconvolution, and even there, if you wish a "stable" solution you must use a deconvolution method that requres a "prior, starting model" (I presume you followed the disertation of Nick Armstrong on this theme). Concerning the fact that Dv and Da are the same although the parameters <R> and "c" of the two size distributions are different, it is not surprisingly. By contrary, it should have been very bad that Dv and Da be dependent on the choosen model of distribution. Dv & Da are quantities "seen" in diffraction. In fact the dispute on this subject started from the doubt of one of the participants that the physical model of microstructure determined ONLY from diffraction is unique! And that is essential to search for physical models, etc. (you can follow in archive). "Dv and Da contain more reliable information about <R> and c than those elaborate approximation...". You mix the planes doing comparison between disjunct things. Once the model choosen (lognormal, gamma, etc.), "those elaborate approximations" give the possibility to find the model parameters in an automatic way, by direct refinement, if these are introduced in the whole pattern fiting (Rietveld in particular). These "elaborate approximations" are doing nothing else than to approximate analytically the exact profile that can be only calculated by numerical integration, then time costly in a whole pattern fitting. > > In new version of DDM (see the following message) I included some > estimations of average crystallite diameter <D> and its dispersion > sigma<D> based on empirical approximations derived from fitting TCH-pV > function to simulated profiles for the model of spherical > crystallites with different size distribution dispersions. For > simulated data (which are supplied with the DDM package) these "magic" > expressions: > > <D> = Da + 0.25(DaDv)^0.5 and sigma<D> = <D>(Dv/Da - 1/2)/2 > > allowed reproducing <D> and sigma<D> with less than 10% deviation in > the interval of relative dispersions 0.05 < c < 0.4 for both gamma and > lognorm distributions. Of course, I don't think that these expressions > are perfect and I would be glad to see better estimations. Well, I don't know where from you taken these formulae but I observe that for spheres of equal radius, then zero dispersion, you have: sigma(D)=5<D>/4, different from zero! Best wishes, Nicolae > > Best regards, > Leonid > > > > > __________________________________ > Do you Yahoo!? > Plan great trips with Yahoo! Travel: Now over 17,000 guides! > http://travel.yahoo.com/p-travelguide >
